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The paper derives the full four-fold angular distributions for Λ_b→Λ(→Nπ)ℓ⁺ℓ⁻ with longitudinally, normally, and transversely polarized leptons, and shows that normal and transverse polarizations introduce six and twelve new angular coeffic

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2026-08-04 13:27 UTC pith:4LOEIHXG

load-bearing objection Genuinely new polarized four-fold angular distributions for Lambda_b->Lambda ell+ell- with a sound analytic core, plus fixable wording and numerical errors. the 3 major comments →

arxiv 2510.00249 v3 pith:4LOEIHXG submitted 2025-09-30 hep-ph

Lepton polarization dependent angular observables and the polarization asymmetries in the four-fold Λ_b rightarrow Λ(rightarrow N π) ell^+ell^- decay

Rana Khan , Faisal Munir Bhutta , Ishtiaq Ahmed , M. Jamil Aslam This is my paper
classification hep-ph
keywords Λ_b baryon decayslepton polarizationangular observablesb→sℓ⁺ℓ⁻ transitionshelicity amplitudestransversity amplitudesflavor-changing neutral currentsnew physics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to establish a complete, polarized-lepton version of the four-fold angular analysis of the rare decay Λ_b→Λ(→Nπ)ℓ⁺ℓ⁻, driven by the flavor-changing neutral-current transition b→sℓ⁺ℓ⁻. It derives the decay distributions for longitudinal, normal, and transverse lepton polarization and shows that longitudinal polarization only reweights the already known angular structures, while normal and transverse polarizations generate six and twelve genuinely new angular coefficients. All coefficients are expressed in closed form in terms of helicity and transversity amplitudes, with lepton-mass effects included. The paper argues that the resulting polarization asymmetries provide a baseline for distinguishing Standard Model predictions from specific vector/axial-vector new-physics scenarios.

Core claim

For an unpolarized Λ_b, polarizing the final-state lepton produces three distinct four-fold angular distributions. Longitudinal polarization does not alter the angular structure: the spin-dependent information is absorbed into the same ten coefficient shapes as the unpolarized distribution. Normal polarization introduces six additional coefficients, of which two are real and four are imaginary, and only two of the imaginary ones enter physical observables directly. Transverse polarization introduces twelve new coefficients—eight real and four imaginary—several of which feed into the differential branching ratio and the lepton, hadron, and lepton-hadron forward-backward asymmetries. All polar

What carries the argument

The central mechanism is the lepton spin projector (1+γ₅s̸₁)/2 inserted into the leptonic tensor, which turns the unpolarized lepton pair into one with definite longitudinal, normal, or transverse polarization. Combined with the helicity-formalism decomposition of the four-body amplitude, this yields the three four-fold distributions in cosθ_ℓ, cosθ_Λ, and φ, organized into angular coefficients K. The new coefficients are packaged in the transversity basis—left/right-chirality combinations of hadronic helicity amplitudes—which makes their dependence on the Wilson coefficients explicit and exposes which terms are real or imaginary.

Load-bearing premise

The numerical predictions assume the effective Hamiltonian contains only vector, axial-vector, and photon-dipole operators, with charm-loop effects absorbed in C_9^eff, and evaluate observables over the full q² range without modeling or vetoing the J/ψ and ψ(2S) resonance regions.

What would settle it

Measure the transverse polarization asymmetry P_T(q²) in Λ_b→Λμ⁺μ⁻. The Standard Model prediction is about an order of magnitude smaller than the longitudinal P_L and has a characteristic q² shape; a measured P_T that rises well above this baseline in the low-to-intermediate q² region would contradict the vector/axial-vector-only description.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Longitudinal-polarization asymmetries such as P_L, the spin-dependent lepton forward-backward asymmetries, and the coefficients K2ccL and K2ssL separate Standard Model predictions from new-physics scenarios; in the benchmark scenario S1 the deviations lie outside the Standard Model uncertainty band across most of q².
  • The zero-crossing position of the K2ccL polarization asymmetry is sensitive to shifts in the effective Wilson coefficients C_9 and C_10, so a precise measurement of that crossing could distinguish among new-physics scenarios.
  • Transverse polarization asymmetries such as P_T and the spin-dependent lepton forward-backward asymmetries are suppressed by the lepton mass, giving a complementary, mostly Standard-Model-dominated probe in the low-to-intermediate q² region.
  • The new real angular coefficients from normal and transverse polarization are predicted to be small in the Standard Model, providing baseline predictions that future data can test.
  • Because lepton-mass effects are retained throughout, the same framework applies directly to τ⁺τ⁻ final states, where the mass-suppressed transverse coefficients become more prominent.

Where Pith is reading between the lines

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

  • Editorial inference: the paper evaluates observables over the full q² range without modeling the J/ψ and ψ(2S) charmonium resonances, so a direct comparison with data would require resonance vetoes or a resonance-aware treatment; the low-q² and very-high-q² windows are the cleanest regions for testing the Standard Model baseline.
  • Editorial inference: the new normal and transverse angular coefficients are helicity-sensitive by construction, so they could act as direct probes of the chirality structure of new physics; a measured deviation from the Standard Model baseline would single out chirality-flipped operators.
  • Editorial inference: in the τ channel the m_ℓ suppression of transverse coefficients turns into an enhancement, so P_T and the related asymmetries may become substantially larger and more observable than in the muon mode.
  • Editorial inference: the imaginary coefficients identified in this paper are natural places to look for CP-violating weak phases; combining the polarized angular distribution with CP-asymmetric observables would be a straightforward extension.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper derives the four-fold differential decay distributions for the cascade decay Λ_b → Λ(→ N π) ℓ^+ ℓ^- with the final-state lepton polarized longitudinally, normally, or transversely. For each polarization, it identifies the angular coefficients that appear relative to the unpolarized case, and provides explicit expressions in both helicity and transversity amplitude bases (Appendices E and F). It then constructs differential rates, forward-backward asymmetries, and lepton-polarization asymmetries, and presents SM and new-physics (NP) predictions using lattice-QCD form factors and external NP global fits. The central claimed results are: longitudinal polarization preserves the unpolarized angular structure; normal polarization introduces six additional coefficients; transverse polarization introduces twelve additional coefficients.

Significance. If correct, the paper provides a useful extension of the established Λ_b → Λ(→ N π) ℓ^+ ℓ^- angular analysis, with a complete catalogue of polarization-dependent coefficients including lepton-mass effects. The use of lattice-QCD form factors and external global NP fits is appropriate, and the unpolarized limit appears to be reproduced. However, the numerical and interpretive parts currently contain inconsistencies — in particular, the repeated classification of coefficients as 'imaginary' contradicts the explicit real expressions in Appendix E, and the full-q² analysis ignores charmonium resonances. These issues currently prevent the results from being taken at face value, though they appear fixable.

major comments (3)
  1. [§III.B.2, Appendix E (Eqs. (E22)–(E27)), Abstract, Conclusions] The paper repeatedly states that four of the six normal-polarization coefficients (K1sN, K2sN, K4cN, K4N) are 'imaginary'. The explicit expressions in Appendix E are manifestly real: each is a real prefactor |N3|^2 (times βℓ, α, mℓ/√q² as appropriate) multiplying Re[...] or Im[...] of helicity amplitudes. Since these coefficients multiply real angular functions, an imaginary coefficient would make the differential distribution complex, which is impossible. Thus the classification is internally inconsistent and is presented as a central result in the abstract, body, and conclusions. At minimum the wording must be corrected; more importantly, the entire coefficient catalogue in Appendices E/F should be independently verified by re-expanding the squared amplitude.
  2. [§V, Figs. 1–6; Eq. (3)] The phenomenological analysis plots and compares observables over the full q² range up to (mΛb−mΛ)²≈20.3 GeV², including the charmonium resonance region q²≈9–14 GeV² (J/ψ, ψ(2S)). The effective Hamiltonian of Eq. (3) with C9^eff evaluated in perturbation theory (Appendix B) does not include these narrow resonances, and no resonance veto or model is introduced. Therefore the SM/NP predictions in the resonance bins are not predictions of the stated effective theory. The authors should either restrict the numerical analysis to resonance-free q² windows or include a resonant contribution with associated uncertainties.
  3. [Table II, Eqs. (46), (50), (54), (56)] TABLE II lists B(Λ→pπ−)=0.64% and B(Λ→nπ0)=0.36%. The established PDG values are approximately 63.9% and 35.8%, respectively — a factor of 100 difference. Since these branching ratios multiply every differential rate (e.g., Eq. (46)), using the quoted values would suppress all absolute branching-ratio predictions by two orders of magnitude. If the intended values are 0.64 and 0.36 as fractions, the percent signs must be removed and the numerical analysis checked; if the table values were actually used, all absolute-rate results need to be recomputed.
minor comments (5)
  1. [§II.B] Typo: 'the helicity amplitudes are gievn as' should be 'given'.
  2. [Appendix A] The Källén function is typeset with broken characters in several places; please fix the encoding.
  3. [Figures 1 and 2] The label 'dβ/dq²' is used but β is not defined in the text or captions. Also 'Dotted region represent ... dashed region represent ...' should clarify that these are uncertainty bands, not regions.
  4. [Table III] The mapping of the NP scenarios S1–S4 to the best-fit scenarios of Ref. [35] should be stated explicitly; the current table uses shorthand that is not self-contained.
  5. [Appendix B] The expressions for C7^eff and C9^eff require the charmed-quark pole mass and bottom-quark pole mass, but Table II lists only the bottom pole mass. Please specify the value of m_c^pole used in the numerical analysis.

Circularity Check

0 steps flagged

No significant circularity: the polarized angular distributions are derived from an external effective Hamiltonian, external lattice form factors, and external global-fit Wilson coefficients; nothing is fitted to the predicted observables.

full rationale

The derivation chain is self-contained. The unpolarized four-fold distribution (Eq. 39) is taken from the external literature [59,62], and the polarized distributions (Eqs. 42, 44, 45) are obtained by inserting the standard lepton spin projector (1+γ5/s1)/2 into the leptonic tensor. All angular coefficients, including the new N- and T-polarization ones, are then expressed as algebraic combinations of helicity/transversity amplitudes in Appendices E and F. No step defines an input in terms of a target observable or fits a parameter to the quantity being predicted. The numerical inputs—SM Wilson coefficients, lattice QCD form factors [86], and NP global-fit scenarios [35]—are external, and the paper explicitly compares its SM/NP results against those inputs without re-fitting them. Self-citations occur (e.g., Refs. [38,39,72,73]) but they are contextual and are not load-bearing for the central angular-distribution derivation. The reviewer-flagged issue that four normal-polarization coefficients are labeled 'imaginary' while their Appendix E expressions are real prefactors times Im(...) is an internal consistency/correctness concern, not a circularity: it does not make the derivation equivalent to its inputs. The paper also notes its own limitation that for N and T polarizations 'the potential NP contributions are indistinguishable from the SM expectations,' which is a numerical finding rather than an assumption that smuggles in the result.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

No new particles, forces, or symmetries are introduced. The central derivation is self-contained given the standard effective Hamiltonian, while the numerical output relies on external lattice-QCD form factors, an external global fit for NP coefficients, and literature values for the Λ→Nπ vertex.

free parameters (3)
  • z-expansion coefficients for Λ_b→Λ form factors (a_0^f, a_1^f, a_2^f) = from lattice QCD fit in Ref. [86]
    The numerical predictions for all observables depend on the form-factor parametrization; the paper uses but does not list the fit coefficients.
  • NP Wilson coefficients C_9^NP, C_10^NP, C_9'^NP, C_10'^NP = best-fit values in Table III from Ref. [35]
    The NP asymmetry predictions are evaluated at these external best-fit points; they are not derived in this paper.
  • Λ→Nπ parameters ω, ξ (or α) = not given in the paper; taken from Refs. [59,62]
    The cascades decay amplitude uses ω and ξ, and the angular distribution uses the parity-violating parameter α; no numerical values are tabulated.
axioms (4)
  • domain assumption The b→sℓℓ transition is described by the weak effective Hamiltonian containing only O7, O9, O10 plus chirality-flipped and vector/axial-vector NP counterparts (Eq. 3).
    Scalar, pseudoscalar and tensor operators are neglected, which restricts the angular distribution to the derived basis.
  • domain assumption The cascade decay Λ→Nπ is treated via the simple weak vertex (ω+ξγ5) and the intermediate Λ is taken on-shell with the propagator omitted (footnote 2, Sec. II.D).
    This is the narrow-width approximation; it is needed to factorize the decay amplitude.
  • standard math The polarized lepton is described by inserting the spin projector (1+γ5s̸)/2 into the leptonic tensor (Sec. III.B).
    This is a standard density-matrix treatment for a fully polarized lepton.
  • domain assumption The form factors are parametrized by a two- or three-parameter z-expansion (Eqs. 65 and 67).
    The numerical results assume this specific parametrization and the lattice-QCD inputs of Ref. [86].

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read the original abstract

The rare decays mediated by flavor-changing neutral current processes, such as $b \to s \ell^{+}\ell^{-}$, provide powerful probes of the Standard Model and potential windows into new physics. Particularly, the angular observables in these exclusive decays are valuable because of their sensitivity to short-distance dynamics and their reduced dependence on hadronic uncertainties, which mainly arise from form factors. In this work, we analyze the $\Lambda_b \to \Lambda(\to N\pi)\ell^{+}\ell^{-}$ (with $N\pi=\{p\pi^-,n\pi^0\}$) decay with polarized final-state lepton and derive the corresponding four-fold differential decay distributions. For the longitudinal, normal, and transverse polarization states, we systematically identify the additional angular coefficients that emerge relative to the unpolarized case. We find that the longitudinal polarization preserves the structure of the unpolarized distribution, while the normal and transverse polarizations introduce some new additional angular coefficients. The analytical expressions of all polarized and unpolarized angular coefficients are explicitly derived in terms of the helicity and transversity amplitudes. To compare the variation in the polarized and unpolarized angular observables, we have plotted them against the square of the momentum transfer $q^2$. Additionally, the Standard Model predictions of the polarization asymmetry observables are provided and their sensitivity to new physics is explored under different new physics scenarios. The obtained results, in the current study, for longitudinal and transverse polarization cases, provide a baseline for the lepton polarization dependent observables, which may serve as sensitive probes to test the Standard Model in these decays.

Figures

Figures reproduced from arXiv: 2510.00249 by Faisal Munir Bhutta, Ishtiaq Ahmed, M. Jamil Aslam, Rana Khan.

Figure 1
Figure 1. Figure 1: FIG. 1: Differential branching ratios and various observables vs. [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Differential branching ratios and various observables vs. [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Predictions for NP sensitivity in asymmetry of the longitudinal polarized observables as a function of [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Predictions for NP sensitivity in asymmetry of the transverse polarized observables as a function of [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: SM predictions for polarization asymmetries of the new real angular coefficients, [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Kinematics of Λ [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

83 extracted references · 71 linked inside Pith

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    Longitudinal four-fold angular distribution For the case of longitudinal lepton polarization, L polarized four-fold angular decay distribution is obtained as d4ΓL (⃗ sℓ− =±ˆ eL) dq2dcosθ ℓdcosθ Λdϕ = 3 8π B(Λ→N π) h K1ssL sin2 θℓ +K 1ccL cos2 θℓ +K 1cL cosθ ℓ + K2ssL sin2 θℓ +K 2ccL cos2 θℓ +K 2cL cosθ ℓ cosθ Λ + K3scL sinθ ℓ cosθ ℓ +K 3sL sinθ ℓ sinϕsinθ...

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