$S$-wave contribution to rare ${D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ decays in the standard model and sensitivity to new physics

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$S$ -wave contribution to rare $D^{0} \to π^{+} π^{-} ℓ^{+} ℓ^{-}$ decays in the standard model and sensitivity to new physics

Svjetlana Fajfer, Eleftheria Solomonidi, and Luiz Vale Silva

Phys. Rev. D 109, 036027 – Published 26 February 2024

Abstract

Physics of the up-type flavor offers unique possibilities of testing the standard model (SM) compared to the down-type flavor sector. Here, we discuss SM and new physics (NP) contributions to the rare charm-meson decay $D^{0} \to π^{+} π^{-} ℓ^{+} ℓ^{-}$ . In particular, we discuss the effect of including the lightest scalar isoscalar resonance in the SM picture, namely, the $f_{0} (500)$ , which manifests in a big portion of the allowed phase space. Other than showing in the total branching ratio at an observable level of about 20%, the $f_{0} (500)$ resonance manifests as interference terms with the vector resonances, such as at high invariant mass of the leptonic pair in distinct angular observables. Recent data from LHCb optimize the sensitivity to $P$ -wave contributions that we analyze in view of the inclusion of vector resonances. We propose the measurement of alternative observables that are sensitive to the $S$ -wave and are straightforward to implement experimentally. This leads to a new set of null observables that vanish in the SM due to its gauge and flavor structures. Finally, we study observables that depend on the SM interference with generic NP contributions from semileptonic four-fermion operators in the presence of the $S$ -wave.

Received 17 December 2023
Accepted 24 January 2024

DOI:https://doi.org/10.1103/PhysRevD.109.036027

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Branching fraction Cabibbo–Kobayashi–Maskawa matrix Effective field theory Electroweak interaction Flavor changing neutral currents Particle decays Phenomenology Rare decays

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Svjetlana Fajfer ¹, Eleftheria Solomonidi ², and Luiz Vale Silva ²

¹Jožef Stefan Institute, Jamova 39, P. O. Box 3000, 1001 Ljubljana, Slovenia and Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
²Departament de Física Teòrica, Instituto de Física Corpuscular, Universitat de València—Consejo Superior de Investigaciones Científicas, Parc Científic, Catedrático José Beltrán 2, E-46980 Paterna, Valencia, Spain

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Vol. 109, Iss. 3 — 1 February 2024

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Figure 1
Phase space allowed in the decay $D^{0} \to π^{+} π^{-} μ^{+} μ^{-}$ ; the invariant mass of the pion (muon) pair is denoted $p^{2}$ (respectively, $q^{2}$ ). Some scalar (blue), vectorial (red), and tensorial (green) resonant contributions are shown (the very narrow pseudoscalar resonances $η^{(}'^{)}$ , leading to the lepton pair via two-photon exchange, are omitted); the bands correspond to $(m \pm Γ / 2)^{2}$ , with $Γ$ taken from Refs. [27, 28, 38, 39]. The “high-energy window” referred to in the plot corresponds to $m_{ρ^{0}}^{2} ≲ q^{2} ≲ 1.5 {GeV}^{2}$ , for which only $f_{0} (500) \equiv σ$ gives an important contribution among the $S$ -wave contributions and is indicated by a hashed pattern delimited by dashed vertical lines. Cascade decays are not indicated.
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Figure 2
Quasi-two-body topologies; the lepton (pion) pair comes from electromagnetic (respectively, strong) decays of the intermediate resonances; from top to bottom: W-type factorization contribution, J-type factorization contribution, A-type factorization contribution (i.e., annihilation topology); pairs of empty squares represent the two quark color-neutral bilinears that are factorized. Bottom: contributions for which the lepton pair comes from an effective semileptonic contact interaction, represented by a solid square.
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Figure 3
The prediction for the differential decay rate $d Γ / d m$ and LHCb data over the dihadron invariant mass $m (π^{+} π^{-}) \equiv \sqrt{p^{2}}$ [2, 3, 4]. Top: the contributions from the $S$ -wave (dotted red curve) and the $P$ -wave (dashed magenta curve) add up to the full resonant contribution (solid blue curve). Bottom: components of the $S$ -wave contribution: $σ ρ^{0}$ (dashed red curve), $σ ω$ (dot-dashed magenta curve), and $σ ϕ$ (dotted orange curve, multiplied by 4 for an easier comparison).
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Figure 4
The differential decay rate $d Γ / d m$ (in blue) and LHCb data over the dilepton invariant mass $m (μ^{+} μ^{-}) \equiv \sqrt{q^{2}}$ [2, 3, 4]. Top: the dashed (dotted) red curve displaying nonoptimal phases corresponds to the optimal $Δ_{1}$ added with $π / 2$ ( $- 3 π / 4$ ). Middle: the dashed red curve displaying nonoptimal phases corresponds to the optimal $Δ_{4}$ added with $5 π / 4$ . Bottom: $P$ - and $S$ -wave components, in dashed magenta and dotted red, respectively; interference terms are set to zero.
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Figure 5
The differential decay rate, after integration of dilepton energies over the range $r^{(ρ : \sup)} \cup r^{(ϕ : \inf)} \cup r^{(ϕ : \sup)} = [0.7 8^{2}, {1.1}^{2}] {GeV}^{2}$ , as a function of $\cos (θ_{π})$ . In dashed magenta the observable is shown in the absence of the $S$ -wave contribution [rescaled such that $\int_{- 1}^{1} d \cos (θ_{π}) d Γ / d \cos (θ_{π}) / Γ = 1$ ]. The solid blue and dotted orange lines correspond to extreme cases reached for certain values of the phase difference $Δ_{S P}$ between the $S$ - and $P$ -waves that maximize their interference. As it is clear from the figure, the interference of the $S$ - and $P$ -waves can generate a distinguished asymmetry.
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Physical Review D

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$S$ -wave contribution to rare $D^{0} \to π^{+} π^{-} ℓ^{+} ℓ^{-}$ decays in the standard model and sensitivity to new physics

Svjetlana Fajfer, Eleftheria Solomonidi, and Luiz Vale Silva

Phys. Rev. D 109, 036027 – Published 26 February 2024

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Physical Review D

covering particles, fields, gravitation, and cosmology

S-wave contribution to rare D0→π+π−ℓ+ℓ− decays in the standard model and sensitivity to new physics

Svjetlana Fajfer, Eleftheria Solomonidi, and Luiz Vale Silva

Phys. Rev. D 109, 036027 – Published 26 February 2024

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$S$ -wave contribution to rare $D^{0} \to π^{+} π^{-} ℓ^{+} ℓ^{-}$ decays in the standard model and sensitivity to new physics