I am not very familiar with Broomsedge, but there has been some research (https://warren.ces.ncsu.edu/2019/03/how-to-get-rid-of-broomsedge-broomstraw/
) showing that application of lime and/or phosphorous can temporarily change the soil chemistry enough so Broomsedge has less of an advantage on a given site. With that said, application of lime and/or fertilizer treats the symptom of invading Broomsedge but will not solve the problem. Repeatedly cutting a field for hay and exporting the bales from the field does not allow for proper stimulation of the plants by grazing, exports a great deal of carbon from the soil system, and leaves the soil with little cover. Occasionally cutting a field for hay and exporting the hay to another location is not usually an issue, but doing so year after year will typically result in a decline in soil health and plant species composition.
Plant species composition in a pasture can be permanently improved and maintained by proper grazing. Proper grazing involves the grazing animals removing no more than half of what has grown at the time they graze and then allowing the plants to fully recover before being grazed again. This means the animals must be confined to a small enough area at a time so they bite ALL of the plants, but do not remove more than half of what is there at the time of grazing. To do this, the animals must only be allowed access to a small area at a time, usually only for a few days and then moved to a new area. Portable electric fencing is a great tool to accomplish this. If the livestock are allowed access to the whole pasture for an extended period of time they will only bite the plants they like and return to bite them again and again. This results in plants that die out from not being grazed (and stimulated) and plants that die out from being grazed to death without time to recover. What is left are undesirable plants (such as Broomsedge). By properly grazing, the plants will all be stimulated to exude sugars into the soil to feed the soil biology and the plants that have been grazed are not bitten again until they have recovered. This results in stimulated, healthy plants and well fed soil that has cover on it all the time. You may never totally get rid of Broomsedge, but it will become a minor component as the species diversity increases. The livestock only really want the top third or so of the plants for their nutrition, so the livestock benefit as well. All the manure and urine are returned to the soil to be cycled again.
Once you can start properly grazing your pastures, they can recover and become populated with desirable plants. If you want to apply some lime and or phosphorous to help speed that process along it's your money, but proper grazing can restore things if you are patient and diligent.