Do users change their search patterns if the search is shorter? Do users search less if the search box is too large?
The only article I'd found is a 2009 Smashing Magazine article:
The [Nielsen] study found that the average search box is 18-characters wide. The data showed that 27% of queries were too long to fit into it. Extending the box to 27 characters would accommodate 90% of queries. Remember, you can set widths using ems, not just pixels and points. One em is the width and height of one “m” character (using whatever font size a website is set to). So, use this measure to scale the width of the text input field to 27-characters wide.
In general, search boxes are better too wide than too short, so that users can quickly review, verify and submit the query. This guideline is very simple but unfortunately too often dismissed or ignored. Some padding in the input field can also improve the design and user experience.