Applying the previously developed systematic thermal (imaginary time) perturbative expansion to the relevant effective field theory, we compute the free energy $F$ of the diluted gas of (nonrelativistic) spin-$1/2$ fermions interacting through a spin-independent repulsive two-body potential as a function of the numbers $N_{+}$ and $N_{-}$ of spin-up and spin-down fermions (i.e., as a function of the system's polarization) and the temperature $T$. We give the complete order ${\left(k_{\mathrm{F}}{a}_0\right)}^3$ ($k_{\mathrm{F}}$ is the Fermi wave vector and $a_0$ is the $s$-wave scattering length characterizing the interaction potential) contribution to $F$. We also extend the computation beyond a fixed order by resumming to all orders in the parameter $k_{\mathrm{F}}{a}_0$ the contributions to $F$ of two infinite sets of Feynman diagrams: the so-called particle-particle rings and the particle-hole rings. We find that including the second one of these two contributions has a dramatic consequence for the transition of the system from the paramagnetic to the ferromagnetic phase (the so-called Stoner phase transition): in this approximation the phase transition simply disappears. This result does not contradict the expectation that a transition to the magnetically ordered state should occur in truly repulsive systems. The $p$-wave and higher scattering lengths, as well as other parameters chacterizing the interaction potential, are in such systems generally of the same order of magnitude as $a_0$ and contributions depending on them should be, therefore, also included in $F$. Our results may, however, have implications for the search of the itinerant ferromagnetism of cold atomic gases in which large $a_0$, much larger than all other parameters, is artificially created by exploiting the physics of the Feshbach resonance.

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• Effective field theory
• Ultracold gases