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##### Sensitivity of predictions in an eﬀective model: Application to the chiral critical end point position in the Nambu–Jona-Lasinio model

**Authors**: Alexandre Biguet, Hubert Hansen, Timothée Brugière, Pedro Costa, Pierre Borgnat

**Ref.**: European Physical Journal A **51-9**, 121-19 (2015)

**Abstract**: The measurement of the position of the chiral critical end point (CEP) in the QCD phase diagram is under debate. While it is possible to predict its position by using eﬀective models speciﬁcally built to reproduce some of the features of the underlying theory (QCD), the quality of the predictions (e.g., the CEP position) obtained by such effective models, depends on whether solving the model equations
constitute a well- or ill-posed inverse problem. Considering these predictions as being inverse problems provides tools to evaluate if the problem is ill-conditioned, meaning that inﬁnitesimal variations of the inputs of the model can cause comparatively large variations of the predictions. If it is ill-conditioned, it has major consequences because of ﬁnite variations that could come from experimental and/or theoretical errors. In the following, we shall apply such a reasoning on the predictions of a particular Nambu–Jona-Lasinio model within the mean ﬁeld + ring approximations, with special attention to the prediction of the chiral CEP position in the (T -μ) plane. We ﬁnd that the problem is ill-conditioned (i.e. very sensitive to input variations) for the T -coordinate of the CEP, whereas, it is well-posed for the μ-coordinate of the CEP. As a consequence, when the chiral condensate varies in a 10 MeV range, μ CEP varies far less. As an illustration to understand how problematic this could be, we show that the main consequence when taking into account ﬁnite variation of the inputs, is that the existence of the CEP itself cannot be predicted anymore: for a deviation as low as 0.6% with respect to vacuum phenomenology (well within the estimation
of the ﬁrst correction to the ring approximation) the CEP may or may not exist.