D → Pℓ+ℓ− decays assisted by QCD light-cone sum rules

A bstract We suggest a new method to analyse the rare D ( s ) → Pℓ + ℓ − decays ( P = π , K ), combining QCD light-cone sum rules (LCSR) with hadronic dispersion relations. As our main study case, we consider the D + → π + ℓ + ℓ − mode which attracts much of interest from the point of view of GIM cancellation and potential new sources of the FCNC c → u transitions. The hadronic amplitude of this decay is dominated by the combination of weak annihilation with the emission of a virtual photon. This D + → π + γ ∗ amplitude is calculated at spacelike photon virtualities, − q 2 ≫ $$ {\Lambda}_{\textrm{QCD}}^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Λ</mml:mi> <mml:mi>QCD</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> , using LCSR with pion distribution amplitudes. The LCSR results are then fitted to the hadronic dispersion relation in the variable q 2 . The fit allows us to determine relative phases of the ρ -, ω - and ϕ -meson terms and to estimate the contributions of heavier hadronic states. The latter are parameterized in two different ways: first, with a z -expansion and second, using a model with excited vector-meson resonances. The resulting D + → π + ℓ + ℓ − width obtained from the LCSR-assisted dispersion relation is not much smaller than the current upper bound measured by LHCb collaboration. In comparison with our result for the dominant D + → π + γ ∗ transition, the contribution induced by the short distance c → uℓ + ℓ − quark transition generated by the effective O 9 operator in Standard Model turns out to be practically invisible. To detect its effect, one readily needs an enhancement of the coefficient C 9 from non-standard effects by many orders of magnitude. We also establish new U -spin symmetry relations between the amplitudes of D → Pℓ + ℓ − decays and apply our method to the Cabibbo-favoured modes $$ {D}_s^{+}\to {\pi}^{+}{\ell}^{+}{\ell}^{-} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>π</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mo>−</mml:mo> </mml:msup> </mml:math> and $$ {D}^0\to {\overline{K}}^0{\ell}^{+}{\ell}^{-} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>D</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mover> <mml:mi>K</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mn>0</mml:mn> </mml:msup> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mo>−</mml:mo> </mml:msup> </mml:math> which proceed via the same weak annihilation mechanism.