The recent formation of ultracold bosonic molecules from a Fermi gas of atoms[@regal] by a Feshbach resonance allows for an experimental check of theoretical calculations for physical quantities within the BCS-BEC crossover. In particular, in Ref.  the relation between the composite-boson scattering length $a_B$ and the fermionic scattering length $a_F$ was calculated in the strong-coupling limit of the BCS-BEC crossover. The summation therein of all bosonic T-matrix diagrams has led to the result $a_B=0.75 a_F$. This result corrects the value $a_B= 2 a_F$ obtained within the Born approximation for the effective residual bosonic interaction[@Haussmann; @PS-96; @epjb]. The result $a_B=0.75 a_F$ could be tested experimentally in the near future, by measuring at the same time the molecule-molecule scattering length and the fermion-fermion scattering length while scanning the magnetic field through the Feshbach resonance.

In this manuscript, we provide a condensed version of the material published in Ref. , focusing specifically on the calculation of the bosonic scattering length. We hope that this short summary of our previous work could be useful to the scientific community at the present time.

Building blocks of the diagrammatic structure for composite bosons
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In this section, we discuss the diagrammatic structure that describes generically the composite bosons in terms of the constituent fermions. Our theory rests on a judicious choice of the fermionic interaction, which (without loss of generality) greatly reduces the number and considerably simplifies the expressions of the Feynman diagrams to be taken into account.

Regularization of the fermionic interaction
-------------------------------------------

We begin by considering the following Hamiltonian for interacting fermions (we set Planck $\hbar$ and Boltzmann $k_{B}$ constants equal to unity throughout): $$\begin{aligned}
& & H  =  \sum_{\sigma} \int d{\bf r} \, \psi_{\sigma}^{\dagger}({\bf r}) \left(
- \frac{\nabla^2}{2m} - \mu \right) \psi_{\sigma}({\bf r})  \nonumber  \\
& & +  \frac{1}{2} \sum_{\sigma, \sigma'} \int d{\bf r} \, d{\bf r'}
  \psi_{\sigma}^{\dagger}({\bf r}) \psi_{\sigma'}^{\dagger}({\bf r'})
  V_{{\mathrm eff}}({\bf r}-{\bf r'}) 
  \psi_{\sigma'}({\bf r'}) \psi_{\sigma}({\bf r}) . 
                                                         \label{Eq:Hamiltonian}\end{aligned}$$ Here, $\psi_{\sigma}({\bf r}$) is the fermionic field operator with spin projection $\sigma = (\uparrow, \downarrow$), $m$ the fermionic mass, $\mu$ the fermionic chemical potential, and $V_{{\mathrm eff}}({\bf r}-{\bf r'})$ the [*effective potential*]{} that provides the *attraction* between fermions. For the application to atomic gases, the two spin states correspond to two different hyperfine states of the fermionic atoms.

To simplify the ensuing many-body diagrammatic structure (yet preserving the physical effects we are after), we adopt for $V_{{\mathrm eff}}$ the form of a “contact” potential [@footnote-contact] $$V_{{\mathrm eff}}({\bf r}-{\bf r'}) = v_{0} \,\, \delta ({\bf r}-{\bf r'})
                                         \label{Eq:deltafunc}$$ where $v_{0}$ is a negative constant. With this choice, the interaction affects only fermions with opposite spins in the Hamiltonian (\[Eq:Hamiltonian\]) owing to Pauli principle. A suitable *regularization* of the potential (\[Eq:deltafunc\]) is, however, required to get accurate control of the many-body diagrammatic structure. In particular, the equation (in the center-of-mass frame) $$\frac{m}{4 \pi a_{F}} \, = \, \frac{1}{v_{0}} \, + \, \int \! \frac{d{\bf k}}
{(2 \pi)^{3}} \frac{m}{{\bf k}^{2}}                    \label{ferm-scatt-ampl}$$ for the *fermionic scattering length* $a_{F}$ associated with the potential (\[Eq:deltafunc\]) is ill-defined, since the integral over the three-dimensional wave vector ${\bf k}$ is ultraviolet divergent. The delta-function potential (\[Eq:deltafunc\]) is then regularized, by introducing an ultraviolet cutoff $k_{0}$ in the integral of Eq. (\[ferm-scatt-ampl\]) and letting $v_{0} \, \rightarrow \, 0$ as $k_{0} \, \rightarrow \, \infty$, in order to keep $a_{F}$ fixed at a *finite* value. The required relation between $v_{0}$ and $k_{0}$ is obtained directly from Eq. (\[ferm-scatt-ampl\]). One finds: $$v_{0} \, = \, - \, \frac{2 \pi^{2}}{m k_{0}} \, - \, \frac{\pi^{3}}{m a_{F} k_{0}^{2}}
                                                               \label{vo}$$ when $k_{0} |a_{F}| \gg 1$.

With the regularization (\[vo\]) for the potential, the classification of the many-body diagrams gets considerably simplified, since only specific sub-structures of these diagrams survive when the limit $k_{0} \, \rightarrow \, \infty$ is eventually taken. In particular, in order to obtain a finite result for a given Feynman diagram, the vanishing strength $v_0$ of the potential should be compensated by an ultraviolet divergence in some internal wave-vector integration. For the particle-particle ladder of Fig. \[fig:pplad\], the internal wave-vector integration associated with each rung diverges in the limit $k_0\to\infty$ and compensates the vanishing of $v_0$, yielding the finite result: $$\begin{aligned}
& &\Gamma_{0}(q) = - \left\{ \frac{m}{4 \pi a_{F}} + \right.
\int \! \frac{d{\bf k}}{(2 \pi)^{3}}\nonumber\\ 
& &\times\left. \left[\frac{\tanh(\beta \xi({\bf k})/2) 
+\tanh(\beta \xi({\bf k-q})/2)}{2(\xi({\bf k})+\xi({\bf k-q})-i
\Omega_{\nu})} 
- \frac{m}{{\bf k}^{2}} \right] \right\}^{-1}\; .
\label{p-p ladder}\end{aligned}$$ Here, $\xi({\bf k}) = {\bf k}^{2} /(2m) - \mu$ and $q=({\bf q},\Omega_{\nu})$ is a four-momentum, with wave vector ${\bf q}$ and Matsubara frequency $\Omega_{\nu}$ ($\nu$ integer). In an analogous way, one can show that in the particle-particle channel the contributions of the vertex corrections and of the two-particle effective interactions other than the rung vanish for our choice of the potential.

It is thus evident from these considerations that, with our choice of the fermionic interaction, in the strong-coupling limit the *skeleton structure* of the diagrammatic theory can be constructed only with the particle-particle ladder (\[p-p ladder\]) plus an infinite number of interaction vertices. A careful diagrammatic analysis considered in detail in Ref.  then shows that: (a) Bare composite bosons correspond to fermionic particle-particle ladders; (b) The interaction among bare composite bosons is described by 4-point, 6-point vertices, and so on, which correspond to the product of $4,6,\ldots$, fermionic bare Green’s functions (with one internal four-momenta integration). The correspondence rules for the bosonic Green’s function and the 4-point vertex are shown in Fig. 2.

In particular, in the strong-coupling limit (whereby $\beta |\mu| \gg 1$),[@footnote-mu] the particle-particle ladder $\Gamma_0(q)$ has the following *polar structure*:[@Haussmann] $$\Gamma_0(q)\approx-\frac{4 \pi}{m^{2} a_{F}}\frac{ 1 + 
\sqrt{1 + \left(-i\Omega_{\nu} +
\frac{{\bf q}^{2}}{4 m} - \mu_{B}\right)\epsilon_{0}^{-1}}} 
{i\Omega_{\nu} - \left(\frac{{\bf q}^{2}}{4m} - \mu_{B}\right)}
\label{pp-sc}$$ where we have used the definition $\mu_{B} = 2\mu + \epsilon_{0}$ for the bosonic chemical potential ($\epsilon_0=1/(m a_F^2)$ being the bound-state energy of the fermionic two-body problem). Note that (apart from the residue being different from unity) the expression (\[pp-sc\]) resembles a “free” boson propagator with mass $2 m$. The (four-point) *effective two-boson interaction* reads instead $$\begin{aligned}
& &\tilde{u}_{2}(q_{1} \dots q_{4}) \, = \, \delta_{q_1+q_2,q_3+q_4} 
\left(\frac{{\cal V}}{\beta}\right)^{2}  \frac{2}{\beta {\cal V}}
\label{two-body-potential}\\
& &\times \sum_k 
{\cal G}^0(-k){\cal G}^0(k+q_2){\cal G}^0(-k+q_1-q_4){\cal G}^0(k+q_4)
\nonumber
\;\; .\end{aligned}$$ Here ${\cal G}^{0}(k) = [i\omega_n -{\bf k}^2/(2m)-\mu]^{-1}$ is a bare fermionic Green’s function ($\omega_n$ being a fermionic Matsubara frequency). Note that the interaction (\[two-body-potential\]) depends on wave vectors as well as on Matsubara frequencies, revealing in this way the composite nature of the bosons. The factor of $2$ in the definition (\[two-body-potential\]) corresponds to the two different sequences of spin labels that can be attached to the four fermionic Green’s functions, as shown graphically in Fig. \[fig:cobos\](b).

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In the strong-coupling limit $\beta |\mu| \gg 1$, a typical value of the two-boson effective interaction is obtained by setting $q_1 = \cdots = q_4 = 0$ in Eq. (\[two-body-potential\]). One gets: $$\tilde{u}_{2}(0) \, = \, 2 \, \left( \frac{{\cal V}}{\beta} \right)^{2}
    \left( \frac{m^{2} a_{F}}{8 \pi} \right)^{2} \, u_{2}(0)  \label{u2-tilde-limit}$$ where [@PS-96] $$u_{2}(0) \, = \, \frac{4 \pi (2a_{F})}{2m} \, .                \label{u2-limit}$$ The factor $m^{2} a_{F}/(8 \pi)$ in Eq. (\[u2-tilde-limit\]) reflects the difference between the true bosonic propagator and the particle-particle ladder in the strong-coupling limit \[cf. also Eq. (\[pp-sc\])\]. Owing to this difference, $u_{2}(0)$ given by Eq. (\[u2-limit\]) (and not $\tilde{u}_{2}(0)$ given by Eq. (\[u2-tilde-limit\])) has to be identified with the boson-boson interaction at zero four-momenta. Note that $u_{2}(0)$ is *positive* in the strong-coupling limit, thus ensuring the *stability* of the bosonic system.

Recall further that the scattering length $a^{\mathrm{Born}}_{B}$ within the Born approximation, obtained for a pair of true bosons (each of mass $2m$) mutually interacting via a two-body potential with Fourier transform $u_{2}(0)$ at zero wave vector, is given by $a^{\mathrm{Born}}_{B} = 2m u_{2}(0)/(4 \pi)$. Equation (\[u2-limit\]) then yields the following relation between the bosonic and fermionic scattering lengths within the Born approximation: $$a^{\mathrm{Born}}_{B} \, = \, 2 \, a_{F} \, .                             
\label{a-Born}$$ The result (\[a-Born\]) was also obtained in Ref.  within the fermionic self-consistent T-matrix approximation (which corresponds to the bosonic Hartree-Fock approximation in the strong-coupling limit). In that reference, the result (\[a-Born\]) was erroneously regarded to be the full value of the scattering length $a_{B}$ for a “dilute” system of composite bosons. We will, in fact, show in the next section that the result (\[a-Born\]) differs from $a_{B}$ when *all* bosonic T-matrix diagrams are properly taken into account.

Besides the four-point vertex (\[two-body-potential\]), the composite nature of the bosons produces (an infinite set of) additional vertices. One can show that all interaction vertices can be neglected in comparison with the four-point vertex in the strong-coupling limit.[@PS00] In this limit, one can thus construct all diagrams representing the two-particle Green’s function in the particle-particle channel in terms only of the bare ladder and of the four-point interaction vertex. This is precisely what one would expect on physical grounds, since the interactions involving more than two bodies become progressively less effective as the composite bosons overlap less when approaching the strong-coupling limit.

Numerical results for the scattering length of composite bosons
===============================================================

In three dimensions the *scattering length* $a$ characterizes the low-energy collisions for the scattering from an ordinary potential. For the mutual scattering of two particles (each of mass $M$), $a$ can be expressed by the relation $t(0) = 4\pi a/M$ in terms of the ordinary T-matrix $t(0)$ in the limit of vanishing wave vector. In particular, within the Born approximation $t(0)$ is replaced by the Fourier trasform $u(0)$ of the interparticle potential for vanishing wave vector.

For composite bosons, the T-matrix $\bar{t}_{B}(q_{1},q_{2},q_{3},q_{4})$ is defined by the following integral equation (cf. Fig. \[fig:comboslo\]): $$\begin{aligned}
& &\bar{t}_{B}(q_{1},q_{2},q_{3},q_{4}) =\bar{u}_{2}(q_{1},q_{2},q_{3},q_{4}) 
\nonumber\\ 
& &-\frac{1}{\beta {\cal V}} \, \sum_{q_{5}} \,
 \bar{u}_{2}(q_{1},q_{2},q_{5},q_{1}+q_{2}-q_{5}) 
\Gamma_{0}(q_{5}) \, \Gamma_{0}(q_{1}+q_{2}-q_{5})\nonumber\\
& &\times \bar{t}_{B}(q_{1}+q_{2}-q_{5},q_{5},q_{3},q_{4}) 
\label{bosonic-t-matrix}\end{aligned}$$ where $\bar{u}_{2}$ is proportional to the effective two-boson interaction of Eq. (\[two-body-potential\]): $$\begin{aligned}
& &\bar{u}_{2}(q_{1} \dots q_{4}) \, = \,  \frac{1}{\beta {\cal V}} \, 
\sum_{k} \nonumber\\
& &\times
{\cal G}^0(-k){\cal G}^0(k+q_2){\cal G}^0(-k+q_1-q_4){\cal G}^0(k+q_4) \; .
\label{two-body-potential-bar}     \end{aligned}$$

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Similarly to the problem of point-like bosons, we *define* the scattering length $a_{B}$ for the composite bosons (each of mass $2m$) in the strong-coupling limit and for vanishing density, by setting $t_{B}(0) = 4 \pi  a_{B}/(2m)$. Here $t_{B}(0) = (8 \pi/(m^{2} a_{F}))^{2} \, \bar{t}_{B}(0)$ and $\bar{t}_{B}(0) \equiv \bar{t}_{B}(0,0,0,0)$. The rescaling between $t_{B}(0)$ and $\bar{t}_{B}(0)$ is due to the difference between the true bosonic propagator and the particle-particle ladder in the strong-coupling limit. The same rescaling is consistently used when defining the boson-boson interaction \[cf. Eqs. (\[u2-tilde-limit\]) and (\[two-body-potential\])\].

To the lowest order in the effective interaction between the composite bosons, we can replace $t_{B}(0)$ by $u_{2}(0)$ and write $u_{2}(0)= 4 
\pi a^{\mathrm{Born}}_{B}/(2m)$, within the Born approximation. Comparison with Eq. (\[u2-limit\]) yields then the value $a^{\mathrm{Born}}_{B} = 2 \, a_{F}$, as anticipated by Eq. (\[a-Born\]).

In order to obtain the exact value of $\bar{t}_{B}(0)$ (and hence of the scattering length $a_B$), it is convenient to determine first the auxiliary quantity $\bar{t}_{B}(q,-q,0,0)$ by solving the following *closed-form* equation $$\begin{aligned}
& &\bar{t}_{B}(q,-q,0,0) = \bar{u}_{2}(q,-q,0,0) \label{bosonic-t-matrix-0}\\ 
& & - \frac{1}{\beta {\cal V}} \, \sum_{q'} \,
     \bar{u}_{2}(q,-q,q',-q')\Gamma_{0}(q') \, \Gamma_{0}(-q') \,
     \bar{t}_{B}(q',-q',0,0)   \,\, , \nonumber \end{aligned}$$ obtained from Eq. (\[bosonic-t-matrix\]) by setting $q_{1}=-q_{2}=q$ and $q_{3}=q_{4}=0$. This integral equation can be solved by standard numerical methods, e.g., by reverting it to the solution of a system of coupled linear equations. Note that, since we are here interested in the calculation of a two-boson quantity, the zero-density limit has to be taken in Eq. (\[bosonic-t-matrix-0\]). This implies that $\mu_B=0$ (or equivalently $2\mu=\epsilon_0$) and $T=0$.[@footnote-mu] The discrete bosonic frequency $\Omega_{\nu}$ becomes a continuous variable accordingly.

Numerical calculation of Eq. (\[bosonic-t-matrix-0\]) requires us to introduce a finite-size mesh for the variables ($|{\bf q}|$, $\Omega$) and ($|{\bf q}'|$, $\Omega'$), with the angular integral over $\hat{q'}$ affecting only the function $\bar{u}_{2}(q,-q,q',-q')$. Equation (\[bosonic-t-matrix-0\]) is thus reduced to a set of coupled equations for the unknowns $\bar{t}_{B}(|{\bf q}|,\Omega;|{\bf q}|,-\Omega;0;0)$, which were solved by the Newton-Ralphson algorithm with a linear interpolation for the integrals over $|{\bf q}'|$ and $\Omega'$. In this way, we are led to the result $$a_{B} = 0.75 \, a_F$$ within an estimated $5\%$ numerical accuracy.

To summarize, we have shown that, in the strong-coupling limit of the fermionic attraction, the value $a_{B} = 0.75 \, a_{F}$ is obtained by a correct summation of the bosonic T-matrix diagrams for the composite bosons which form as bound-fermion pairs. This theoretical prediction could be tested against the experimental data with ultracold Fermi atoms in a trap, when bosonic molecules are obtained by a Feshbach resonance.

C.A. Regal, C. Ticknor, J.L. Bohn, and D. Jin, Nature [**424**]{}, 47 (2003). P. Pieri and G.C. Strinati, Phys. Rev. B [**61**]{}, 15370 (2000); see also cond-mat/9811166. R. Haussmann, Z. Phys. B [**91**]{}, 291 (1993).

F. Pistolesi and G.C. Strinati, Phys. Rev. B [**53**]{}, 15168 (1996).

N. Andrenacci, P. Pieri, and G.C. Strinati, Eur. J. Phys. B [**13**]{}, 637 (2000).

The choice of the “contact” potential (\[Eq:deltafunc\]) does not allow the fermionic attraction to extend over a *finite* range. If a finite-range fermionic attraction would instead been adopted, the effective boson-boson potential in the strong-coupling limit would acquire an *attractive* part which would dominate over the usual repulsive part due to Pauli principle, in the sense that the bosonic scattering length associated with the attractive part would not vanish in that limit. The attractive part would thus lead to an instability of the bosonic system when the finite-range fermionic attraction gets sufficiently strong. To avoid this instability, a condition of the type $\epsilon_{0} \ll k_{0}^{2}/(2m)$ has to be imposed on the BE limit, where $k_{0}$ is the wave vector specifying the finite range of the potential and $\epsilon_0$ is the bound-state energy of the fermionic two-body problem \[cf. footnote 43 of Ref. \]. In this context, it is worth mentioning the work by G. Röpke *et al.* \[Phys. Rev. Lett. [**80**]{}, 3177 (1998)\] where it has been shown that, in the strong-coupling limit, there exists a competition between pair and quartet condensation in a Fermi liquid with finite-range attraction. It is then clear that adopting a finite-range fermionic potential merely makes the BCS-BE crossover more involved, and it is thus not relevant for the present purposes.

A precise definition of what is meant by strong coupling is as follows. In this limit, the binding energy $\epsilon_{0}$ is much larger than the density $\rho$ and temperature $T$. Under this condition, the fermionic chemical potential approaches the value $-\epsilon_{0}/2$ and $\beta |\mu| \gg 1$.
