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Title: Exploring the space of jets with CMS open data

Abstract

We explore the metric space of jets using public collider data from the CMS experiment. Starting from 2.3 fb - 1 of proton-proton collisions at s = 7 TeV collected at the Large Hadron Collider in 2011, we isolate a sample of 1,690,984 central jets with transverse momentum above 375 GeV. To validate the performance of the CMS detector in reconstructing the energy flow of jets, we compare the CMS Open Data to corresponding simulated data samples for a variety of jet kinematic and substructure observables. Even without detector unfolding, we find very good agreement for track-based observables after using charged hadron subtraction to mitigate the impact of pileup. We perform a range of novel analyses, using the “energy mover’s distance” (EMD) to measure the pairwise difference between jet energy flows. The EMD allows us to quantify the impact of detector effects, visualize the metric space of jets, extract correlation dimensions, and identify the most and least typical jet configurations. To facilitate future jet studies with CMS Open Data, we make our datasets and analysis code available, amounting to around two gigabytes of distilled data and one hundred gigabytes of simulation files.

Authors:
ORCiD logo; ; ORCiD logo; ORCiD logo; ORCiD logo
Publication Date:
Research Org.:
Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Nuclear Physics (NP); USDOE Office of Science (SC), High Energy Physics (HEP); Heising-Simons Foundation; Simons Foundation
OSTI Identifier:
1599516
Alternate Identifier(s):
OSTI ID: 1630642
Grant/Contract Number:  
SC0011090; SC0012567; SC0019128
Resource Type:
Published Article
Journal Name:
Physical Review D
Additional Journal Information:
Journal Name: Physical Review D Journal Volume: 101 Journal Issue: 3; Journal ID: ISSN 2470-0010
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
73 NUCLEAR PHYSICS AND RADIATION PHYSICS; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

Citation Formats

Komiske, Patrick T., Mastandrea, Radha, Metodiev, Eric M., Naik, Preksha, and Thaler, Jesse. Exploring the space of jets with CMS open data. United States: N. p., 2020. Web. doi:10.1103/PhysRevD.101.034009.
Komiske, Patrick T., Mastandrea, Radha, Metodiev, Eric M., Naik, Preksha, & Thaler, Jesse. Exploring the space of jets with CMS open data. United States. doi:10.1103/PhysRevD.101.034009.
Komiske, Patrick T., Mastandrea, Radha, Metodiev, Eric M., Naik, Preksha, and Thaler, Jesse. Tue . "Exploring the space of jets with CMS open data". United States. doi:10.1103/PhysRevD.101.034009.
@article{osti_1599516,
title = {Exploring the space of jets with CMS open data},
author = {Komiske, Patrick T. and Mastandrea, Radha and Metodiev, Eric M. and Naik, Preksha and Thaler, Jesse},
abstractNote = {We explore the metric space of jets using public collider data from the CMS experiment. Starting from 2.3fb-1 of proton-proton collisions at s=7TeV collected at the Large Hadron Collider in 2011, we isolate a sample of 1,690,984 central jets with transverse momentum above 375 GeV. To validate the performance of the CMS detector in reconstructing the energy flow of jets, we compare the CMS Open Data to corresponding simulated data samples for a variety of jet kinematic and substructure observables. Even without detector unfolding, we find very good agreement for track-based observables after using charged hadron subtraction to mitigate the impact of pileup. We perform a range of novel analyses, using the “energy mover’s distance” (EMD) to measure the pairwise difference between jet energy flows. The EMD allows us to quantify the impact of detector effects, visualize the metric space of jets, extract correlation dimensions, and identify the most and least typical jet configurations. To facilitate future jet studies with CMS Open Data, we make our datasets and analysis code available, amounting to around two gigabytes of distilled data and one hundred gigabytes of simulation files.},
doi = {10.1103/PhysRevD.101.034009},
journal = {Physical Review D},
number = 3,
volume = 101,
place = {United States},
year = {2020},
month = {2}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1103/PhysRevD.101.034009

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Cited by: 3 works
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margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Collaboration, Cms</span> </li> <li> CERN Open Data Portal</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.7483/OPENDATA.CMS.WKRR.DCJP" class="text-muted" target="_blank" rel="noopener noreferrer">10.7483/OPENDATA.CMS.WKRR.DCJP<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1088/1748-0221/12/01/P01020" target="_blank" rel="noopener noreferrer" class="name">The CMS trigger system<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-01-01">January 2017</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Khachatryan, V.; Sirunyan, A. M.; Tumasyan, A.</span> </li> <li> Journal of Instrumentation, Vol. 12, Issue 01</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1088/1748-0221/12/01/P01020" class="text-muted" target="_blank" rel="noopener noreferrer">10.1088/1748-0221/12/01/P01020<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1088/1126-6708/1997/08/001" target="_blank" rel="noopener noreferrer" class="name">Better jet clustering algorithms<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1997-08-01">August 1997</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Dokshitzer, Yu. 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D.; Moretti, S.</span> </li> <li> Journal of High Energy Physics, Vol. 1997, Issue 08</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1088/1126-6708/1997/08/001" class="text-muted" target="_blank" rel="noopener noreferrer">10.1088/1126-6708/1997/08/001<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> </div> <div class="pagination-container small"> <a class="pure-button prev page" href="#" rel="prev"><span class="fa fa-angle-left"></span></a><ul class="pagination d-inline-block" style="padding-left:.2em;"></ul><a class="pure-button next page" href="#" rel="next"><span class="fa fa-angle-right"></span></a> </div> </div> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a href="" class="reference-type-filter tab-nav" data-tab="biblio-references" data-filter="type" data-pattern="*"><span class="fa fa-angle-right"></span> All References</a></li> <li class="small" style="margin-left:.75em; text-transform:capitalize;"><a href="" class="reference-type-filter tab-nav" data-tab="biblio-references" data-filter="type" data-pattern="book"><span class="fa fa-angle-right"></span> book<small class="text-muted"> (1)</small></a></li> <li class="small" style="margin-left:.75em; text-transform:capitalize;"><a href="" class="reference-type-filter tab-nav" data-tab="biblio-references" data-filter="type" data-pattern="dataset"><span class="fa fa-angle-right"></span> dataset<small class="text-muted"> (4)</small></a></li> <li class="small" style="margin-left:.75em; text-transform:capitalize;"><a href="" class="reference-type-filter tab-nav" data-tab="biblio-references" data-filter="type" data-pattern="journal"><span class="fa fa-angle-right"></span> journal<small class="text-muted"> (97)</small></a></li> <li class="small" style="margin-left:.75em; text-transform:capitalize;"><a href="" class="reference-type-filter tab-nav" data-tab="biblio-references" data-filter="type" data-pattern="null"><span class="fa fa-angle-right"></span> null<small class="text-muted"> (18)</small></a></li> </ul> <div style="margin-top:2em;"> <form class="pure-form small text-muted reference-search"> <label for="reference-search-text" class="sr-only">Search</label> <input class="search form-control pure-input-1" id="reference-search-text" placeholder="Search" style="margin-bottom:10px;" /> <fieldset> <div style="margin-left:1em; font-weight:normal; line-height: 1.6em;"><input type="radio" class="sort" name="references-sort" data-sort="name" style="position:relative;top:2px;" id="reference-search-sort-name"><label for="reference-search-sort-name" style="margin-left: .3em;">Sort by title</label></div> <div style="margin-left:1em; font-weight:normal; line-height: 1.6em;"><input type="radio" class="sort" name="references-sort" data-sort="date" data-order="desc" style="position:relative;top:2px;" id="reference-search-sort-date"><label for="reference-search-sort-date" style="margin-left: .3em;">Sort by date</label></div> </fieldset> <div class="text-left" style="margin-left:1em;"> <a href="" class="filter-clear clearfix" title="Clear filter / sort" style="font-weight:normal; float:none;">[ × clear filter / sort ]</a> </div> </form> </div> </div> </div> </section> <section id="biblio-related" class="tab-content tab-content-sec " data-tab="biblio"> <div class="row"> <div class="col-sm-9 order-sm-9"> <section id="biblio-similar" class="tab-content tab-content-sec active" data-tab="related"> <div class="padding"> <p class="lead text-muted" style="font-size: 18px; margin-top:0px;">Similar Records in DOE PAGES and OSTI.GOV collections:</p> <aside> <ul class="item-list" itemscope itemtype="http://schema.org/ItemList" style="padding-left:0; list-style-type: none;"> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="1" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1576530-measurement-jet-mass-distribution-top-quark-mass-hadronic-decays-boosted-top-quarks-pp-collisions" itemprop="url">Measurement of the Jet Mass Distribution and Top Quark Mass in Hadronic Decays of Boosted Top Quarks in <math> <mrow> <mi> p </mi> <mi> p </mi> </mrow> </math> Collisions at <math> <mrow> <msqrt> <mrow> <mi> s </mi> </mrow> </msqrt> <mo> = </mo> <mn> 13 </mn> <mtext></mtext> <mtext></mtext> <mi> TeV </mi> </mrow> </math></a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Sirunyan, A. M.</span> ; <span class="author">Tumasyan, A.</span> ; <span class="author">Adam, W.</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Physical Review Letters</span> </span> </div> <div class="abstract">A<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> measurement is reported of the jet mass distribution in hadronic decays of boosted top quarks produced in <math> <mi> p </mi><mi> p </mi> </math> collisions at <math> <msqrt> <mi> s </mi> </msqrt><mo> = </mo><mn> 13 </mn><mtext> </mtext><mtext> </mtext><mi> TeV </mi> </math>. The data were collected with the CMS detector at the LHC and correspond to an integrated luminosity of <math> <mrow> <mn> 35.9 </mn> <mtext> </mtext> <mtext> </mtext> <msup> <mrow> <mi> fb </mi> </mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </math>. The measurement is performed in the <math> <mrow> <mi> lepton </mi> <mo> + </mo> <mtext> jets </mtext> </mrow> </math> channel of <math> <mi> t </mi><mover> <mi> t </mi> <mo> ¯ </mo> </mover> </math> events, where the lepton is an electron or muon. The products of the hadronic top quark decay <math> <mrow> <mi> t </mi> <mo> → </mo> <mi> b </mi> <mi> W </mi> <mo> → </mo> <mi> b </mi> <mi> q </mi> <msup> <mrow> <mover> <mrow> <mi> q </mi> </mrow> <mrow> <mo> ¯ </mo> </mrow> </mover> </mrow> <mrow> <mo> ' </mo> </mrow> </msup> </mrow> </math> are reconstructed as a single jet with transverse momentum larger than 400 GeV. The <math> <mi> t </mi><mover> <mi> t </mi> <mo> ¯ </mo> </mover> </math> cross section as a function of the jet mass is unfolded at the particle level and used to extract a value of the top quark mass of <math> <mn> 172.6 </mn><mo> ± </mo><mn> 2.5 </mn><mtext> </mtext><mtext> </mtext><mi> GeV </mi> </math>. novel jet reconstruction technique is used for the first time at the LHC, which improves the precision by a factor of 3 relative to an earlier measurement. This highlights the potential of measurements using boosted top quarks, where the new technique will enable future precision measurements.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevLett.124.202001" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1576530" data-product-type="Journal Article" data-product-subtype="AM" >10.1103/PhysRevLett.124.202001</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1576530" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1576530" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="2" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1594409-semi-inclusive-studies-semileptonic-bs-decays-belle" itemprop="url">Semi-inclusive studies of semileptonic <math> <msub> <mi> B </mi> <mi> s </mi> </msub> </math> decays at Belle</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Oswald, C.</span> ; <span class="author">Urquijo, P.</span> ; <span class="author">Dingfelder, J.</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Physical Review. D, Particles, Fields, Gravitation and Cosmology</span> </span> </div> <div class="abstract">We presen<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> t an analysis of the semi-inclusive decays <math> <mrow> <msub> <mrow> <mi> B </mi> </mrow> <mrow> <mi> s </mi> </mrow> </msub> <mo> → </mo> <msubsup> <mrow> <mi> D </mi> </mrow> <mrow> <mi> s </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msubsup> <mi> X </mi> <msup> <mrow> <mo> ℓ </mo> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <mi> ν </mi> </mrow> </math> and <math> <mrow> <msub> <mrow> <mi> B </mi> </mrow> <mrow> <mi> s </mi> </mrow> </msub> <mo> → </mo> <msubsup> <mrow> <mi> D </mi> </mrow> <mrow> <mi> s </mi> </mrow> <mrow> <mo> * </mo> <mo> - </mo> </mrow> </msubsup> <mi> X </mi> <msup> <mrow> <mo> ℓ </mo> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <mi> ν </mi> </mrow> </math>, where <math> <mi> X </mi> </math> denotes a final state that may consist of additional hadrons or photons and <math> <mo> ℓ </mo> </math> is an electron or muon. The studied <math> <msub> <mi> B </mi> <mi> s </mi> </msub> </math> decays are contained in the <math> <mrow> <mn> 121.4 </mn> <mtext> </mtext> <mtext> </mtext> <msup> <mrow> <mi> fb </mi> </mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </math> <math> <mrow> <mi mathvariant="normal"> Υ </mi> <mo> ( </mo> <mn> 5 </mn> <mi> S </mi> <mo> ) </mo> </mrow> </math> data sample collected by the Belle detector at the KEKB asymmetric-energy <math> <msup> <mi> e </mi> <mo> + </mo> </msup><msup> <mi> e </mi> <mo> - </mo> </msup> </math> collider. The branching fractions of the decays are measured to be <math> <mrow> <mi mathvariant="script"> B </mi> <mo> ( </mo> <msub> <mrow> <mi> B </mi> </mrow> <mrow> <mi> s </mi> </mrow> </msub> <mo> → </mo> <msubsup> <mrow> <mi> D </mi> </mrow> <mrow> <mi> s </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msubsup> <mi> X </mi> <msup> <mrow> <mo> ℓ </mo> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <mi> ν </mi> <mo> ) </mo> <mo> = </mo> <mo> [ </mo> <mn> 8.2 </mn> <mo> ± </mo> <mn> 0.2 </mn> <mo> ( </mo> <mi> stat </mi> <mo> ) </mo> <mo> ± </mo> <mn> 0.6 </mn> <mo> ( </mo> <mi> syst </mi> <mo> ) </mo> <mo> ± </mo> <mn> 1.4 </mn> <mo> ( </mo> <mi> ext </mi> <mo> ) </mo> <mo> ] </mo> <mo> % </mo> </mrow> </math> and <math> <mrow> <mi mathvariant="script"> B </mi> <mo> ( </mo> <msub> <mrow> <mi> B </mi> </mrow> <mrow> <mi> s </mi> </mrow> </msub> <mo> → </mo> <msubsup> <mrow> <mi> D </mi> </mrow> <mrow> <mi> s </mi> </mrow> <mrow> <mo> * </mo> <mo> - </mo> </mrow> </msubsup> <mi> X </mi> <msup> <mrow> <mo> ℓ </mo> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <mi> ν </mi> <mo> ) </mo> <mo> = </mo> <mspace></mspace> <mo> [ </mo> <mn> 5.4 </mn> <mo> ± </mo> <mn> 0.4 </mn> <mo> ( </mo> <mi> stat </mi> <mo> ) </mo> <mo> ± </mo> <mn> 0.4 </mn> <mo> ( </mo> <mi> syst </mi> <mo> ) </mo> <mo> ± </mo> <mn> 0.9 </mn> <mo> ( </mo> <mi> ext </mi> <mo> ) </mo> <mo> ] </mo> <mo> % </mo> </mrow> </math>, where the first two uncertainties are statistical and systematic and the last is due to external parameters. The measurement also provides an estimate of the <math> <msubsup> <mi> B </mi> <mi> s </mi> <mrow> <mo> ( </mo> <mo> * </mo> <mo> ) </mo> </mrow> </msubsup><msubsup> <mover> <mi> B </mi> <mo> ¯ </mo> </mover> <mi> s </mi> <mrow> <mo> ( </mo> <mo> * </mo> <mo> ) </mo> </mrow> </msubsup> </math> production cross section, <math> <mrow> <mi> σ </mi> <mo> ( </mo> <msup> <mrow> <mi> e </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <msup> <mrow> <mi> e </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> <mo> → </mo> <msubsup> <mrow> <mi> B </mi> </mrow> <mrow> <mi> s </mi> </mrow> <mrow> <mo> ( </mo> <mo> * </mo> <mo> ) </mo> </mrow> </msubsup> <msubsup> <mrow> <mover> <mrow> <mi> B </mi> </mrow> <mrow> <mo> ¯ </mo> </mrow> </mover> </mrow> <mrow> <mi> s </mi> </mrow> <mrow> <mo> ( </mo> <mo> * </mo> <mo> ) </mo> </mrow> </msubsup> <mo> ) </mo> <mo> = </mo> <mo> [ </mo> <mn> 53.8 </mn> <mo> ± </mo> <mn> 1.4 </mn> <mo> ( </mo> <mi> stat </mi> <mo> ) </mo> <mo> ± </mo> <mn> 4.0 </mn> <mo> ( </mo> <mi> syst </mi> <mo> ) </mo> <mo> ± </mo> <mn> 3.4 </mn> <mo> ( </mo> <mi> ext </mi> <mo> ) </mo> <mo> ] </mo> <mtext> </mtext> <mtext> </mtext> <mi> pb </mi> </mrow> </math>, at the center-of-mass energy <math> <mrow> <msqrt> <mrow> <mi> s </mi> </mrow> </msqrt> <mo> = </mo> <mn> 10.86 </mn> <mtext> </mtext> <mtext> </mtext> <mi> GeV </mi> </mrow> </math>.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <span class="fa fa-book text-muted" aria-hidden="true"></span> Cited by 3<div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevD.92.072013" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1594409" data-product-type="Journal Article" data-product-subtype="AM" >10.1103/PhysRevD.92.072013</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1594409" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1594409" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="3" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1648392-muon-connection" itemprop="url">Muon <math> <mi> g </mi> <mo> − </mo> <mn> 2 </mn> </math> and <math> <mi mathvariant="normal"> Δ </mi> <mi> α </mi> </math> connection</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Keshavarzi, Alexander</span> ; <span class="author">Marciano, William J.</span> ; <span class="author">Passera, Massimo</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Physical Review D</span> </span> </div> <div class="abstract">T<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> he discrepancy between the Standard Model theory and experimental measurement of the muon magnetic moment anomaly, <math> <msub> <mi> a </mi> <mi> μ </mi> </msub><mo> = </mo><mrow> <mo> ( </mo> <msub> <mi> g </mi> <mi> μ </mi> </msub> <mo> - </mo> <mn> 2 </mn> <mo> ) </mo> </mrow><mo> / </mo><mn> 2 </mn> </math>, is connected to precision electroweak (EW) predictions via their common dependence on hadronic vacuum polarization effects. he same data for the total <math> <mrow> <msup> <mrow> <mi> e </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <msup> <mrow> <mi> e </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> <mo> → </mo> <mtext> hadrons </mtext> </mrow> </math> cross section, <math> <msub> <mi> σ </mi> <mrow> <mtext> had </mtext> </mrow> </msub><mo> ( </mo><mi> s </mi><mo> ) </mo> </math>, are used as input into dispersion relations to estimate the hadronic vacuum polarization contributions, <math> <msubsup> <mi> a </mi> <mi> μ </mi> <mrow> <mtext> had </mtext> <mo> , </mo> <mi> VP </mi> </mrow> </msubsup> </math>, as well as the five-flavor hadronic contribution to the running QED coupling at the <math> <mi> Z </mi> </math>-pole, <math> <mi mathvariant="normal"> Δ </mi><msubsup> <mi> α </mi> <mrow> <mtext> had </mtext> </mrow> <mrow> <mo> ( </mo> <mn> 5 </mn> <mo> ) </mo> </mrow> </msubsup><mo> ( </mo><msubsup> <mi> M </mi> <mi> Z </mi> <mn> 2 </mn> </msubsup><mo> ) </mo> </math>, which enters natural relations and global EW fits. he EW fit prediction of <math> <mi mathvariant="normal"> Δ </mi><msubsup> <mi> α </mi> <mrow> <mtext> had </mtext> </mrow> <mrow> <mo> ( </mo> <mn> 5 </mn> <mo> ) </mo> </mrow> </msubsup><mo> ( </mo><msubsup> <mi> M </mi> <mi> Z </mi> <mn> 2 </mn> </msubsup><mo> ) </mo><mo> = </mo><mn> 0.02722 </mn><mo> ( </mo><mn> 41 </mn><mo> ) </mo> </math> agrees well with <math> <mi mathvariant="normal"> Δ </mi><msubsup> <mi> α </mi> <mrow> <mtext> had </mtext> </mrow> <mrow> <mo> ( </mo> <mn> 5 </mn> <mo> ) </mo> </mrow> </msubsup><mo> ( </mo><msubsup> <mi> M </mi> <mi> Z </mi> <mn> 2 </mn> </msubsup><mo> ) </mo><mo> = </mo><mn> 0.02761 </mn><mo> ( </mo><mn> 11 </mn><mo> ) </mo> </math> obtained from the dispersion relation approach, but exhibits a smaller central value suggestive of a larger discrepancy <math> <mi mathvariant="normal"> Δ </mi><msub> <mi> a </mi> <mi> μ </mi> </msub><mo> = </mo><msubsup> <mi> a </mi> <mi> μ </mi> <mi> exp </mi> </msubsup><mo> - </mo><msubsup> <mi> a </mi> <mi> μ </mi> <mrow> <mi> SM </mi> </mrow> </msubsup> </math> than currently expected. Postulating that the <math> <mi mathvariant="normal"> Δ </mi><msub> <mi> a </mi> <mi> μ </mi> </msub> </math> difference may be due to unforeseen missing <math> <msub> <mi> σ </mi> <mrow> <mtext> had </mtext> </mrow> </msub><mo> ( </mo><mi> s </mi><mo> ) </mo> </math> contributions, implications for <math> <msub> <mi> M </mi> <mi> W </mi> </msub> </math>, <math> <msup> <mi> sin </mi> <mn> 2 </mn> </msup><msubsup> <mi> θ </mi> <mi> eff </mi> <mrow> <mi> lep </mi> </mrow> </msubsup> </math> and <math> <msub> <mi> M </mi> <mi> H </mi> </msub> </math> obtained from global EW fits are investigated. Shifts in <math> <msub> <mi> σ </mi> <mrow> <mtext> had </mtext> </mrow> </msub><mo> ( </mo><mi> s </mi><mo> ) </mo> </math> needed to bridge <math> <mi mathvariant="normal"> Δ </mi><msub> <mi> a </mi> <mi> μ </mi> </msub> </math> are found to be excluded above <math> <msqrt> <mi> s </mi> </msqrt><mo> ≳ </mo><mn> 0.7 </mn><mtext> </mtext><mtext> </mtext><mi> GeV </mi> </math> at the 95% C.L. Moreover, prospects for <math> <mi mathvariant="normal"> Δ </mi><msub> <mi> a </mi> <mi> μ </mi> </msub> </math> originating below that energy are deemed improbable given the required increases in the hadronic cross section. Such hypothetical changes to the hadronic data are also found to affect other related observables, such as the electron anomaly, <math> <msubsup> <mi> a </mi> <mi> e </mi> <mtext> SM </mtext> </msubsup> </math>, the rescaled ratio <math> <mrow> <msub> <mrow> <mi> R </mi> </mrow> <mrow> <mi> e </mi> <mo> / </mo> <mi> μ </mi> </mrow> </msub> <mo> = </mo> <mspace></mspace> <mo> ( </mo> <msub> <mrow> <mi> m </mi> </mrow> <mrow> <mi> μ </mi> </mrow> </msub> <mo> / </mo> <msub> <mrow> <mi> m </mi> </mrow> <mrow> <mi> e </mi> </mrow> </msub> <msup> <mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> </mrow> </msup> <mo> ( </mo> <msubsup> <mrow> <mi> a </mi> </mrow> <mrow> <mi> e </mi> </mrow> <mrow> <mtext> had </mtext> <mo> , </mo> <mi> LO </mi> <mtext> </mtext> <mi> VP </mi> </mrow> </msubsup> <mo> / </mo> <msubsup> <mrow> <mi> a </mi> </mrow> <mrow> <mi> μ </mi> </mrow> <mrow> <mtext> had </mtext> <mo> , </mo> <mi> LO </mi> <mtext> </mtext> <mi> VP </mi> </mrow> </msubsup> <mo> ) </mo> </mrow> </math>, and the running of the weak mixing angle at low energies, although the consequences of these are currently less constraining.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevD.102.033002" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1648392" data-product-type="Journal Article" data-product-subtype="PA" >10.1103/PhysRevD.102.033002</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="6" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1594415-amplitude-analysis-e+e-ns-sqrt-gev" itemprop="url">Amplitude analysis of <em>e<sup>+</sup>e<sup>-</sup>→Υ(nS)π<sup>+</sup>π<sup>-</sup></em> at $$\sqrt{s}$$ = 10.886 GeV</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Garmash, A.</span> ; <span class="author">Bondar, A.</span> ; <span class="author">Kuzmin, A.</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Physical Review. D, Particles, Fields, Gravitation and Cosmology</span> </span> </div> <div class="abstract">We repor<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> t results on studies of the <math> <msup> <mi> e </mi> <mo> + </mo> </msup><msup> <mi> e </mi> <mo> - </mo> </msup> </math> annihilation into three-body <math> <mrow> <mi mathvariant="normal"> Υ </mi> <mo> ( </mo> <mi> n </mi> <mi> S </mi> <mo> ) </mo> <msup> <mrow> <mi> π </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <msup> <mrow> <mi> π </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> </mrow> </math> ( <math> <mi> n </mi><mo> = </mo><mn> 1 </mn><mo> , </mo><mn> 2 </mn><mo> , </mo><mn> 3 </mn> </math>) final states including measurements of cross sections and the full amplitude analysis. The cross sections measured at <math> <mrow> <msqrt> <mrow> <mi> s </mi> </mrow> </msqrt> <mo> = </mo> <mn> 10.866 </mn> <mtext></mtext> <mtext></mtext> <mi> GeV </mi> </mrow> </math> and corrected for the initial state radiation are <math> <mrow> <mi> σ </mi> <mo> ( </mo> <msup> <mrow> <mi> e </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <msup> <mrow> <mi> e </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> <mo> → </mo> <mi mathvariant="normal"> Υ </mi> <mo> ( </mo> <mn> 1 </mn> <mi> S </mi> <mo> ) </mo> <msup> <mrow> <mi> π </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <msup> <mrow> <mi> π </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> <mo> ) </mo> <mo> = </mo> <mspace></mspace> <mo> ( </mo> <mn> 2.27 </mn> <mo> ± </mo> <mn> 0.12 </mn> <mo> ± </mo> <mn> 0.14 </mn> <mo> ) </mo> <mtext></mtext> <mtext></mtext> <mi> pb </mi> </mrow> </math>, <math> <mrow> <mi> σ </mi> <mo> ( </mo> <msup> <mrow> <mi> e </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <msup> <mrow> <mi> e </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> <mo> → </mo> <mi mathvariant="normal"> Υ </mi> <mo> ( </mo> <mn> 2 </mn> <mi> S </mi> <mo> ) </mo> <msup> <mrow> <mi> π </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <msup> <mrow> <mi> π </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> <mo> ) </mo> <mo> = </mo> <mo> ( </mo> <mn> 4.07 </mn> <mo> ± </mo> <mn> 0.16 </mn> <mo> ± </mo> <mn> 0.45 </mn> <mo> ) </mo> <mtext></mtext> <mtext></mtext> <mi> pb </mi> </mrow> </math>, and <math> <mrow> <mi> σ </mi> <mo> ( </mo> <msup> <mrow> <mi> e </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <msup> <mrow> <mi> e </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> <mo> → </mo> <mi mathvariant="normal"> Υ </mi> <mo> ( </mo> <mn> 3 </mn> <mi> S </mi> <mo> ) </mo> <msup> <mrow> <mi> π </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <msup> <mrow> <mi> π </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> <mo> ) </mo> <mo> = </mo> <mspace></mspace> <mo> ( </mo> <mn> 1.46 </mn> <mo> ± </mo> <mn> 0.09 </mn> <mo> ± </mo> <mn> 0.16 </mn> <mo> ) </mo> <mtext></mtext> <mtext></mtext> <mi> pb </mi> </mrow> </math>. Amplitude analysis of the three-body <math> <mrow> <mi mathvariant="normal"> Υ </mi> <mo> ( </mo> <mi> n </mi> <mi> S </mi> <mo> ) </mo> <msup> <mrow> <mi> π </mi> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <msup> <mrow> <mi> π </mi> </mrow> <mrow> <mo> - </mo> </mrow> </msup> </mrow> </math> final states strongly favors <math> <mrow> <msup> <mrow> <mi> I </mi> </mrow> <mrow> <mi> G </mi> </mrow> </msup> <mo> ( </mo> <msup> <mrow> <mi> J </mi> </mrow> <mrow> <mi> P </mi> </mrow> </msup> <mo> ) </mo> <mo> = </mo> <msup> <mrow> <mn> 1 </mn> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <mo> ( </mo> <msup> <mrow> <mn> 1 </mn> </mrow> <mrow> <mo> + </mo> </mrow> </msup> <mo> ) </mo> </mrow> </math> quantum-number assignments for the two bottomonium-like <math> <msubsup> <mi> Z </mi> <mi> b </mi> <mo> ± </mo> </msubsup> </math> states, recently observed in the <math> <mrow> <mi mathvariant="normal"> Υ </mi> <mo> ( </mo> <mi> n </mi> <mi> S </mi> <mo> ) </mo> <msup> <mrow> <mi> π </mi> </mrow> <mrow> <mo> ± </mo> </mrow> </msup> </mrow> </math> and <math> <msub> <mi> h </mi> <mi> b </mi> </msub><mo> ( </mo><mi> m </mi><mi> P </mi><mo> ) </mo><msup> <mi> π </mi> <mo> ± </mo> </msup> </math> ( <math> <mi> m </mi><mo> = </mo><mn> 1 </mn><mo> , </mo><mn> 2 </mn> </math>) decay channels. The results are obtained with a <math> <mrow> <mn> 121.4 </mn> <mtext> </mtext> <mtext> </mtext> <msup> <mrow> <mi> fb </mi> </mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </math> data sample collected with the Belle detector at the KEKB asymmetric-energy <math> <msup> <mi> e </mi> <mo> + </mo> </msup><msup> <mi> e </mi> <mo> - </mo> </msup> </math> collider.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <span class="fa fa-book text-muted" aria-hidden="true"></span> Cited by 29<div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevD.91.072003" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1594415" data-product-type="Journal Article" data-product-subtype="AM" >10.1103/PhysRevD.91.072003</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1594415" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1594415" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="8" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1674996-studies-psi-production-bar-collisions-tev" itemprop="url">Studies of $X(3872)$ and $$\psi(2S)$$ production in $$p\bar{p}$$ collisions at 1.96 TeV</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Abazov, V.  M.</span> ; <span class="author">Abbott, B.</span> ; <span class="author">Acharya, B.  S.</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Physical Review D</span> </span> </div> <div class="abstract">We presen<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> t various properties of the production of the <math> <mi> X </mi><mo> ( </mo><mn> 3872 </mn><mo> ) </mo> </math> and <math> <mrow> <mi> ψ </mi> <mo> ( </mo> <mn> 2 </mn> <mi> S </mi> <mo> ) </mo> </mrow> </math> states based on <math> <mrow> <mn> 10.4 </mn> <mtext> </mtext> <mtext> </mtext> <msup> <mrow> <mi> fb </mi> </mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </math> collected by the D0 experiment in Tevatron <math> <mi> p </mi><mover> <mi> p </mi> <mo> ¯ </mo> </mover> </math> collisions at <math> <msqrt> <mi> s </mi> </msqrt><mo> = </mo><mn> 1.96 </mn><mtext> </mtext><mtext> </mtext><mi> TeV </mi> </math>. For both states, we measure the nonprompt fraction <math> <msub> <mi> f </mi> <mrow> <mi> N </mi> <mi> P </mi> </mrow> </msub> </math> of the inclusive production rate due to decays of <math> <mi> b </mi> </math>-flavored hadrons. We find the <math> <msub> <mi> f </mi> <mrow> <mi> N </mi> <mi> P </mi> </mrow> </msub> </math> values systematically below those obtained at the LHC. The <math> <msub> <mi> f </mi> <mrow> <mi> N </mi> <mi> P </mi> </mrow> </msub> </math> fraction for <math> <mi> ψ </mi><mo> ( </mo><mn> 2 </mn><mi> S </mi><mo> ) </mo> </math> increases with transverse momentum, whereas for the <math> <mi> X </mi><mo> ( </mo><mn> 3872 </mn><mo> ) </mo> </math> it is constant within large uncertainties, in agreement with the LHC results. The ratio of prompt to nonprompt <math> <mi> ψ </mi><mo> ( </mo><mn> 2 </mn><mi> S </mi><mo> ) </mo> </math> production, <math> <mo> ( </mo><mn> 1 </mn><mo> - </mo><msub> <mi> f </mi> <mrow> <mi> N </mi> <mi> P </mi> </mrow> </msub><mo> ) </mo><mo> / </mo><msub> <mi> f </mi> <mrow> <mi> N </mi> <mi> P </mi> </mrow> </msub> </math>, decreases only slightly going from the Tevatron to the LHC, but for the <math> <mi> X </mi><mo> ( </mo><mn> 3872 </mn><mo> ) </mo> </math>, this ratio decreases by a factor of about 3. We test the soft-pion signature of the <math> <mi> X </mi><mo> ( </mo><mn> 3872 </mn><mo> ) </mo> </math> modeled as a weakly bound charm-meson pair by studying the production of the <math> <mi> X </mi><mo> ( </mo><mn> 3872 </mn><mo> ) </mo> </math> as a function of the kinetic energy of the <math> <mi> X </mi><mo> ( </mo><mn> 3872 </mn><mo> ) </mo> </math> and the pion in the <math> <mi> X </mi><mo> ( </mo><mn> 3872 </mn><mo> ) </mo><mi> π </mi> </math> center-of-mass frame. For a subsample consistent with prompt production, the results are incompatible with a strong enhancement in the production of the <math> <mi> X </mi><mo> ( </mo><mn> 3872 </mn><mo> ) </mo> </math> at the small kinetic energy of the <math> <mi> X </mi><mo> ( </mo><mn> 3872 </mn><mo> ) </mo> </math> and the <math> <mi> π </mi> </math> in the <math> <mi> X </mi><mo> ( </mo><mn> 3872 </mn><mo> ) </mo><mi> π </mi> </math> center-of-mass frame expected for the <math> <mrow> <mi> X </mi> <mo> + </mo> <mi> soft </mi> </mrow> </math>-pion production mechanism. For events consistent with being due to decays of <math> <mi> b </mi> </math> hadrons, there is no significant evidence for the soft-pion effect, but its presence at the level expected for the binding energy of 0.17 MeV and the momentum scale <math> <mi mathvariant="normal"> Λ </mi><mo> = </mo><mi> M </mi><mo> ( </mo><mi> π </mi><mo> ) </mo> </math> is not ruled out.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/physrevd.102.072005" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1674996" data-product-type="Journal Article" data-product-subtype="AM" >10.1103/physrevd.102.072005</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1674996" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1674996" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> </ul> </aside> </div> </section> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a class="tab-nav disabled" data-tab="related" style="color: #636c72 !important; opacity: 1;"><span class="fa fa-angle-right"></span> Similar Records</a></li> </ul> </div> </div> </section> </div></div> </div> </div> </section> <footer class="" style="background-color:#f9f9f9; /* padding-top: 0.5rem; */"> <div class="footer-minor"> <div class="container"> <hr class="footer-separator" /> <div class="text-center" style="margin-top:1.25rem;"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list" id="footer-org-menu"> <li class="pure-menu-item"> <a href="https://energy.gov" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-us-doe-min" alt="U.S. Department of Energy" /> </a> </li> <li class="pure-menu-item"> <a href="https://www.energy.gov/science/office-science" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-office-of-science-min" alt="Office of Science" /> </a> </li> <li class="pure-menu-item"> <a href="/"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-osti-min" alt="Office of Scientific and Technical Information" /> </a> </li> </ul> </div> </div> <div class="text-center small" style="margin-top:0.5em;margin-bottom:2.0rem;"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list"> <li class="pure-menu-item"><a href="/disclaim" class="pure-menu-link"><span class="fa fa-institution"></span> Website Policies <span class="hidden-xs">/ Important Links</span></a></li> <li class="pure-menu-item"><a href="/pages/contact" class="pure-menu-link"><span class="fa fa-comments-o"></span> Contact Us</a></li> <li class="d-block d-md-none"></li> <li class="pure-menu-item"><a href="https://www.facebook.com/ostigov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-facebook" style=""></span></a></li> <li class="pure-menu-item"><a href="https://twitter.com/OSTIgov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-twitter" style=""></span></a></li> <li class="pure-menu-item"><a href="https://www.youtube.com/user/ostigov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-youtube-play" style=""></span></a></li> </ul> </div> </div> </div> </div> </footer> <link href="/pages/css/pages.fonts.200912.1307.css" rel="stylesheet"> <script src="/pages/js/pages.200912.1307.js"></script><noscript></noscript> <script src='https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js?config=TeX-MML-AM_CHTML' async> MathJax.Hub.Config({ tex2jax: {displayMath: [], inlineMath: [['$$','$$'], ['\\(','\\)']]}, asciimath2jax: {delimiters: []}, skipStartupTypeset: true, webFont: null, CommonHTML: { scale: 84, // mathjx wants to scale everything to 119% by default? 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