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We present the results of the application of the Dispersion Matrix approach to exclusive semileptonic B-meson decays. This method allows to determine the hadronic form factors in a non-perturbative and completely modelindependent way. Starting from lattice results available at large values of the momentum transfer, the behaviour of the form factors in their whole kinematical range is obtained without introducing any parameterization of their momentum dependence. We will focus on the determination of the Cabibbo-Kobayashi- Maskawa matrix elements |Vcb| and |Vub| through the analysis of B(s) → D(s)(*)lv and B(s) → π(K)ℓν decays. New theoretical determinations of the Lepton Flavour Universality ratios relevant for these transitions will be also presented.

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The DM method allows a non-perturbative, model-independent determination of the momentum dependence of the semileptonic form factors to be achieved. Starting from lattice results available at large values of the 4-momentum transfer and implementing non-perturbative unitarity bounds, the behaviour of the form factors in their whole kinematical range is obtained without introducing any explicit parameterization of their momentum dependence. We consider the four exclusive semileptonic B (s) → D ( * ) (s) ν decays and extract |V cb | from the experimental data for each transition. The average over the four channels iswhich is compatible with the latest inclusive determination at 1σ level. We address also the issue of Lepton Flavour Universality by computing pure theoretical estimates of the τ/ ratios of the branching fractions for each channel, where is a light lepton. In the case of a light spectator quark we obtain R(D * ) = 0.275(8) and R(D) = 0.296(8), which are compatible with the corresponding experimental values within 1.3σ. In the case of a strange spectator quark we obtain R(D * s ) = 0.2497(60) and R(D s ) = 0.298(5). The different values for R(D * s ) and R(D * ) may reflect SU(3) F symmetry breaking effects, which seem to be present in some of the lattice form factors, especially at large values of the recoil.

An important aspect in the study of quark gluon matter is represented by the computation of its transport coefficients. Indeed, they contain important information about thermodynamics and the phase diagram of QCD. Furthermore, they are used as input parameters in hydrodynamical simulations of heavy-ion collision experiments. The electric conductivity is a transport coefficient which parameterizes the charge transport phenomena. It is expected to play a central role in the dynamics of the Quark Gluon Plasma (QGP), since electric and magnetic fields are generated in heavy-ion collisions. The computation of this quantity can be done using the so-called Kubo formulas where spectral functions of electric current-current correlation functions and conductivity are directly related. The spectral functions can be extracted from the correlators on the lattice. The two quantities are related by an integral relation which has to be inverted to extract spectral functions. This inversion can be done by using smearing techniques to look for approximate solutions. We performed the study of the conductivity in two cases. On the one hand, we present the lattice QCD study of the electromagnetic conductivity dependence on baryon density. A first discussion can be found in [1]. On the other hand, we studied the electromagnetic conductivity in presence of strong magnetic fields, namely eB = 4, 9 GeV. This gives an evidence of the CME.

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We examine the semileptonic 𝐵 → 𝐷 ( * ) ℓ𝜈 ℓ and 𝐵 → 𝜋ℓ𝜈 ℓ decays adopting the unitarity-based Dispersive Matrix (DM) method, which allows to determine the shape of the relevant hadronic form factors (FFs) in their whole kinematical range, using only lattice QCD results available at large values of the 4-momentum transfer without making any assumption on their momentum dependence. Moreover, the experimental data are not used to constrain the shape of the FFs, but only to obtain our final exclusive determination of |𝑉 𝑐𝑏 | and |𝑉 𝑢𝑏 |, namely: |𝑉 𝑐𝑏 | • 10 3 = 41.1 ± 1.0 and |𝑉 𝑢𝑏 | • 10 3 = 3.88 ± 0.32, which are consistent with the latest inclusive determinations at the 1𝜎 level or better. Our calculation of the FFs allows to obtain pure theoretical estimates of the 𝜏/𝜇 ratios of differential decay rates, 𝑅(𝐷) = 0.296 ± 0.008 and 𝑅(𝐷 * ) = 0.275 ± 0.008, which turn out to be compatible with the experimental world averages within ≃ 1.4 standard deviations.

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