Capital punishment is usually defended on the basis of a theory of deterrence or retribution, but there are some other, less common arguments as well. There’s for example the argument that capital punishment is necessary for “closure” and “healing” of the victim’s surviving family and friends. Capital punishment is therefore viewed as a therapy. Apart from the doubts that capital punishment can serve this purpose – what does closure and healing mean and do they necessarily require an execution? – there’s a strong case that it shouldn’t be used for this purpose even if it can be: it would amount to crude instrumentalization of the criminal, even more than in the case of deterrence. Moreover, there’s a problem with cause and effect: if people are told that they need an execution in order to accomplish closure, then perhaps they’ll start to believe there’s no other way.
Another argument in favor of capital punishment is based on guesses about the harm that would result from failing to use this type of punishment. If we don’t satisfy the public’s blood lust – or call it “punitive emotions” if you want – the public will seek to satisfy it in ways that we wouldn’t like (e.g. lynching). However, there’s again a problem with cause and effect in this argument: the justice system does not merely reflect opinion about appropriate punishment, but also shapes it. Far from reducing blood lust, capital punishment may instead promote it. This is the so-called brutalization effect.
The basis of blood lust is moral outrage, and such outrage – contrary to blood lust – is often completely justified. And it should be recognized, but it can be in ways that don’t involve executions.
More on capital punishment is here.