Why do people want to harm those who harmed them? Several social psychologists try to explain the science behind revenge.

For example, Ian McKee of Adelaide University in Australia writes in Social Justice Research (Vol. 138, No. 2), linking vengeful tendencies primarily with two social attitudes: right-wing authoritarianism and social dominance.

Popular social psychologist Dan Arieli in The Upside of Irrationality: The Unexpected Benefits of Defying Logic at Work and at Home (New York:  Harper Collins, 2010) suggests that revenge has a biological basis and feels pleasurable.

I am looking for new research explaining this topic.
There are at least two additional domains you can investigate.

The first is reciprocity as a means towards justice. In The Fourth Arm of Justice: The Art and Science of Revenge (1997), Judi Parks looks at this from the perspective an employee, but cites Susan Jacoby's report on the survivors of the Majdanek concentration camp in her 1983 Wild justice: The evolution of revenge:


  A desire for retribution does not require rationalization. "A victim wants to see an assailant punished not only for reasons of pragmatic deterrence but also as a means to repairing a damaged sense of civic order and personal identity. Deterrence and retribution are hardly identical, but the former invariably involves an element of the latter.


A more recent book that explores this dimension is Payback: Why We Retaliate, Redirect Aggression, and Take Revenge (Barash &amp; Lipton, 2011).

Since deterrence is quite salient in game theory, this aspect shows up there, too. See, for example, The Paradox of Revenge in Conflicts (Amegashie &amp; Runkel, 2012).

The second is identity theory. An example in “If you wrong us, shall we not revenge?” Social identity salience moderates support for retaliation in response to collective threat (Fischer, Haslam, &amp; Smith, 2010)):


  Specifically, we found that a threat to national identity (the 7/7/2005 London bombings) led to greater aggression and greater support for revenge when national rather than gender identity was salient. In contrast, a threat to gender identity (Taliban misogyny) led to greater aggression and greater support for revenge when gender rather than national identity was salient.