A chiral effective theory of quarks and diquarks is formulated and applied to exotic tetraquarks and pentaquarks. The effective theory is similar to the chiral quark effective theory with the addition of diquark degrees of freedom and couplings to quarks. Chiral symmetry through generalized Goldberger-Treiman relations, fixes the mass splitting between the even and odd parity diquarks to be about 600 MeV. We provide an estimate of the parameters of the effective theory from both the random instanton model and known data on the low lying scalar nonet. We assess the static properties of the exotic baryons, such as their masses, magnetic moments and decay widths. We show that the small decay widths of the newly reported exotics are largely due to a large tunneling suppression of a quark between a pair of diquarks. We find Γ (Θ + → K + n) ≃ 2.5 ∼ 7.0 MeV and Γ (Ξ −− → Ξ − π − ) ≃ 1

Solving the Schwinger-Dyson equations, we analyze the pairing of quarks in asymmetric quark matter where quarks have different chemical potentials. We show that in the asymmetric quark matter a crystalline color-superconducting gap opens when the quark coupling is stronger than a critical value. The critical coupling is nonzero, since the infrared divergence is lessened when the momenta of pairing quarks are not opposite. The superconducting gaps and the critical couplings are calculated both at high and intermediate densities.

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