The police have had a long standing common law power to arrest for breach of the peace. In 1982 the court of appeal said of a breach of the peace:-
"even in these days when affrays, riotous behaviour and other disturbances happen all too frequently, we cannot accept that there can be a breach of the peace unless there has been an act done or threatened to be done which either actually harms a person, or in his presence his property, or is likely to cause such harm, which puts someone in fear if such harm being done."
Hence the courts have attempted to define this often very unclear concept, and establish that it denotes some harm, actual of prospective, against persons or property.
Accepting the above as the best definition of a breach of the peace, the powers of arrest for it can be stated as:
NB - The power of arrest here is essentially therefore a private one. Of course, after arrest under the common law power there is nothing to prevent the person being charged under S4 or 5 of the Public Order Act 1986 or new sections of the CJA.