Rare $ \Lambda_c $ decays and new physics effects

Cong-Feng Qiao; Sheng-Qi Zhang

REVIEW 4 major objections 5 minor 89 references

Rare $ \Lambda_c $ decays and new physics effects

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper derives all ten form factors for the rare Lambda_c -> p l+l- decay from QCD sum rules, turning it into a new-physics test.

desk verdict A legitimate new QCD sum-rule calculation of the complete Lambda_c -> p form-factor set, but the resonance-dominated rates are calibrated to data, the no-resonance rate is two orders below lattice, and the Borel-window checks are missing for most form factors. read the letter →

arxiv 2411.15857 v2 pith:3DQUOVNV submitted 2024-11-24 hep-ph

Sheng-Qi Zhang , Cong-Feng Qiao This is my paper · ORCID

classification hep-ph

keywords rarecharmbaryondecaysLambda_c->pl+lQCDsumrulesbaryonicformfactorsflavor-changingneutralcurrentsangularobservablesnewphysicsXi_c

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the rare baryon decay $\Lambda_c \to p \ell^+\ell^-$ can be described from first principles well enough to act as a new-physics probe. It derives the complete set of ten form factors for the $\Lambda_c \to p$ transition in the large-recoil region using QCD sum rules, then extends them to the full kinematic range with a $z$-series parametrization. With these form factors it computes branching fractions for $\Lambda_c \to p e^+e^-$ and $\Lambda_c \to p \mu^+\mu^-$, and for related $\Xi_c$ modes through flavor-symmetry relations. The key phenomenological payoff is that the lepton forward-backward asymmetry is predicted to be exactly zero in the Standard Model, so any measured nonzero value would indicate new physics, while the fraction of longitudinally polarized dileptons would acquire visible resonance peaks only under new-physics effects.

What carries the argument

The load-bearing machinery is the three-point QCD sum rule. Two interpolating currents with the quark content of $\Lambda_c$ and the proton are contracted with a weak transition current, and the resulting correlation function is evaluated both phenomenologically, through hadronic states and decay constants $\lambda_{\Lambda_c},\lambda_p$, and in QCD, through an operator-product expansion truncated at dimension six that includes quark, gluon, mixed, and four-quark condensates. Matching the two representations by quark-hadron duality and a double Borel transform yields the form factors; the extraction is kept only in a Borel window where the ground-state pole contribution exceeds 40% and the highest-dimension condensate stays below 30%. The $z$-series parametrization then carries the ten form factors from the large-recoil region to the full physical $q^2$ range, and helicity amplitudes built from the form factors produce the differential width and the angular observables $A_{FB}$ and $F_L$. A useful internal identity is $f_{T2}(0)=g_{T2}(0)$, which emerges from the sum rules and matches the covariant quark-model result.

What would settle it

A direct check would be to plot, for each of the ten form factors, the pole-contribution ratio and the relative $\langle \bar q q\rangle^2$ contribution across the Borel window; any form factor violating the stated 40%/30% criteria would break the extraction. An independent lattice QCD computation of $g_2(0)$ would also settle the issue, since the paper's value $-0.25\pm0.02$ differs sharply from the existing lattice estimate near zero.

Watch

Extended reading notes

Core claim

The central claim is that a first-principles QCD-sum-rule calculation yields a complete hadronic description of $\Lambda_c \to p$: the ten form factors $f_1,f_2,f_3,f_{T1},f_{T2},g_1,g_2,g_3,g_{T1},g_{T2}$ at $q^2=0$, extrapolated across the physical region by a $z$-series fit. From these the paper predicts the differential and total branching fractions for $\Lambda_c \to p e^+e^-$ and $\Lambda_c \to p \mu^+\mu^-$ with and without long-distance $\rho,\omega,\phi$ resonance contributions, and shows that the no-resonance rate lies below the present experimental upper limit. It also finds $A_{FB}=0$ throughout the physical range in the Standard Model, $F_L=1/3$ at both kinematic endpoints, and uses U-spin relations to estimate $\Xi_c \to (\Sigma,\Lambda)\ell^+\ell^-$ branching fractions. These results are offered as concrete tests: a nonzero forward-backward asymmetry, or resonance structure in $F_L$ absent in the Standard Model, would signal new physics in $c \to u \ell^+\ell^-$ transitions.

Load-bearing premise

The calculation assumes that in the chosen Borel window for $\Lambda_c \to p$ the truncated operator-product expansion converges and ground-state baryons, not excited states, dominate the sum rule; the paper sets these criteria as a pole contribution above 40 percent and a dimension-six condensate below 30 percent, and it reports that the same criteria fail for $\Lambda_b \to n$. If either condition fails for any of the ten form factors, the extracted form factors and the observables built from them would not be reliable.

Editorial extensions

If this is right

The predicted no-resonance branching fractions for $\Lambda_c \to p e^+e^-$ and $\Lambda_c \to p \mu^+\mu^-$, $(3.2\pm2.3)\times10^{-13}$ and $(2.4\pm1.8)\times10^{-13}$, sit below the current experimental upper limit and can be tested with more data.
$A_{FB}$ is predicted to vanish across the full $q^2$ range in the Standard Model, so any nonzero measurement is a clean new-physics signal.
$F_L$ takes the parameter-free endpoint values $1/3$ at $q^2=4m_\ell^2$ and at $q^2=(M_{\Lambda_c}-M_p)^2$, independent of the form-factor uncertainties.
Under new physics, $F_L$ acquires resonance peaks at the $\rho,\omega,\phi$ poles that are absent in the Standard Model, providing a distinguishing signature.
U-spin relations extend the $\Lambda_c\to p$ form factors to six $\Xi_c$ rare modes, for example predicting ${\rm Br}(\Xi_c^+\to\Sigma^+\mu^+\mu^-)=(5.2\pm3.7)\times10^{-13}$ without resonance contributions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Our inference: because the same sum rules fail their convergence test for $\Lambda_b\to n$, where the $\langle\bar q q\rangle^2$ contribution reaches 50-90%, the method should not be expected to transfer directly to $b$-baryon decays; a light-cone or heavy-quark-effective-theory formulation would be needed there.
Our inference: a first measurement of $\Xi_c^+\to\Sigma^+\mu^+\mu^-$ could validate the U-spin-extended form factors more cheaply than $\Lambda_c$ decays, since the predicted rate sits in a range future datasets might reach.
Our inference: the roughly 60% uncertainty attributed to the charm-scale choice implies that improved Wilson coefficients, not just better form factors, are the limiting input for precision tests of $c\to u\ell^+\ell^-$ transitions.
Our inference: the endpoint identity $F_L=1/3$ is a parameter-free prediction that a future angular analysis can test immediately; a measured deviation would point to new physics or to missing Standard Model contributions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a real calculation, not a toy, but it is being sold as more predictive than it is. The genuinely new piece is the three-point QCDSR evaluation of all ten Lambda_c -> p FCNC form factors at q^2=0, the z-series extrapolation to the physical region, and the rough U-spin relations for Xi_c decays. That is a useful alternative to lattice QCD and the quark models, and the q^2=0 values for f1, g1, f2, g2 line up reasonably with existing LCSR and lattice numbers. The authors are also honest that the same method fails for Lambda_b -> n; the contrast is informative and the formalism itself looks standard.

The soft spots are serious but not all equal. First, the Borel-window and OPE-convergence criteria in Sec. 3.1 are stated but not shown. The text gives the 40% pole and 30% ^2 criteria, yet only f1 is displayed in Fig. 1, and there are no RPC curves or Borel windows for the other nine structures. Given that Table 1 shows g2(0) = -0.25 +/- 0.02 versus lattice 0.003 +/- 0.052, this is not a cosmetic omission; one wants to know whether g2 is a failed sum rule or a convention difference. The stress-test note is right about this.

Second, the resonance-included branching fractions in Table 3 are calibrated, not predicted. Eq. (26) fixes a_omega and a_phi from measured Br(Lambda_c -> p V) Br(V -> mu+mu-). The resonant part then dominates, so the agreement with other groups in Table 3 mostly means the fit reproduces the input. The paper should say that plainly.

Third, the non-resonant rate is 2.4e-13 for mu+mu-, roughly two orders below the lattice number 4.1e-11 and also below the relativistic quark model's 2.8e-12. The authors attribute the spread to Wilson coefficient choice, but a 100x gap is hard to explain by C9 variation alone, and the z-series slopes in Table 2 carry huge uncertainties (f1 a1 = -0.36 +/- 4.44), so the extrapolated form factors have more slack than the quoted central values suggest.

There are smaller issues: no numerical comparison with Ref. [42], which did a related QCDSR study of Lambda_c -> p FCNC, and the NP section is over-sold. The rates are resonance-dominated and short-distance physics is far below current sensitivity, so statements that new physics "may be testified" in these modes need more careful hedging.

Who gets value: people working on charm FCNC phenomenology who want an independent QCDSR form-factor set and a map of existing approaches. I would cite it as a cross-check, not as a primary input. It deserves a serious referee, but the report should ask for the missing Borel-window plots, an explicit discussion of the lattice discrepancy, and a cleaner separation between calibrated resonance rates and genuine predictions. With those changes it could become a reliable reference.

Referee Report

4 major / 5 minor

Summary. The paper derives, within the QCD sum-rule framework, a complete set of ten form factors for the transition Λc → p (f1,f2,f3,g1,g2,g3,fT1,fT2,gT1,gT2). The form factors are computed at q^2 = 0 and in a small spacelike interval q^2 ∈ [−0.4,0.4] GeV^2, then extrapolated to the full physical region using a z-series parametrization. These form factors are used to compute branching fractions for Λc → p e+e− and Λc → p μ+μ−, both with and without long-distance vector-resonance contributions, as well as angular observables (AFB and FL). Flavor-symmetry relations are used to estimate branching fractions for Ξc decays. The paper also studies the sensitivity of the observables to new physics in the Wilson coefficients C7 and C10.

Significance. If the form-factor set is correct, this would be a useful ingredient for charm-baryon rare-decay phenomenology, complementing lattice QCD and quark-model results. The paper has several positive features: the QCD sum-rule framework is standard, the comparison with existing LQCD and RQM predictions is informative, and the z-series formalism is appropriate in spirit. The no-resonance branching fractions are genuine predictions and correctly respect the LHCb upper limit on Λc → pμ+μ−. The angular observables proposed as new-physics diagnostics, especially the vanishing SM AFB, are well motivated. However, the central claim of a complete and reliable form-factor set is not fully supported by the evidence presented: the Borel-window/OPE-convergence checks are not shown for most form factors, one form factor disagrees sharply with lattice QCD without being discussed, and the z-series fits have very large slope uncertainties. The resonance-included branching fractions are also normalized to the same data they are meant to predict, which weakens their significance. These issues are reparable, but they need to be addressed before the paper can be accepted.

major comments (4)

[Sec. 3.1, Eqs. (17)–(18) and Fig. 1] The paper states two criteria for choosing the Borel window: the pole contribution must exceed 40% and the ⟨qq̄⟩^2 condensate contribution must remain below 30%. However, only the second criterion is displayed, and only for the single form factor f1. No RPC curves or Borel-window values are shown for the other nine form factors, and Fig. 1 shows only the ⟨qq̄⟩^2 fraction for f1. Since the fits in Table 2 use points in q^2 ∈ [−0.4,0.4] GeV^2, convergence must hold at each fitted point and for each Lorentz structure. Without these checks, the claim of a complete and reliable set of form factors is unsupported.
[Table 1] The value g2(0) = −0.25 ± 0.02 disagrees with lattice QCD, g2(0) = 0.003 ± 0.052 [46], by approximately 5σ, yet the paper's list of form factors with reasonable agreement (f3, fT1, gT1) does not include g2. This discrepancy is either a sign/convention error or a sign that the sum rule fails for the g2 structure. The authors need to investigate and either correct the calculation or explain the discrepancy, because g2 enters the helicity amplitudes in Eq. (29) and affects the branching fractions and FL.
[Sec. 3.1, Eq. (19) and Table 2] The z-series fits have extremely large slope uncertainties (for example, f1: a1 = −0.36 ± 4.44; gT1: a1 = −23.25 ± 16.28), despite the fits being performed over the narrow interval [−0.4,0.4] GeV^2. Using these fits to extrapolate to q^2 ≈ 1.8 GeV^2, the entire physical range, is uncontrolled. The paper should either justify that a single slope parameter suffices, add constraints from the endpoint behavior, or provide error bands that faithfully propagate the slope uncertainties into the branching fractions.
[Sec. 3.2, Eqs. (25)–(28) and Table 3] The resonance couplings aω and aφ are determined from Eq. (26), which equates the contribution from C9^R alone to the measured Br(Λc → pV)Br(V → μ+μ−). Consequently, the resonance-included branching fraction for Λc → pμ+μ− in Table 3 is essentially the experimental input reflected back and is not an independent prediction. The agreement with the LHCb limit in Fig. 3 is therefore not a test of the calculation. The paper should clearly separate genuine predictions (no-resonance rates and off-resonance bins) from input-normalized estimates, and the new-physics analysis in Sec. 3.3, which uses C9 = C9^R, should be framed accordingly.

minor comments (5)

[Abstract and Introduction] There are several typographical and grammatical issues, e.g., "exhibit an heavily dependence" in the Introduction and "the the four-vector" near Eq. (1). The abstract and text should be carefully proofread.
[Fig. 3 caption] The caption writes "the √q2 region" in a way that is hard to read; it should say "the region of sqrt(q^2) excluding ±40 MeV intervals around mω and mφ" for clarity.
[Fig. 2 and Table 2] The text says the symbol points in Fig. 2 denote the fitted points, but the figure appears to show only central curves; the caption should clarify what the points and error bands represent, especially given the large a1 uncertainties.
[Sec. 3.2, Flavor-symmetry relations] The Xi_c predictions in Table 3 rely on U-spin symmetry and on adopting aω from Λc → p; this should be stated as an order-of-magnitude estimate rather than a precise prediction, and the large systematic uncertainty from the symmetry assumption should be acknowledged in the discussion.
[References] Ref. [75] is used both as a source of input parameters and as a comparison for LCSR form factors; the text should be explicit about which quantities are taken from which reference to avoid ambiguity.

Circularity Check

2 steps flagged · score 6.0 of 10

Resonance-included branching fractions are normalized to the experimental two-body inputs via Eq. (26), so Table 3's resonance entries largely return those inputs; the QCDSR form factors and no-resonance results remain independent.

fitted input called prediction [Section 3.2, Eqs. (25)-(28) and Table 3 (Lambda_c resonance rows)]
"To fix the two couplings aω and aϕ, the following approximation is employed: Br(Λc(→ pV ) → pµ+µ−) = Br(Λc → pV )Br(V → µ+µ−) , V = ω, ϕ . (26) Here, the left-hand side is directly calculated using only CR9(q2), considering only the contributions from the ω and ϕ, while the right-hand side is evaluated by the latest experimental data [27,73]: As a result, we obtain aω = 0.072 ± 0.015, aϕ = 0.104 ± 0.010."

The couplings aω and aϕ are determined by requiring the resonance-only CR9(q2) calculation of Λc→pμ+μ− to reproduce the measured products Br(Λc→pV)Br(V→μ+μ−). The 'resonance' branching fractions in Table 3 are then obtained by integrating the same CR9(q2)-dominated amplitudes, so their central values are essentially the normalization inputs reflected back. This is a consistency check of the Breit-Wigner normalization, not an independent prediction for the total rate. The no-resonance branching fractions and the QCD sum-rule form factors do not participate in this reduction.
fitted input called prediction [Section 3.2, Xi_c paragraph and Table 3 (Xi_c resonance rows)]
"Regarding the couplings aϕ and aω, only one relevant two-body decay mode, Ξ0c → Λϕ, has been observed. The branching fraction for this decay is Br(Ξ0c → Λϕ) = (4.9 ± 1.5) × 10−4 [88], which suggest aϕ = 0.20 ± 0.04 in Ξc decay mode. There is no sufficient experimental data to determine aω, so we adopt the same value as in the Λc → p process."

The Ξc resonance predictions use aϕ fixed from Br(Ξ0c→Λϕ), so the Ξ0c→Λℓ+ℓ− resonance entries in Table 3 largely return that measured two-body rate multiplied by V→ℓ+ℓ−, and the Ξ+c→Σ+ entries inherit the same fitted aϕ via U-spin. The listed 'rough estimates' are therefore normalized to the very data they purport to predict. Only the treatment of ω and the no-resonance estimates remain independent of this particular input.

full rationale

The central form-factor derivation is not circular: the ten form factors are computed from three-point QCD sum rules with the input parameters listed in Eq. (15); the z-series fit is performed on the sum-rule outputs over q²∈[-0.4,0.4] GeV² and is checked against the directly computed f_i(0). No external branching-fraction data enter that part of the chain, and the self-citations (e.g. Refs. [57], [64]) are not load-bearing for the main claims. The circularity is confined to the resonance-included branching fractions. Equations (25)-(28) fix aω and aφ by matching the CR9-only Λc→pμ+μ− rate to Br(Λc→pV)Br(V→μ+μ−), and Table 3's resonance rows then integrate the same CR9-dominated amplitudes; likewise aφ for Ξc is set by Br(Ξ0c→Λφ) and the Ξc resonance rates inherit that input. These entries are normalizations returned as predictions, so the paper receives score 6 under the fitted-input-called-prediction pattern. The no-resonance branching fractions, the angular observables, and the new-physics sensitivity plots are not reduced by these normalizations and retain independent content. The missing per-form-factor Borel-window/OPE-convergence curves noted in Section 3.1 are a reproducibility and correctness risk, not a demonstrated circular step.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

No new particles or interactions are introduced. The calculation leans on a chain of inherited inputs: condensates, decay constants, threshold parameters, Wilson coefficients, and the fitted resonance couplings. The most consequential fitted quantities are a_omega and a_phi, because they control the resonance-dominated branching fractions.

free parameters (6)

Lambda_c decay constant lambda_Lambda_c = 0.0119 GeV^3
Input from two-point sum rules in Refs. [75,76]; enters Eq. (14) as normalization of the form factors and has no uncertainty quoted.
proton decay constant lambda_p = 0.02 GeV^3
Input from two-point sum rules; used in Eq. (14) to normalize the form factors.
threshold parameters s0_1, s0_2 = s0_1 = 9.5-10.5 GeV^2, s0_2 = 2.4-3.0 GeV^2
Varied to determine the Borel window; inherited from two-point sum rules but tuned here to stabilize the form factors.
z-series slope parameters a1 for 10 form factors = Table 2 (e.g., f1: -0.36 +/- 4.44; gT1: -23.25 +/- 16.28)
Fitted to the QCDSR points over q^2 in [-0.4,0.4] GeV^2; large uncertainties propagate into the extrapolated form factors.
resonance couplings a_omega, a_phi = 0.072 +/- 0.015, 0.104 +/- 0.010 (Lambda_c); 0.20 +/- 0.04 (Xi_c phi)
Fixed using Eq. (26) to reproduce measured Br(Lambda_c -> p V) Br(V -> l+l-) and Br(Xi_c0 -> Lambda phi); these inputs dominate the resonance-included branching fractions.
strong phases delta_omega, delta_phi = varied over [0, 2pi]
Unknown phases in the resonance amplitude Eq. (25); scanned to build the blue uncertainty bands in Fig. 3.

assumptions (7)

domain assumption Quark-hadron duality: the integrated spectral density with continuum subtraction equals the hadronic sum over ground states and resonances.
Invoked in Eq. (14) to connect the QCD and phenomenological sides of the three-point function; standard but not proven for this channel.
domain assumption Truncated OPE at dimension 6 is convergent, with the <qqbar>^2 term below 30%.
Section 3.1 states this criterion; Fig. 1 shows it is satisfied for Lambda_c -> p but not for Lambda_b -> n, where the OPE fails.
domain assumption Pole contribution exceeds 40% in the Borel window.
Eqs. (17)-(18) define RPC; the authors assert this criterion selects the Borel window but do not show the resulting RPC curves.
ad hoc to paper The z-series with a single slope a1 plus pole factor is adequate to extrapolate form factors from q^2 in [-0.4,0.4] GeV^2 to the full physical region.
Eq. (19) uses this parameterization; no cross-validation in the physical region is possible.
domain assumption Factorization approximation of Eq. (26): Br(Lambda_c(->pV) -> p l+l-) = Br(Lambda_c -> pV) Br(V -> l+l-).
Used to fix the couplings a_omega and a_phi; neglects off-shell and interference effects.
domain assumption Flavor symmetry relations: f(Lambda_c->p) = f(Xi_c+->Sigma+) = sqrt(2) f(Xi_c0->Sigma0) = sqrt(6) f(Xi_c0->Lambda).
Section 3.2 uses U-spin and isospin relations to estimate Xi_c form factors from the Lambda_c ones; these are approximate, broken by quark mass differences.
domain assumption The interpolating currents of Eqs. (5)-(6) couple only to the ground-state baryons with decay constants lambda_Lambda_c and lambda_p.
This is the standard QCDSR saturation assumption needed for Eq. (7) to have a simple pole form.

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Cite this review

Pith. "Pith review of Rare $ \Lambda_c $ decays and new physics effects." pith.science (2026). https://pith.science/paper/3DQUOVNV

@misc{pith2026241115857,
  author       = {Pith},
  title        = {Pith review of: Rare $ \Lambda_c $ decays and new physics effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3DQUOVNV}},
  note         = {Machine review of arXiv:2411.15857}
}

abstract

Recent experimental progress on baryonic rare decays has spurred a deeper investigation on flavor-changing neutral current transitions in the baryon sector. Within the framework of QCD sum rules, we derive a complete set of form factors for the $ \Lambda_c\to p $ process in the large recoil region and use the $z$-series parametrization to extrapolate them across the full physical range. Employing these form factors and flavor symmetries, we compute branching fractions for the decays $\Lambda_c \to p e^+ e^-$ and $\Lambda_c \to p \mu^+ \mu^-$, as well as for rare $ \Xi_c $ decay modes. We examine as well the new physics effects through specific angular observables such as the lepton forward-backward asymmetry and the fraction of longitudinally polarized dileptons. Results indicate that new physics models may be testified in baryonic rare decays, with immense data collected in running and future colliders.

Figures

Figures reproduced from arXiv: 2411.15857 by the authors.

**Figure 2.** The q 2 dependence of form factors with the central value of fitting parameters listed in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Predictions for the q 2 dependence of dBr/dq 2 , AF B and FL for Λc → pµ+µ − decay mode with and without the vector resonance contributions. The red line represents the contribution excluding resonances, while the red error band reflects uncertainties from the Wilson coefficients and form factors. The blue lines denote the contribution including resonances. The blue error bands represent uncertainties from strong p… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: New physics effects for the differential branching fraction, the forward-backward asym [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

Discussion (0). Continue with ORCID to comment.

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