Spread the tuna, scallions, olives and capers on a plate and freeze for 5 minutes. Transfer to a food processor and pulse until the tuna is finely chopped. Transfer the mixture to a bowl; season with salt and pepper. Form the mixture into four 4-inch patties.
Light a grill or preheat a grill pan. Brush the burgers with olive oil and grill over moderately high heat, turning once, until golden and crusty and just cooked through, about 6 minutes.
In a bowl, mix the mayonnaise and anchovy paste; spread on the buns. Top with the burgers, tomato and arugula, close and serve.