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Implications of the evidence for direct $C P$ violation in $D \to π^{+} π^{-}$ decays

Rahul Sinha ^1,2,*, Thomas E. Browder ^1,†, N. G. Deshpande ^3,‡, Dibyakrupa Sahoo ^4,§, and Nita Sinha^2,∥

¹Department of Physics and Astronomy, University of Hawaii, Honolulu, Hawaii 96822, USA
²The Institute of Mathematical Sciences, Taramani, Chennai 600113, India
³Institute for Fundamental Science, University of Oregon, Eugene, Oregon 94703, USA
⁴Faculty of Physics, Institute of Theoretical Physics, University of Warsaw, ulica Pasteura 5, 02-093 Warsaw, Poland

^*Contact author: rsinha@hawaii.edu, sinha@imsc.res.in
^†Contact author: teb@phys.hawaii.edu
^‡Contact author: desh@uoregon.edu
^§Contact author: Dibyakrupa.Sahoo@fuw.edu.pl
^∥Contact author: nita@imsc.res.in

Phys. Rev. D 113, 093003 – Published 6 May, 2026

DOI: https://doi.org/10.1103/lppq-czdd

Abstract

The observation of $C P$ violation in the difference of $C P$ asymmetries between $D \to K^{+} K^{-}$ and $D \to π^{+} π^{-}$ has raised a debate whether the observed asymmetries can be regarded as a signal of physics beyond the standard model (SM). In this paper we obtain all the topological amplitudes and isospin amplitudes directly from measured observables for $D \to π π$ . These results unambiguously imply a very large penguin contribution, having a central value 4.74 times the magnitude of the amplitude for $D^{0} \to π^{+} π^{-}$ . This fitted central value differs from a reasonable SM estimate of 10% with a significance greater than $3.3 σ$ . In contrast to previous studies, we present model-independent arguments based only on unitarity of rescattering amplitudes to show that large penguins cannot arise from rescattering alone and likely indicate physics beyond the SM. In a model-independent approach we show how a very small contribution from physics beyond the SM with a large weak phase alleviates the problem.

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Phys. Rev. D 113, 093003– Published 6 May, 2026

Vol. 113, Iss. 9 — 1 May 2026

Received 25 June 2025
Accepted 27 March 2026

DOI: https://doi.org/10.1103/lppq-czdd

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.

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Implications of the evidence for direct $C P$ violation in $D \to π^{+} π^{-}$ decays

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Implications of the evidence for direct CP violation in D→π+π− decays

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Implications of the evidence for direct $C P$ violation in $D \to π^{+} π^{-}$ decays