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two components will behave essentially as ideal Bose and Fermi gases. Bosons will condense at T$_{c}$ = 3 . 31 n 2 / 3 B /m$_{B}$ , while fermions will fill out (at sufficienty low tempera- ture) a Fermi sphere with radius k 0 $_{F}$, the chemical po- tentials µ$_{B}$ and µ$_{F}$ being modified with respect to their noninteracting values by the mean-field shifts 2 πn$_{F}$ a/m$_{r}$ , and 2 πn$_{B}$a/m$_{r}$ , respectively. In the opposite strong- coupling limit where a is small and positive (such that ( k$_{F}$ a ) − 1 ≫ 1 ), bosons will pair with fermions to form tightly bound fermionic molecules, with binding energy ϵ 0 = 1 / (2 m$_{r}$a $^{2}$). In particular, in systems where n$_{B}$ ≤ n$_{F}$ , on which we focus in the present paper, all bosons will eventually pair with fermions into molecules, sup- pressing condensation completely. 

In a complete analogy with the BCS-BEC crossover problem in a two-component Fermi gas [24], the choice of the self-energy diagrams will be guided by the criterion that a single set of diagrams should recover the correct physical description of the two above opposite limits. In the normal phase above the condensation critical tem- perature T$_{c}$ , to which we restrict in the present paper, this condition is met (as we shall see below) by the T- matrix choice of diagrams represented in Fig. 1, yielding the expressions: 

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for the bosonic and fermionic self-energies Σ$_{B}$ and Σ$_{F}$ , where G 0 B and G 0 F are bare bosonic and fermionic Green’s functions and Γ( P ) is the many-body T-matrix: 

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In the above expressions q = ( q , ω$_{ν}$ ), k = ( k , ω$_{n}$ ), 

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P = ( P , Ω$_{m}$), where ω$_{ν}$ = 2 πν/β and ω$_{n}$ = (2 n + 1) π/β , Ω$_{m}$ = (2 m + 1) π/β are bosonic and fermionic Matsub- ara frequencies, respectively ( β = 1 /T being the inverse temperature and ν, n, m integer), while f ( x ) and b ( x ) are the Fermi and Bose distribution functions and ξ$_{s}$ ( p ) = p $^{2}$/ (2 m$_{s}$ ) − µ$_{s}$ . The above self-energies determine finally the dressed Green’s functions G − 1 s = G 0 − 1 s − Σ$_{s}$ entering the particle numbers equations: 

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Equations (2 − 6) fully determine the thermodynamic properties of the Bose-Fermi mixture in the normal phase. This phase is characterized by the condition µ$_{B}$ − Σ$_{B}$( q = 0) < 0; upon lowering the temperature, condensation will then occur when µ$_{B}$ = Σ$_{B}$( q = 0). 

Analytic results can be obtained in the two opposite limits of the boson-fermion coupling [25]. For weak- coupling Γ( P ) ≃ − 2 πa m r , yielding Σ$_{B}$ ≃ 2 πn$_{F}$ a/m$_{r}$ and Σ$_{F}$ ≃ 2 πn$_{B}$a/m$_{r}$ , in accordance with the expected mean- field shifts. For strong-coupling Γ( P ) gets instead pro- portional to the molecular propagator: 

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where µ$_{M}$ ≡ µ$_{B}$ + µ$_{F}$ + ϵ 0 is the molecular chemical poten- tial and M = m$_{B}$ + m$_{F}$ . Insertion of this propagator in Eqs. (2 − 6) leads, for n$_{B}$ < n$_{F}$ , to µ$_{M}$ ≃ (6 π $^{2}$n$_{B}$ ) 2 / $^{3}$/ 2 M , µ$_{F}$ ≃ [6 π $^{2}$( n$_{F}$ − n$_{B}$ )] 2 / $^{3}$/ 2 m$_{F}$ and µ$_{B}$ = µ$_{M}$ − µ$_{F}$ − ϵ 0 ≃ − ϵ 0 , as expected when all bosons pair with fermions. 

For intermediate values of the boson-fermion coupling Eqs. (2 − 6) need to be solved numerically. The results for the condensation critical temperature T$_{c}$ vs the boson- fermion dimensionless coupling ( k$_{F}$ a ) − 1 are presented in 

