arxiv: 2604.25875 · v1 · submitted 2026-04-28 · ✦ hep-ph · hep-ex

Recognition: unknown

CP violation in singly Cabibbo suppressed Dto π a₀(980) decays

Yu-Kuo Hsiao , Shu-Ting Cai , Yan-Li Wang

Authors on Pith no claims yet

Pith reviewed 2026-05-07 15:41 UTC · model grok-4.3

classification ✦ hep-ph hep-ex

keywords CP violationD meson decaysa0(980) resonancerescattering effectssingly Cabibbo suppressedbranching fraction ratios

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The pith

Rescattering in singly Cabibbo-suppressed D to pi a0 decays produces direct CP asymmetries at the level of a few parts per thousand.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Measurements of D to pi a0(980) decays show branching-fraction ratios far larger than short-distance quark-level predictions, signaling that long-distance effects must dominate. The key mechanism is the rescattering chain D to K star K to a0 pi, which supplies an amplitude component comparable in size to the direct term and carrying large strong phases. These phases, when combined with the small weak-phase difference from the CKM factors, generate direct CP asymmetries of order 10 to the minus 3. The paper calculates explicit values for several charge modes and concludes that these decays therefore open a new experimental window on CP violation in the charm sector.

Core claim

The process D to K star K to a0 pi supplies an amplitude M_s whose magnitude is comparable to the direct amplitude M_d and whose strong phase is nontrivial. Inserted into the full decay amplitude M equals lambda_d M_d plus lambda_s M_s, this rescattering produces direct CP asymmetries A_CP of order 10 to the minus 3, with central values A_CP(D0 to pi minus a0 plus) approximately minus 0.7 times 10 to the minus 3, A_CP(D plus to pi0 a0 plus) approximately minus 1.4 times 10 to the minus 3, and A_CP(D0 to pi plus a0 minus) approximately minus 2.1 times 10 to the minus 3.

What carries the argument

The rescattering chain D to K star K to a0 pi, which generates an amplitude M_s comparable to M_d together with large strong phases in the total amplitude M = lambda_d M_d + lambda_s M_s.

If this is right

The observed branching-fraction ratios r^{+/-} and r^{+/0} are explained by the same rescattering that produces the CP asymmetries.
Direct CP violation at the few-per-mil level becomes accessible in existing and forthcoming charm-factory data samples.
Singly Cabibbo-suppressed D to pi a0 modes join the set of channels useful for testing the CKM picture in the charm sector.
The same rescattering mechanism can be applied to related final states involving other light scalars.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the predicted asymmetries are confirmed, similar rescattering effects should be searched for in other SCS D decays involving scalars or vectors.
Updated global fits of charm CP violation could incorporate these modes as independent constraints once experimental precision improves.
Lattice or dispersion-relation calculations of the strong phases in the K star K to a0 pi transition would provide an independent cross-check.

Load-bearing premise

The rescattering process D to K star K to a0 pi must produce an amplitude component whose size is comparable to the direct term and whose strong phase difference is large enough to yield observable CP violation.

What would settle it

A precision measurement showing all direct CP asymmetries in these modes to be consistent with zero at the 10 to the minus 4 level or below would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.25875 by Shu-Ting Cai, Yan-Li Wang, Yu-Kuo Hsiao.

**Figure 1.** Figure 1: FIG. 1. Rescattering processes: (a) view at source ↗

**Figure 2.** Figure 2: FIG. 2 view at source ↗

read the original abstract

The singly Cabibbo suppressed (SCS) decays $D\to \pi a_0$, with $a_0\equiv a_0(980)$, have been measured with the branching-fraction ratios $r^{+/-}_{\rm ex}\equiv {\cal B}(D^0\to\pi^- a_0^+)/{\cal B}(D^0\to\pi^+ a_0^-)=7.5^{+2.5}_{-0.8}\pm 1.7$ and $r^{+/0}_{\rm ex}\equiv {\cal B}(D^+\to\pi^0 a_0^+)/{\cal B}(D^+\to\pi^+ a_0^0)=2.6\pm 0.6\pm 0.3$, deviating significantly from the short-distance expectations $(r^{+/-},r^{+/0})\simeq (0.07,0.2)$. This discrepancy indicates the necessity of long-distance rescattering effects. In particular, the process $D\to K^*K\to a_0\pi$ generates ${\cal M}_s$ comparable in magnitude to ${\cal M}_d$ in the amplitude ${\cal M}=\lambda_d{\cal M}_d+\lambda_s{\cal M}_s$, with $\lambda_q\equiv V_{cq}^*V_{uq}$, accompanied by nontrivial strong phases essential for $CP$ violation. Consequently, the direct $CP$ asymmetries naturally arise at the ${\cal O}(10^{-3})$ level, for example, ${\cal A}_{CP}(D^0\to\pi^- a_0^+)=(-0.7\pm 0.1\pm 0.1\pm 0.1)\times 10^{-3}$, ${\cal A}_{CP}(D^+\to\pi^0 a_0^+)=(-1.4\pm 0.1\pm 0.1\pm 0.1)\times 10^{-3}$, and ${\cal A}_{CP}(D^0\to\pi^+ a_0^-)=(-2.1\pm 0.9\pm 1.1\pm 0.4)\times 10^{-3}$. These results establish SCS $D\to \pi a_0$ decays as a new avenue for probing $CP$ violation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper fits rescattering amplitudes to branching ratio discrepancies in D to pi a0 decays and outputs O(10^{-3}) CP asymmetries as a result, with the numbers tied closely to those same inputs.

read the letter

The main thing to know is that this work takes the large measured deviations in the branching ratio ratios r^{+/-}_ex and r^{+/0}_ex for D to pi a0(980) and uses them to argue that rescattering through D to K* K to a0 pi makes the s-quark and d-quark amplitudes comparable in size with a sizable strong phase difference. From there it derives direct CP asymmetries at the few times 10^{-3} level for the three modes listed in the abstract. That is the concrete new output: specific numerical A_CP values with propagated errors rather than a general statement that effects could be present. The paper does a reasonable job of showing why short-distance expectations fail badly for these ratios and why the long-distance piece supplies the phase needed for CP violation, which is a standard move in charm phenomenology. It correctly identifies these channels as a possible additional place to hunt for small CP violation beyond the usual modes. The soft spot is exactly the one flagged in the stress-test: the |M_s| ~ |M_d| relation and the phase are not fixed by independent data or lattice input but are set by reproducing the same branching fractions that motivated the rescattering story in the first place. This makes the quoted A_CP numbers more like model-consistent outputs than standalone predictions. The multiple error bars look like they come from the experimental inputs on the rates, but the underlying amplitude model (K* lineshape, a0 coupling, etc.) adds its own uncertainty that is harder to quantify. If the full text spells out the explicit parameterization and shows some variation under reasonable changes, that would help, but the circularity remains the central limitation. This is aimed at people working on charm decays, CP violation searches, and hadronic effects in weak processes. Experimental groups at LHCb or Belle II might find the suggested channels worth checking, and theorists who model rescattering could use it as a test case. It is coherent on its own terms and engages the existing literature on long-distance contributions, so it deserves a serious referee rather than a desk reject even though the model dependence will need discussion in review.

Referee Report

2 major / 1 minor

Summary. The manuscript examines CP violation in singly Cabibbo-suppressed D→π a0(980) decays. It notes that the measured branching-fraction ratios r^{+/-}_ex≈7.5 and r^{+/0}_ex≈2.6 deviate strongly from short-distance expectations (≈0.07, 0.2), indicating dominant long-distance rescattering via D→K^*K→a0π. This generates an amplitude component M_s comparable in magnitude to M_d in M=λ_d M_d + λ_s M_s, together with O(1) strong phases. The paper concludes that direct CP asymmetries therefore arise naturally at O(10^{-3}), and quotes explicit values with uncertainties: A_CP(D^0→π^- a0^+)=(-0.7±0.1±0.1±0.1)×10^{-3}, A_CP(D^+→π^0 a0^+)=(-1.4±0.1±0.1±0.1)×10^{-3}, and A_CP(D^0→π^+ a0^-)=(-2.1±0.9±1.1±0.4)×10^{-3}.

Significance. If the underlying rescattering model holds, the work identifies a new, experimentally accessible set of charm decays in which direct CP violation can be probed at the few-per-mille level. It illustrates how final-state interactions can convert branching-ratio discrepancies into concrete CP-asymmetry predictions, a technique that has proven useful in other non-leptonic charm channels. The provision of numerical targets with quoted uncertainties supplies clear benchmarks for future measurements at LHCb, Belle II, or BESIII.

major comments (2)

[Abstract] Abstract (and the amplitude construction that produces the quoted A_CP numbers): the central claim that |M_s| is comparable to |M_d| with a nontrivial relative strong phase is inferred directly from the same measured ratios r^{+/-}_ex and r^{+/0}_ex that the model is then used to reproduce. No independent constraint (lattice input, other D decays, or external lineshape data) is cited to fix |M_s/M_d| or δ_s-d outside the branching-fraction fit. Consequently the quoted A_CP values are outputs of that fit rather than robust, falsifiable predictions; the error budget (±0.1, ±0.1, ±0.1 etc.) therefore reflects only the experimental uncertainties on the input ratios, not the model uncertainty in the rescattering mechanism itself.
[Amplitude model] The mapping from the observed ratios to concrete values of |M_s/M_d| and the strong phase requires a specific model of the K^*K intermediate state, its coupling to a0π, and the a0(980) lineshape. Without an explicit derivation or sensitivity study of these modeling choices (e.g., variation of the K^* width or a0 lineshape parameters), it is impossible to judge whether the O(10^{-3}) scale of A_CP is stable or an artifact of the chosen parametrization.

minor comments (1)

[Abstract] The abstract states numerical results but does not indicate where in the manuscript the explicit amplitude expressions, the fitting procedure, or the error propagation are given; a brief pointer to the relevant section or equation would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important aspects of our amplitude model and the nature of our predictions. We address each major comment below and have revised the manuscript to include additional clarifications, explicit derivations, and a sensitivity study as requested.

read point-by-point responses

Referee: Abstract (and the amplitude construction that produces the quoted A_CP numbers): the central claim that |M_s| is comparable to |M_d| with a nontrivial relative strong phase is inferred directly from the same measured ratios r^{+/-}_ex and r^{+/0}_ex that the model is then used to reproduce. No independent constraint (lattice input, other D decays, or external lineshape data) is cited to fix |M_s/M_d| or δ_s-d outside the branching-fraction fit. Consequently the quoted A_CP values are outputs of that fit rather than robust, falsifiable predictions; the error budget (±0.1, ±0.1, ±0.1 etc.) therefore reflects only the experimental uncertainties on the input ratios, not the model uncertainty in the rescattering mechanism itself.

Authors: We agree that the parameters |M_s/M_d| and the strong phase δ are determined by fitting the model to the experimental branching-fraction ratios. This is inherent to our approach, as the large deviations from short-distance expectations necessitate the inclusion of the rescattering amplitude M_s to explain the data. The CP asymmetries are then derived as predictions within this framework. While there is no external lattice or other decay data used to fix these parameters independently, the model is falsifiable through direct measurements of the CP asymmetries themselves. To incorporate model uncertainties, we have revised the manuscript to include an estimate of theoretical errors arising from the rescattering mechanism. Specifically, we now quote an additional uncertainty component based on variations in the model parameters. The central values remain at O(10^{-3}), and the scale is stable provided the rescattering contribution is significant, as required by the branching ratios. revision: partial
Referee: The mapping from the observed ratios to concrete values of |M_s/M_d| and the strong phase requires a specific model of the K^*K intermediate state, its coupling to a0π, and the a0(980) lineshape. Without an explicit derivation or sensitivity study of these modeling choices (e.g., variation of the K^* width or a0 lineshape parameters), it is impossible to judge whether the O(10^{-3}) scale of A_CP is stable or an artifact of the chosen parametrization.

Authors: We acknowledge the need for greater transparency in the modeling choices. The original manuscript employs a standard parametrization for the rescattering via the K^*K channel, with couplings and lineshapes taken from established literature on a0(980) and related decays. In response to this comment, we have added an appendix to the revised version providing the explicit expressions for the amplitudes M_d and M_s, including the treatment of the intermediate states and the a0 lineshape. Furthermore, we have performed a sensitivity analysis by varying the K^* width within its experimental uncertainty (±10%) and the a0(980) lineshape parameters (e.g., pole position and width) within their allowed ranges. The resulting variations in the CP asymmetries are found to be smaller than or comparable to the quoted uncertainties, confirming that the O(10^{-3}) magnitude is not an artifact but a robust feature of the model when |M_s| is comparable to |M_d| with O(1) phase difference. revision: yes

Circularity Check

1 steps flagged

CP asymmetry values obtained by fitting rescattering parameters to the same branching-ratio data used to infer |Ms|~|Md| and strong phases

specific steps

fitted input called prediction [Abstract]
"This discrepancy indicates the necessity of long-distance rescattering effects. In particular, the process D→K^*K→a_0π generates M_s comparable in magnitude to M_d in the amplitude M=λ_d M_d + λ_s M_s ... accompanied by nontrivial strong phases essential for CP violation. Consequently, the direct CP asymmetries naturally arise at the O(10^{-3}) level, for example, A_CP(D^0→π^- a_0^+)=(-0.7±0.1±0.1±0.1)×10^{-3}, A_CP(D^+→π^0 a_0^+)=(-1.4±0.1±0.1±0.1)×10^{-3}, and A_CP(D^0→π^+ a_0^-)=(-2.1±0.9±1.1±0.4)×10^{-3}."

The quoted A_CP values with uncertainties are presented as a consequence of the rescattering that was introduced to explain the measured branching ratios r^{+/-}_ex≈7.5 and r^{+/0}_ex≈2.6. The specific |Ms/Md| ratio and relative strong phase needed to produce those A_CP numbers are determined by fitting the same branching-fraction data; the CP asymmetries are therefore recomputed outputs of the fit rather than independent predictions.

full rationale

The paper observes that measured r^{+/-}_ex and r^{+/0}_ex deviate from short-distance expectations, attributes this to D→K^*K→a0π rescattering making |Ms| comparable to |Md| with O(1) relative phase, and then quotes specific A_CP numbers at O(10^{-3}). These numbers are outputs of the model that was adjusted to reproduce the branching ratios; no independent external constraint (lattice, other decays, or parameter-free calculation) fixes |Ms/Md| or δ_s-d. The central claim therefore reduces to a fit-to-data followed by recomputation of a related observable, matching the fitted-input-called-prediction pattern.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that rescattering amplitudes are comparable in magnitude and carry independent strong phases; these are introduced to explain branching ratios and then used for CP predictions. No new particles or forces are postulated.

free parameters (2)

magnitude ratio |M_s / M_d|
Assumed comparable to account for the large observed branching fraction ratios.
strong phase difference between M_d and M_s
Nontrivial phases are required for CP violation and are implicitly fitted or chosen to produce the quoted asymmetry values.

axioms (2)

standard math Standard CKM factors λ_d = V_cd^* V_ud and λ_s = V_cs^* V_us govern the weak amplitudes.
Invoked in the amplitude decomposition M = λ_d M_d + λ_s M_s.
domain assumption Long-distance rescattering through K^*K can be treated as an effective contribution with its own magnitude and phase.
Used to explain the branching ratio discrepancy and generate CP violation.

pith-pipeline@v0.9.0 · 5744 in / 1629 out tokens · 36963 ms · 2026-05-07T15:41:14.448851+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

80 extracted references · 5 canonical work pages · 3 internal anchors

[1]

= 2 . 6 ± 0. 6 ± 0. 3, deviating signiﬁcantly from the short- distance expectations ( r+/ − , r +/ 0) ≃ (0. 07, 0. 2). This discrepancy indicates the necessity of long- distance rescattering eﬀects. In particular, the process D → K ∗K → a0π generates Ms comparable in magnitude to Md in the amplitude M = λ dMd + λ sMs, with λ q ≡ V ∗ cqVuq, accompanied by ...
[2]

$CP$ violation in singly Cabibbo suppressed $D\to \pi a_0(980)$ decays

1 ± 0. 4) × 10− 3. These results establish SCS D → πa 0 decays as a new avenue for probing CP violation. ∗ yukuohsiao@gmail.com † 18734581917@163.com ‡ ylwang0726@163.com 1 arXiv:2604.25875v1 [hep-ph] 28 Apr 2026 I. INTRODUCTION To account for the matter–antimatter extreme asymmetry of th e universe, CP violation has been proposed as one of the essential ...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[3]

29) × 10− 3 [2]. Whether the measured ∆ ACP can be fully accommodated within the Standard Model (SM) [3–16] or instead points to possible new physic s contributions [17–24] remains an open question [25]. Addressing this issue has induced exte nsive searches for ACP . However, no signal with a signiﬁcance comparable to ∆ ACP has yet been observed in the D ...
[4]

Includ ing LD rescattering eﬀects signiﬁcantly alleviates these discrepancies [40–43]

[38], and D+ s → ρ0a+ 0 [39] exceed theoretical estimates based solely on SD contributions by one to two orders of magnitude. Includ ing LD rescattering eﬀects signiﬁcantly alleviates these discrepancies [40–43]. Motivated by these successful resolutions, we incorporate LD re scattering eﬀects in the SCS D → πa 0 decays. In particular, rescattering proces...
[5]

= ia1(m2 D − m2 a0)fπ +F D+→ a0 0 . (2) Here, the eﬀective parameters a1, 2 arise from the factorization framework, the decay con- stants (fa0, f π ) correspond to the vacuum-produced mesons, and the form fac tors F D→ (a0,π ) parameterize the D-meson transition matrix elements. The subscripts S and P denote the cases where the scalar ( a0) or pseudoscala...
[6]

Using a2/a 1 ∼ − 0

λ d (T, C ), where GF is the Fermi constant and λ q ≡ V ∗ cqVuq denotes the relevant Cabibbo–Kobayashi– Maskawa (CKM) matrix element. Using a2/a 1 ∼ − 0. 5 [47], fa0/f π ∼ 0. 01 [36, 48], and F D0→ π − /F D0→ a− 0 ∼ 2. 0 [49, 50], we obtain r+/ − SD = 0 . 065 and r+/ 0 SD = 0 . 16, as quoted in the introduction. Even after including additional SD contribu...
[7]

However, owing to the tiny scalar decay constant fa0 ∼ 0

also contribute. However, owing to the tiny scalar decay constant fa0 ∼ 0. 01 fπ , these contributions are negligible compared to their counterparts CP(D+ → π 0a+ 0 ) and TP(D+ → π +a0 0), respectively. Second, the W - annihilation ( Wan) and W -exchange ( Wex) processes induce the transitions D+ → u ¯d → π +(0)a0(+) 0 and D0 → d ¯d → π ∓ a± 0 , respectiv...
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The form factor FP3(q2

, (5) where m1, m2, and m3 are the masses of the intermediate particles V1, P2, and the exchanged particle P3, respectively. The form factor FP3(q2
[9]

The total amplitu de of D → πa 0, by combining the SD and LD contributions in Eqs

≡ (Λ 2 P3 − m2 3)/ (Λ 2 P3 − q2 3), with the cutoﬀ parameter Λ P3, is introduced to regularize the loop integral [54]. The total amplitu de of D → πa 0, by combining the SD and LD contributions in Eqs. (2) and (4), is given as MT (D → πa 0) = MSD + M(ρη+ρη ′) LD eiδ1 + M (K ∗ 1 K1+K ∗ 2 K2) LD eiδ2 , (6) where ( δ1, δ 2) denote the relative phases among t...
[10]

For the D → a0 form factors, we take F D0(+)→ a− (+) 0 = √ 2F D+→ a0 0 = 0

05) [47] and fπ + = √ 2fπ 0 = 130 MeV [48] as theoretical inputs. For the D → a0 form factors, we take F D0(+)→ a− (+) 0 = √ 2F D+→ a0 0 = 0 . 44 ± 0. 03, as determined from semileptonic D decays [49]. In this framework, treating the a0 meson as an exotic tetraquark ( q2 ¯q2) state leads to B(D0 → a− 0 e+νe, a − 0 → π − η) = (0 . 8 ± 0. 1 ± 0. 1) × 10− 4,...
[11]

= ( − 163. 3 ± 4. 3, − 50. 5 ± 4. 2)◦ , (δ+ 1 , δ + 2 ) = ( − 98. 2 ± 4. 3, 45. 3 ± 3. 6)◦ , (7) where the phases δ0 1, 2 and δ+ 1, 2 correspond to the D0 and D+ decay modes, respectively, and provide essential strong-phase inputs for the direct CP asymmetry. Using the deﬁnition ACP (D → πa 0) = B(D → πa 0) − B ( ¯D → ¯π ¯a0) B(D → πa 0) + B( ¯D → ¯π ¯a0)...
[12]

As a result, BT (D0 → π +a− 0 ) is reduced to a value consistent with the experimental data

lead to destructive interference. As a result, BT (D0 → π +a− 0 ) is reduced to a value consistent with the experimental data. Nonetheless, these sizable contributions propagate relatively large uncertainties, especially given the reduced central value. The inclusion of SD and LD contributions, together with their interfe rence, brings BT (D+ → π 0a+ 0 ) ...
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into agreement with the data. In particular, since BSD(D+ → π +a0 0), B(D+ → ρ+η′ → π +a0 0), and B(D+ → (K ∗+ ¯K 0, ¯K ∗0K +) → π +a0 0) 8 are comparable to or exceed the experimental value, destructive interference is required, thereby reducing BT (D+ → π +a0 0). As a consequence, the propagated uncertainties become comparable to the central value. In c...
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With the amplitude written as M = λ d|Md|iδd + λ s|Ms|eiδs, the direct CP asymmetry in Eq

1σ. With the amplitude written as M = λ d|Md|iδd + λ s|Ms|eiδs, the direct CP asymmetry in Eq. (8) can be recast as ACP ≃ FWRM sin ∆ 1 + R2 M − 2RM cos ∆ , (9) where, for SCS D decays, we have used the approximation λ s ≃ − λ d. We deﬁne FW = − 2 Im(λ dλ ∗ s)/ |λ d|2, RM = |Ms|/ |Md|, and ∆ = δd − δs. The value FW ≃ − 1. 3 × 10− 3 reﬂects the small weak p...
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= (0 . 2 ± 0. 0 ± 0. 1 ± 0. 1) × 10− 3, respectively. Clearly, RM ≃ 1 indicates comparable magnitudes of |Ms| and |Md|, leading to ACP at the 10 − 3 level. In contrast, for RM ≃ 2. 7, the imbalance between |Ms|and |Md|suppresses ACP (D+ → π +a0 0) to the 10 − 4 level. Owing to the absence of severe destructive interference for the branching fractions, the...
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1, − 2. 1 ± 0. 9 ± 1. 1 ± 0. 4) × 10− 3 and ACP (D+ → π 0a+ 0 ) = ( − 1. 4 ± 0. 1 ± 0. 1 ± 0. 1) × 10− 3, which naturally reach the O(10− 3) level and are accessible to experiments at BESIII, Belle II, and LHCb. These results open a new avenue for probing CP violation in SCS D → πa 0 decays. 10 ACKNOWLEDGMENTS The authors would like to thank Dr. Yu Lu for...
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