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We microscopically study the effect of the magnetic field (Zeeman splitting)on the superconducting state in a model for quasi-one-dimensional organicsuperconductors (TMTSF)$_{2}X$. We investigate the competition between spinsinglet and spin triplet pairings and theFulde-Ferrell-Larkin-Ovchinnikov(FFLO) state by random phase approximation.While we studied the competition by comparison with the eigenvalue of the gapequation at a fixed temperature in our previous study (Phys. Rev. Lett.\textbf{102} (2009) 016403), here we obtain both the $T_c$ for each pairingstate and a phase diagram in the $T$(temperature)-$h_z$(field)-$V_y$(strengthof the charge fluctuation) space. The phase diagram shows that consecutivetransitions from singlet pairing to the FFLO state and further to $S_z=1$triplet pairing can occur upon increasing the magnetic field when $2k_{F}$charge fluctuations coexist with $2k_{F}$ spin fluctuations. In the FFLO state,the singlet d-wave and $S_{z}=0$ triplet $f$-wave components are strongly mixedespecially when the charge fluctuations are strong.

Pairing competition in a quasi-one-dimensional model of organic superconductors (TMTSF)$_{2}X$ in magnetic field