From effective java, 2nd ed.,
The sole exception to the previous rule concerns “constant fields,”
whose names should consist of one or more uppercase words separated by
the underscore character, for example, VALUES or NEGATIVE_INFINITY. A
constant field is a static final field whose value is immutable. If a
static final field has a primitive type or an immutable reference type
(Item 15), then it is a constant field. For example, enum constants
are constant fields. If a static final field has a mutable reference
type, it can still be a constant field if the referenced object is
immutable.
In summary, constant == static final, plus if it's a reference (vs. a simple type), immutability.
Looking at the slf4j logger,
http://www.slf4j.org/api/org/slf4j/Logger.html
It is immutable. On the other hand, the JUL logger is mutable. The log4j logger is also mutable. So to be correct, if you are using log4j or JUL, it should be "logger", and if you are using slf4j, it should be LOGGER.
Note that the slf4j javadocs page linked above has an example where they use "logger", not "LOGGER".
These are of course only conventions and not rules. If you happen to be using slf4j and you want to use "logger" because you are used to that from other frameworks, or if it is easier to type, or for readability, go ahead.