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Logarithmic behaviour of connected correlation function in CFT

We study $(m)$-type connected correlation functions of OPE blocks with respect to one spatial region in two dimensional conformal field theory. We find logarithmic divergence for these correlation functions. We justify the logarithmic behaviour from three different approaches: massless free scalar theory, Selberg integral and conformal block. Cutoff independent coefficients are obtained from analytic continuation of conformal blocks. A UV/IR relation has been found in connected correlation functions. We could derive a formal ``first law of thermodynamics'' for a subsystem using deformed reduced density matrix. Area law of connected correlation function in higher dimensions is also discussed briefly.