no trace of critical phenomena are observed in | r ( t ) | – the shape of this magnitude is smooth around λ = λ (1) c . Moreover, Fig. 6 confirms that r$_{max}$ ( λ ) also behaves in a smooth an size-independent way. The conclusion is, thus, that the first-order ESQPT does not affect the decoherence induced in the central qubit. The decoherence induced in a single qubit by its interaction with the environment, mod- elled as a scalar two-level boson model, is studied. The environment presents a quantum phase transition from symmetric to non-symetric phases at around α = 4 / 5, which can be first order ( ω = 0) or second order ( ω = 0). In the non-symmetric phase, the environment also presents excited state quantum phase transitions (ESQPTs): a second order one for any ω value at E (2) c = 0, and also a first order one for ω = 0 at an energy E (1) c < 0. We have shown that the second order ESQPT affects dramatically the decoherence factor which goes rapidly to zero. A finite size scaling study shows that in that case the decoherence factor goes to zero at the critical point following a power law. On the other hand, the first order ESQPT does not affect the decoherence of the central qubit. We have also shown that a mean field treatment provides a good description of the decoherence factor r ( t ), except in the regions around the critical points. Therefore, more sophisticated approximations are needed to obtain an analytical description of the critical behavior of r ( t ), and, particulary, to estimate the critical exponent λ . This work has been partially supported by the Spanish Ministerio de Educaci´ on y Ciencia and by the European regional development fund (FEDER) under projects num- ber FIS2008-04189, FIS2006-12783-C03-01 FPA2006-13807-C02-02 and FPA2007-63074, by CPAN-Ingenio, by Comunidad de Madrid under project 200650M012, CSIC and by Junta de Analuc´ıa under projects FQM160, FQM318, P05-FQM437 and P07-FQM-02962. A.R. is supported by the Spanish program ”Juan de la Cierva” and P. P-F. is supported by a FPU 22 Acknowledgements V. SUMMARY AND CONCLUSIONS