If you neither want to use InfoPath nor Workflow, then you have to use SharePoint web services to fetch corresponding data in Column B for a selected value in dropdown Column B.
You can achieve this functionality using SPServices library. Its a js library used to call SharePoint web services and perform operation very easily and in efficient way.
You have to write script in your Script editor webpart in Newform.aspx page
<script language="javascript" type="text/javascript" src="/address to document library/jquery-1.6.2.min.js"></script>
<script language="javascript" type="text/javascript" src="/address to document library/jquery.SPServices-0.0.6.min.js"></script>
<script language="javascript" type="text/javascript">
$(document).ready(function() {
$().SPServices.SPCascadeDropdowns({
relationshipList: "Cities",
relationshipListParentColumn: "Country",
relationshipListChildColumn: "Title",
parentColumn: "Country",
childColumn: "City",
debug: true
});
});
</script>
Here is the link http://www.uccorner.com/253/sharepoint/cascading-drop-down-sharepoint-foundation-2010/ for the same.
Try to download SPSerives-2013 version and JQuery-1.10.1 version. Refrence these in your webpage. As there are compatible to each other.
Alternatively, SPServices-0.6 and JQuery 1.6.2 version are compatible to each other.
After referencing these libraries, try simple function SPGetCurrentUser
var thisUserAccount = $().SPServices.SPGetCurrentUser({
fieldName: "Name",
debug: false
});
And alert this thisUserAccount, If it return value then SPServices is working otherwise not. Then implement your functionality of cascade.
Reference to this link for simple example:-http://spservices.codeplex.com/wikipage?title=%24().SPServices.SPGetCurrentUser