So, “the title for the index page” in your original post led me to believe you were referring to the title on the actual page, not the browser tab. The title in the header is created by a theme function which returns the “Site Title” set in Settings > General. This data is retrieved from the “blogname” record in the wp_options table. I couldn’t find any specific documentation on the browser tab but, based on some very unscientific testing, it appears that the browser tab text is derived from the site title and the page that is being displayed. In the default theme configuration, on the home page, this would be “site name – site tagline”. These values are retrieved from the database by a default WP function and placed between the <title> tags in the header. However, your header is showing this:
<title></title>
Again, based on a bit of testing, if the site title and tagline are missing, the default for the browser tab is the site URL. This would seem to indicate that for your site either (a) it can’t find the site title and/or the tagline, or (b) there is something else (SEO plugin?) overriding the default processing.