Commercial Fixed Wing
Flight Crew Last 90 Days 61; Flight Crew Total 4330; Flight Crew Type 4132
Communication Breakdown; Situational Awareness
ATC Issue All Types; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy
Air Traffic Control Issued New Clearance; Flight Crew Returned To Clearance
Environment - Non Weather Related; Human Factors
On a routine flight from ZZZ to ZZZ1 conducting Re-qualification training for a former Captain returning to the aircraft FO seat we were only 30 minutes into the flight when ZZZ Center issued direct ZZZ1 and shortly thereafter we were issued direct ZZZZZ (on the ZZZZZ4 arrival which was our filed route).  For almost 2 hours; we were flying direct to the initial fix for the STAR we had been filed for.  In ZZZ1 Center airspace; the second Controller issued a descent clearance to FL240 (pilot discretion).  The PF (FO in training) had been descending in a cruise descent based on a previous Controller's issuance of 'Descend to FL300.' I was discussing what was happening to our speed due to the re-cruise with the low cost index when the Controller issued 'descend via the ZZZZZ5 Arrival' (which we had not been given as we had been filed/cleared via the ZZZZZ4).  I repeated the clearance and even set the appropriate altitude on direction from the PF.  Something seemed off but I did not query ATC.  Moments later; ZZZ1 Controller asked if we were turning towards ZZZZZ2 or 'It looks like you are headed towards ZZZZZ6 (which is on the ZZZZZ4).  I answered that indeed it appeared we were turning towards ZZZZZ6 and that I'd take a look.  We were given a heading and altitude (10 degrees left) and told to contact Approach and that they would have vectors.  Upon check-in with ZZZ1 Approach; we were told expect vectors to XXL and sensed all was fine.  The approach etc.; was non-eventful.  I believe; given the low traffic environment and ATC being so helpful a thousand miles or so earlier may have contributed to the confusion on descent as we should have been given the ZZZZZ5 arrival but were never issued the arrival change (as we passed through ZZZ2/ZZZ3/ZZZ2 Center.  Being cleared direct ZZZZZ so far out may have caused an assumption that a STAR change had been issued (ZZZZZ is on both STARS).  Traffic was light and no loss of separation occurred.
Air carrier Captain reported a track heading deviation during arrival and cited communication issues with ATC as a contributing factor.