Improving the upper bound on the conformal dimension in primary fields

Modular invarinat, constraints the spectrum of the theory. Using the medum temprature expansion, for first and third order of derivative, a universal upper bound on the lowest primary field has been obtained in recent researches. In this paper, we will improve the upper bound on the scaling dimension of the lowest primary field. We use by the medium temprature expansion for an arbitrary orders of derivative. We show that the upper bound depends on the order of derivative. In this research, we obtain the optimal values of the order of derivatives which leads to the best upper bound.