High-temperature cuprate superconductors show remarkable deviations from Fermi liquid behaviour in their normal state, in particular in the underdoped region. One of the most striking features is the opening of a pseudogap above the critical temperature $T_c$ and below another temperature $T^*$ that increases when the doping is reduced.[1] This pseudogap appears to have the same angular dependence and magnitude as the superconducting gap below $T_c$, with which it seems to merge at the phase transition. Moreover, thermodynamic quantities and transport coefficients also deviate from Fermi liquid behaviour.[2] 

To explain these anomalies, different scenarios can be invoked. One of them is based on the formation in the pseudogap region of incoherent Cooper pairs, also called ``preformed'' pairs.[3] At a lower temperature, phase coherence is established among these pairs leading to the superconducting transition. In this respect, the superconducting transition can be regarded almost as a Bose condensation of preformed Cooper pairs,[4] while the BCS behaviour, in which the formation of the Cooper pairs and their condensation occur at the same temperature, is recovered as one approaches the overdoped region. The fact that the low temperature coherent length is much shorter in the underdoped regime is consistent with the picture of preformed pairs well localized in space.[5]

In this framework, it is interesting to investigate the structure of this preformed pairs. We propose to study the Green function for the operator which creates a pair of particles whose partners are at a certain relative distance, and study how this internal distance varies as a function of temperature, interaction strenght and density of particle. To this end, we consider a Bethe-Salpeter approach and introduce a $T-$matrix formalism, which takes into account the relevant interaction channel of repeated two-particle scattering. In the diluite case, which is the one of interest to us, the T-matrix approximation enable us to calculate the correlation function for a pair operator, from which is possible to separate the internal degrees of freedom (i.e., the relative distance between particles in a pair) from the center of mass ones. 

[1] T. Timusk, B. Statt, Rep. Prog. Phys. 62, 61 (1999). 

[2] M. Randeira, Proceedings of the International School of Physics ``Enrico Fermi'', CXXXVI Course, Varenna (1997), Cond-Mat/9710223. 

[3] J.V. Emery, S.A. Kivelson, Nature 374, 434 (1998). 

[4] P. Nozieres and S. Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985). 

[5] Y. J. Uemura et al., Phys. Rev. Lett. 66, 2665 (1991)