There is a lot of stuff you can read if you want to find out what the argument's about.
The ORR» 's page on this subject has links to their consultation document (March 2016), all the responses, and their consequent ruling (May 2016). The important items are ORR's two, and the submissions from
DfT» ,
TfL» , and HAL.
But be warned - the legal content of these is pretty chewy.
As far as I can see, the current law applies to all railway owners and train operators irrespective of who they are, public or private, though subject to some tests as to what counts as a railway. ORR can dictate access and charges, and all of that seems to have been accepted. What HAL want, though, is for Crossrail trains to pay an extra (and it seems rather large) sum to pay off the original construction costs. There are rules about that as well, one of which is that if the original plan said where that repayment came from, the owners can't demand the same money again from a new user.
There are (surprise, surprise!) further complications. The rules (i.e. the laws) have changed more than once, and the government at one stage gave HAL a perpetual exemption from something or other. Of course in reality HAL don't have a whole railway at all; it's only useable with rights to run to Paddington - which they have until 2023. (I imagine that's also when their capital repayment plan ends, in theory.) So the rules and exemption apply differently for trains that don't go to Paddington.
ORR expects to be shown evidence about costs to be charged for, and for any capital recovery too. But the construction of the railway under Heathrow was planned jointly with
BR▸ , then the joint company was bought out by HAL (as it is now), which was going to borrow the money. In the end, HAL paid for it out of revenue, and seems to have lost the bills. Of course it never was built as a money-making exercise for HAL (whatever they may say in court), it is one of many things they built to make the airport attractive to their (direct and indirect) customers.
I suspect Heathrow's owners are trying to screw all they can out of the asset while they can get away with it - that is their general approach to running the airport (if it can be called "theirs", given the debt against it). They do not seem to have any allies, at least in the railway business.
RDG‡ didn't comment on the Heathrow case at all, but did argue for "a whole-network approach to the charging framework" - i.e. in their narrow interests as
TOCs▸ , rather than as part of the bigger "companies running privatised services" sector.
Obviously in the end it will be settled - no-one's got a whole train set otherwise. And this may be more a a test case, to see what this law does, and much less antagonistic than it looks. After all, courts can set the level of compensation for what a government has done, but can't really restrict what a new government does in the future (as no parliament can bind another future one).