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The calculation is performed by reducing the relevant Feynman integrals into a canonical basis of master integrals by means of integration-by-parts identities. The resulting integrals are then evaluated by solving a system of differential equations in the kinematic invariants, whose boundary conditions are determined analytically with some care due to the presence of Coulomb singularities. The fully differential soft function is expressed in terms of Goncharov polylogarithms. This result is an essential ingredient for a range of N$^3$LL resummations for key collider observables at lepton colliders, such as the $Q\bar{Q}V$ production cross section at threshold and observables sensitive to the total transverse momentum of the radiation in heavy-quark final states. Moreover, it constitutes the complete final-final dipole contribution to the fully differential soft function needed for the description of $Q\bar{Q}V$ production at hadron colliders, which plays an important role in the LHC physics programme. 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The calculation is performed by reducing the relevant Feynman integrals into a canonical basis of master integrals by means of integration-by-parts identities. The resulting integrals are then evaluated by solving a system of differential equations in the kinematic invariants, whose boundary conditions are determined analytically with some care due to the presence of Coulomb singularities. The fully differential soft function is expressed in terms of Goncharov polylogarithms. This result is an essential ingredient for a range of N$^3$LL resummations for key collider observables at lepton colliders, such as the $Q\bar{Q}V$ production cross section at threshold and observables sensitive to the total transverse momentum of the radiation in heavy-quark final states. Moreover, it constitutes the complete final-final dipole contribution to the fully differential soft function needed for the description of $Q\bar{Q}V$ production at hadron colliders, which plays an important role in the LHC physics programme." property="og:description" />  <meta content="summary" name="twitter:card" /> <style></style> </head> <body class="CERN32Document32Server search" lang="en">  <div id="cern-toolbar"> <h1><a href="http://cern.ch" title="CERN">CERN <span>Accelerating science</span></a></h1> <ul> <li class="cern-accountlinks"><a class="cern-account" href="https://cds.cern.ch/youraccount/login?ln=en&referer=https%3A//cds.cern.ch/record/2917727%3Fln%3Den" title="Sign in to your CERN account">Sign in</a></li> <li><a class="cern-directory" href="http://cern.ch/directory" title="Search CERN resources and browse the directory">Directory</a></li> </ul> </div>   <div role="banner" class="clearfix" id="header"> <div class="header-inner inner"> <hgroup class="clearfix"> <h2 id="site-name"> <a rel="home" title="Home" href="/"><span>CERN Document Server</span></a> </h2> <h3 id="site-slogan">Access articles, reports and multimedia content in HEP</h3> </hgroup> <div role="navigation" id="main-navigation" class="cdsmenu"> <h2 class="element-invisible">Main menu</h2><ul class="links inline clearfix"> <li class="menu-386 first active-trail"><a class="active-trail" href="https://cds.cern.ch/?ln=en">Search</a></li> <li class="menu-444 "><a class="" title="" href="https://cds.cern.ch/submit?ln=en">Submit</a></li> <li class="menu-426 "><a class="" href="https://cds.cern.ch/help/?ln=en">Help</a></li> <li class="leaf hassubcdsmenu"> <a hreflang="en" class="header" href="https://cds.cern.ch/youraccount/display?ln=en">Personalize</a> <ul class="subsubcdsmenu"><li><a href="https://cds.cern.ch/youralerts/list?ln=en">Your alerts</a></li><li><a href="https://cds.cern.ch/yourbaskets/display?ln=en">Your baskets</a></li><li><a href="https://cds.cern.ch/yourcomments?ln=en">Your comments</a></li><li><a href="https://cds.cern.ch/youralerts/display?ln=en">Your searches</a></li></ul></li> </ul> </div> </div> </div>  <table class="navtrailbox"> <tr> <td class="navtrailboxbody"> <a href="/?ln=en" class="navtrail">Home</a> > The two-loop fully differential soft function for $Q\bar{Q}V$ production at lepton colliders </td> </tr> </table> </div> <div class="pagebody"><div class="pagebodystripemiddle"> <div class="detailedrecordbox"> <div class="detailedrecordtabs"> <div> <ul class="detailedrecordtabs"><li class="on first"><a href="/record/2917727/?ln=en">Information </a></li><li class=""><a href="/record/2917727/files?ln=en">Files </a></li></ul> <div id="tabsSpacer" style="clear:both;height:0px"> </div></div> </div> <div class="detailedrecordboxcontent"> <div class="top-left-folded"></div> <div class="top-right-folded"></div> <div class="inside">  <abbr class="unapi-id" title="2917727"></abbr>  <link href="/img/download_and_embed_buttons.css" rel="stylesheet" type="text/css" /> <style type="text/css">  .video_talk_wrapper{ max-width: 640px; min-height: 360px; } </style> <table class="formatRecordTableFullWidth"> <tr> <td class="formatRecordHeader" style="background-image: url('/img/journals.jpg');" colspan="2">  Preprint </td> </tr> <tr><td class="formatRecordLabel"> Report number </td><td style="padding-left:5px;">CERN-TH-2024-203 ; <a href="http://arxiv.org/abs/arXiv:2411.13466">arXiv:2411.13466</a></td></tr> <tr><td class="formatRecordLabel"> Title </td><td style="padding-left:5px;"><b>The two-loop fully differential soft function for $Q\bar{Q}V$ production at lepton colliders</b></td></tr> <style> .video_control_container, #video_controlbutton { display: none; }#video_detailbox_download {display: none;}#video_detailbox_embed {display: none;}</style> <div class="video_control_container"> <div style="box-sizing: inherit;position: relative;min-height: 1em;margin: 1em 0;background: #f8f8f9;padding: 1em 1.5em;line-height: 1.4285em;transition: opacity .1s ease,color .1s ease,background .1s ease,box-shadow .1s ease,-webkit-box-shadow .1s ease;border-radius: .28571429rem;font-size: 1.1em;margin-top: 0;margin-bottom: 6px;background-color: #f8ffff;color: #276f86;box-shadow: 0 0 0 1px #a9d5de inset,0 0 0 0 transparent;"> <i class="fa fa-info"></i><span style="padding-left: 9px;">If you experience any problem watching the video, click the download button below</span> </div> <a id="video_controlbutton_download" rel="download_button" class="video_controlbutton" href="#">Download</a> <a id="video_controlbutton_embed" class="video_controlbutton" href="#">Embed</a> </div> <div id="video_detailbox_download" style="margin:10px"> </div> <div id="video_detailbox_embed" style="margin:10px"> </div> <tr><td class="formatRecordLabel"> <span style="white-space:nowrap;">Author(s)</span> </td><td style="padding-left:5px;"><a href="https://cds.cern.ch/search?f=author&p=Liu%2C%20Ze%20Long&ln=en">Liu, Ze Long</a> (Beijing, Inst. High Energy Phys.) ; <a href="https://cds.cern.ch/search?f=author&p=Monni%2C%20Pier%20Francesco&ln=en">Monni, Pier Francesco</a> (CERN)</td></tr> <tr><td class="formatRecordLabel"> Document contact </td><td style="padding-left:5px;"><span style="">Contact: </span>arXiv</td></tr>  <script type="text/javascript"> /* * Changes the Imprint title to Date - Duration as requested by RQF0820114 * This is a hack not to touch the general imprint format element */ $(document).ready(function() { var collections = 'PREPRINT'; if (collections.indexOf('E-LEARNING') > -1) { $('.cern-imprint-label').text('Date - Duration'); } }) </script> <tr><td class="formatRecordLabel cern-imprint-label"> Imprint </td><td style="padding-left:5px;">2024-11-20</td></tr> <tr><td class="formatRecordLabel"> Number of pages </td><td style="padding-left:5px;">15</td></tr>  <tr><td class="formatRecordLabel"> Note </td><td style="padding-left:5px;">15 pages, 1 figure; ancillary files for the analytical result</td></tr> <tr><td class="formatRecordLabel"> Subject category </td><td style="padding-left:5px;">hep-ph ; Particle Physics - Phenomenology</td></tr> <tr><td class="formatRecordLabel"> Abstract </td><td style="padding-left:5px;">We consider the production of a pair of heavy quarks $Q\bar{Q}$ in association with a generic colour singlet system $V$ at lepton colliders, and present the first analytic calculation of the two-loop soft function differential in the total momentum of the real radiation. The calculation is performed by reducing the relevant Feynman integrals into a canonical basis of master integrals by means of integration-by-parts identities. The resulting integrals are then evaluated by solving a system of differential equations in the kinematic invariants, whose boundary conditions are determined analytically with some care due to the presence of Coulomb singularities. The fully differential soft function is expressed in terms of Goncharov polylogarithms. This result is an essential ingredient for a range of N$^3$LL resummations for key collider observables at lepton colliders, such as the $Q\bar{Q}V$ production cross section at threshold and observables sensitive to the total transverse momentum of the radiation in heavy-quark final states. Moreover, it constitutes the complete final-final dipole contribution to the fully differential soft function needed for the description of $Q\bar{Q}V$ production at hadron colliders, which plays an important role in the LHC physics programme.</td></tr> <tr><td class="formatRecordLabel"> Other source </td><td style="padding-left:5px;"><a href="http://inspirehep.net/record/2850253">Inspire</a></td></tr> <tr><td class="formatRecordLabel"> Copyright/License </td><td style="padding-left:5px;">preprint: (License: <a href="http://arxiv.org/licenses/nonexclusive-distrib/1.0/">arXiv nonexclusive-distrib 1.0</a>)</td></tr> </table> <br /><div style="max-width:1024px;margin:auto"><div style="overflow-x:auto;display:inline;width:100%;"><a href="/record/2917727/plots#0"><img style="vertical-align:middle;" src="https://cds.cern.ch/record/2917727/files/dia3.png" title=" : " width="200px"/></a> <a href="/record/2917727/plots#1"><img style="vertical-align:middle;" src="https://cds.cern.ch/record/2917727/files/dia1.png" title=" : " width="200px"/></a> <a href="/record/2917727/plots#2"><img style="vertical-align:middle;" src="https://cds.cern.ch/record/2917727/files/dia2.png" title=" : " width="200px"/></a> <a href='/record/2917727/plots'>Show more plots</a></div></div><br /> <br/><div><div style="clear: both;"> </div></div> <script type="text/javascript"> // Initially hide: $(".longCaption").hide(); // Allow to toggle visibility: $(".toggleLongCaption").toggle(function(){ $(this).siblings(".longCaption").show('fast'); var thisElem = $(this); thisElem.text(thisElem.text() === "more" ? 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The calculation is performed by reducing the relevant Feynman integrals into a canonical basis of master integrals by means of integration-by-parts identities. The resulting integrals are then evaluated by solving a system of differential equations in the kinematic invariants, whose boundary conditions are determined analytically with some care due to the presence of Coulomb singularities. The fully differential soft function is expressed in terms of Goncharov polylogarithms. This result is an essential ingredient for a range of N$^3$LL resummations for key collider observables at lepton colliders, such as the $Q\\bar{Q}V$ production cross section at threshold and observables sensitive to the total transverse momentum of the radiation in heavy-quark final states. 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