Hide short product description when not logged in

Hello,

I know that I can hide prices when users are not logged in, using this code:

add_action('after_setup_theme','activate_filter') ;

function activate_filter(){
add_filter('woocommerce_get_price_html', 'show_price_logged');
}

function show_price_logged($price){
if(is_user_logged_in() ){
return $price;
}
else
{
return '<a href="' . get_permalink(woocommerce_get_page_id('myaccount')) . '">Login to See Prices</a>';
remove_action( 'woocommerce_after_shop_loop_item', 'woocommerce_template_loop_add_to_cart' );
remove_action( 'woocommerce_single_product_summary', 'woocommerce_template_single_price', 10 );
remove_action( 'woocommerce_single_product_summary', 'woocommerce_template_single_add_to_cart', 30 );
remove_action( 'woocommerce_after_shop_loop_item_title', 'woocommerce_template_loop_price', 10 );

}
}

Is there a way to hide the short product description when users are not logged in? In my shop this description is filled with information that quests are not allowed to see.

Thanks!!