Absence of a four-body Efimov effect in the 2 + 2 fermionic problem

In the free three-dimensional space, we consider a pair of identical $\uparrow$ fermions of some species or in some internal state, and a pair of identical $\downarrow$ fermions of another species or in another state. There is a resonant $s$-wave interaction (that is of zero range and infinite scattering length) between fermions in different pairs, and no interaction within the same pair. We study whether this $2+2$ fermionic system can exhibit (as the $3+1$ fermionic system) a four-body Efimov effect in the absence of three-body Efimov effect, that is the mass ratio $\alpha$ between $\uparrow$ and $\downarrow$ fermions and its inverse are both smaller than 13.6069{\ldots}. For this purpose, we investigate scale invariant zero-energy solutions of the four-body Schr\''odinger equation, that is positively homogeneous functions of the coordinates of degree {$s-7/2$}, where $s$ is a generalized Efimov exponent {that becomes purely imaginary in the presence of a four-body Efimov effect.} Using rotational invariance in momentum space, it is found that the allowed values of $s$ are such that $M(s)$ has a zero eigenvalue; here the operator $M(s)$, that depends on the total angular momentum $\ell$, acts on functions of two real variables (the cosine of the angle between two wave vectors and the logarithm of the ratio of their moduli), and we write it explicitly in terms of an integral matrix kernel. We have performed a spectral analysis of $M(s)$, analytical and for an arbitrary imaginary $s$ for the continuous spectrum, numerical and limited to $s = 0$ and $\ell \le 12$ for the discrete spectrum. We conclude that no eigenvalue of $M(0)$ crosses zero over the mass ratio interval $\alpha \in [1, 13.6069\ldots]$, even if, in the parity sector $(-1)^{\ell}$, the continuous spectrum of $M(s)$ has everywhere a zero lower border. As a consequence, there is no possibility of a four-body Efimov effect for the 2+2 fermions. We also enunciated a conjecture for the fourth virial coefficient of the unitary spin-$1/2$ Fermi gas,inspired from the known analytical form of the third cluster coefficient and involving the integral over the imaginary $s$-axis of $s$ times the logarithmic derivative of the determinant of $M(s)$ summed over all angular momenta.The conjectured value is in contradiction with the experimental results.