Not all lime is created equal, as with most things. Bottom line is you are looking for it's ability to neutralize the acidity of the soil and raise it to somewhat close to neutral, which is a pH of 7.0. Your one field is above that so this field is actually somewhat alkaline. This field has a lot of broomsedge. Lack of lime or low pH is not the cause for the broom to be here. More on that in a second. Lime's ability to neutralize is evaluated with calcium carbonate equivalents, with 1, or 100, being fully effective. Lime with lower numbers, <1 or <100 will require more lime to raise the soil pH to the same level as a lime with 1 or 100 as the equivalent. Some lime amendments have calcium carbonate equivalents that are >1 or >100 and will require less quantity of the amendment.
Magnesium balance in your soil is something you should consider as well and high mag lime may be something for you to consider. The best help you can get in calculating if you need high mag lime or how much lime to apply is going to come from your extension agent. A note about the quantity of lime to apply: You do not want to apply more than 2 tons of material per acre in the form of a top dressing per year (can incorporate 4 tons per acre per year). The overage will be lost with wind or runoff, or have an adverse effect on your vegetation. If more lime is required, apply it in spit applications over years.
As for the broomsedge or broom. The reason it is present, and it has been noted but not explained, is a lack of phosphorus to the vegetation. The pH is above neutral so your field with the broomsedge is just low in phosphorus, as your test indicates. However, pH affecte the AVAILABILITY of phosphorous to the plant. It is important to note that if the pH is too low, the phosphorous AVAILABILITY to the plant is negatively affected and plants do not have access to phosphorous that is available in the soil. When this is the case, adding phosphorous fertilizer will. NOT correct the broomsedge problem as added phosphorous will not be available to the plant. Correcting or raising the pH of the soil will free up phosphorous that is bound to the soil making it available to the vegetation.
Broomsedge really isn't much of a competitive plant. When soil conditions are right (for desired plants) the desired plants easily out compete the broomsedge. Broomsedge is successful in situations where phosphorous availability is low because broomsedge has a very low phosphorous requirement. In soils wher phosphorous levels are higher, more in line with what is needed, broomsedge simply is not competetive.
