This is called intentional infliction of emotional distress. A cause of action for intentional infliction of emotional distress consists of: (1) outrageous conduct by the defendant with the intention to cause or reckless disregard of the probability of causing emotional distress, (2) severe emotional suffering and (3) actual and proximate causation of the emotional distress.
To be specific, in California intentional infliction of emotional distress exists when person or entity conducts
• Extreme or outrageous conduct
• That intentionally or recklessly causes
• Severe emotional distress (and also possible bodily harm)
Extreme and outrageous conduct goes beyond those acts that are merely malicious, harmful or offensive. Legally speaking, people must have a certain level of thick skin and possess the ability to weather ordinary rude or obnoxious behavior. In order to rise to the level of actionable extreme and outrageous conduct, the behavior must exceed all possible bounds of decency. Normal insults or rudeness don't normally qualify as extreme and outrageous conduct. In addition, ordinary insults or actions can constitute extreme and outrageous behavior if the actor knows that the victim is particularly susceptible to emotional distress because of some physical or mental condition or abnormality.
Emotional distress can take many forms. Many unpleasant emotions qualify as emotional distress, including embarrassment, shame, fright and grief. In order to satisfy the elements of an intentional infliction of emotional distress claim, however, the emotional distress that occurs in response to intentional or reckless extreme and outrageous behavior must reach a "severe" level. The exact definition of "severe emotional distress" is vague, and plaintiffs must prove to a judge or jury that the emotional distress they experienced reached a sufficient level of severity to justify an award for the tort of intentional infliction of emotional distress. Some guidelines do exist to help determine whether an emotional disturbance constitutes severe emotional distress. When extreme and outrageous conduct causes suffering of a type that no reasonable person should have to endure, a judge or jury will likely hold that the experience reached the level of severe emotional distress.
The intensity and duration of the emotional distress also contribute to its severity. The longer the emotional disturbance continues, the more likely it is to constitute severe emotional distress.
A plaintiff must use evidence to demonstrate their emotional distress to a jury.
Finally, bodily harm also acts as an indicator that severe emotional distress has occurred. Ulcers or headaches, for example, can show that the plaintiff has experienced severe emotional distress that has revealed itself through these physical symptoms.
Thus, while you can sue them, you'll need to show some kind of significant harm from the slur used against you. If you were to get treatment from a doctor who could diagnose you and link the symptoms to what was written to you, you'd have a stronger case.