"Have no dog in this race" is a popular idiom that originated from horse racing, but it is often used in various contexts to imply not having a personal stake or interest in a particular matter. Several synonyms can be employed to convey a similar meaning. One alternative is "not be invested", which signifies lacking any emotional or vested interest. Another phrase could be "remain impartial", denoting a neutral standpoint without favoring any side. Additionally, "not having a horse in this race" can be used interchangeably to illustrate the absence of personal involvement. These synonyms effectively convey the sense of detachment and unbiased perspective associated with having no dog in the race.