Starting from the observation that artificial neural networks are uniquely suited to solving optimisation problems, and most physics problems can be cast as an optimisation task, we introduce a novel way of finding a numerical solution to wide classes of differential equations. We find our approach to be very flexible and stable without relying on trial solutions, and applicable to ordinary, partial and coupled differential equations. We apply our method to the calculation of tunnelling profiles for cosmological phase transitions, which is a problem of relevance for baryogenesis and stochastic gravitational wave spectra. Comparing our solutions with publicly available codes which use numerical methods optimised for the calculation of tunnelling profiles, we find our approach to provide at least as accurate results as these dedicated differential equation solvers, and for some parameter choices even more accurate and reliable solutions. In particular, we compare the neural network approach with two publicly available profile solvers, CosmoTransitions and BubbleProfiler, and give explicit examples where the neural network approach finds the correct solution while dedicated solvers do not. We point out that this approach of using artificial neural networks to solve equations is viable for any problem that can be cast into the form F( x) = 0, and is thus applicable to various other problems in perturbative and non-perturbative quantum field theory.

Determining for the first time the Darwin operator contribution for the non-leptonic charm-quark decays and using new non-perturbative results for the matrix elements of ∆C = 0 four-quark operators, including eye-contractions, we present a comprehensive study of the lifetimes of charmed mesons and inclusive semileptonic decay rates as well as the ratios, within the framework of the Heavy Quark Expansion (HQE). We find good agreement with experiment for the ratio τ (D + )/τ (D 0 ), for the total D + s -meson decay rate, for the semileptonic rates of all three mesons D 0 , D + and D + s , and for the semileptonic ratio Γ D + sl /Γ D 0 sl . The total decay rates of the D 0 and D + mesons are underestimated in our HQE approach and we suspect that this is due to missing higher-order QCD corrections to the free charm quark decay and the Pauli interference contribution. For the SU (3) F breaking ratios τ (D + s )/τ (D 0 ) and Γ D + s sl /Γ D 0 sl our predictions lie closer to one than experiment. This might originate from the poor knowledge of the non-perturbative parameters µ 2 G , µ 2 π and ρ 3 D in the D 0 and D + s systems. These parameters could be determined by experimental studies of the moments of inclusive semileptonic D meson decays.

We compute the Darwin operator contribution ($$ 1/{m}_b^3 $$ 1 / m b 3 correction) to the width of the inclusive non-leptonic decay of a B meson (B+, Bd or Bs), stemming from the quark flavour-changing transition b → $$ {q}_1{\overline{q}}_2{q}_3 $$ q 1 q ¯ 2 q 3 , where q1, q2 = u, c and q3 = d, s. The key ideas of the computation are the local expansion of the quark propagator in the external gluon field including terms with a covariant derivative of the gluon field strength tensor and the standard technique of the Heavy Quark Expansion (HQE). We confirm the previously known expressions of the $$ 1/{m}_b^3 $$ 1 / m b 3 contributions to the semi-leptonic decay b → $$ {q}_1\mathrm{\ell}{\overline{\nu}}_{\mathrm{\ell}} $$ q 1 ℓ ν ¯ ℓ , with ℓ = e, μ, τ and of the $$ 1/{m}_b^2 $$ 1 / m b 2 contributions to the non-leptonic modes. We find that this new term can give a sizeable correction of about −4 % to the non-leptonic decay width of a B meson. For Bd and Bs mesons this turns out to be the dominant correction to the free b-quark decay, while for the B+ meson the Darwin term gives the second most important correction — roughly 1/2 to 1/3 of the phase space enhanced Pauli interference contribution. Due to the tiny experimental uncertainties in lifetime measurements the incorporation of the Darwin term contribution is crucial for precision tests of the Standard Model.

Determining for the first time the Darwin operator contribution for the non-leptonic charm-quark decays and using new non-perturbative results for the matrix elements of ∆C = 0 four-quark operators, including eye-contractions, we present a comprehensive study of the lifetimes of charmed mesons and inclusive semileptonic decay rates as well as the ratios, within the framework of the Heavy Quark Expansion (HQE). We find good agreement with experiment for the ratio τ(D+)/τ(D0), for the total $$ {D}_s^{+} $$ D s + -meson decay rate, for the semileptonic rates of all three mesons D0, D+ and $$ {D}_s^{+} $$ D s + , and for the semileptonic ratio $$ {\Gamma}_{sl}^{D^{+}}/{\Gamma}_{sl}^{D^0} $$ Γ sl D + / Γ sl D 0 . The total decay rates of the D0 and D+ mesons are underestimated in our HQE approach and we suspect that this is due to missing higher-order QCD corrections to the free charm quark decay and the Pauli interference contribution. For the SU(3)F breaking ratios $$ \tau \left({D}_s^{+}\right)/\tau \left({D}^0\right) $$ τ D s + / τ D 0 and $$ {\Gamma}_{sl}^{D_s^{+}}/{\Gamma}_{sl}^{D^0} $$ Γ sl D s + / Γ sl D 0 our predictions lie closer to one than experiment. This might originate from the poor knowledge of the non-perturbative parameters $$ {\mu}_G^2 $$ μ G 2 , $$ {\mu}_{\uppi}^2 $$ μ π 2 and $$ {\rho}_D^3 $$ ρ D 3 in the D0 and $$ {D}_s^{+} $$ D s + systems. These parameters could be determined by experimental studies of the moments of inclusive semileptonic D meson decays.