Neither $search_form nor $search_box is available in Zen or, likely, any other themes in Drupal 7. Instead, there's a block.

If you need to print the search box in page.tpl.php, you'll need to render the block and then print it like so:

To break that down, 'search' is the module invoking the block. 'block_view' is the operation we want to call, that is, we want to view the block, and 'form' is the machine name of the block you want to print.