I'll use California penal code 837 as an example, though most other states have similar statutes:
A private person may arrest another:
- For a public offense committed or attempted in his presence...
Any person making an arrest may orally summon as many persons as he deems necessary to aid him therein.
Generally, someone making an arrest is allowed to use "reasonable force" to effect the arrest.
The question then becomes, is the act of interrupting a football game a public offense? Once a fan at a football game enters the field, assuming it is a violation of the license granted to the fan, they are trespassing. These fans are often drunk when performing their midfield dance so that is another public offense for which they could be arrested.
Once arrested, the interloper must be turned over as soon as possible to a magistrate or peace officer.
The person making the arrest is always subject to being sued. It is a question for a trier of fact to determine if unreasonable force was used in effecting the arrest.
My guess is that in most of these cases security simply ejects the exuberant fan from the premises and the fan never looks back. If a lawsuit were to be filed it would be based on unreasonable force being applied during the arrest.
California penal code 240 defines assault as
"an unlawful attempt, coupled with a present ability, to commit a violent injury on the person of another."
Certainly, one could be charged with assault in effecting a citizen's or private arrest but it would go back to the definition of reasonable force and what force was necessary to effect the arrest. If someone resisted arrest I think it more likely that that person could face an assault charge.