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SCET sum rules for $\Lambda_b \to \Lambda \ell^+\ell^-$, $\Lambda \gamma$ decays

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read One soft form factor, computed at NLO by SCET sum rules, governs the rare decays $\Lambda_b\to\Lambda\ell^+\ell^-$ and $\Lambda_b\to\Lambda\gamma$.

desk verdict Solid, technically careful SCET sum-rule analysis of Λb→Λ form factors: the new C-type one-loop hard functions are useful, the soft-dominance conclusion holds up, but the central numerics are calibrated to a single exponential LCDA model and the quoted uncertainties omit shape error. read the letter →

arxiv 2506.21419 v2 pith:C2OKULS7 submitted 2025-06-26 hep-ph

Long-Shun Lu , Cai-Dian Lü , Yue-Long Shen , Yan-Bing Wei This is my paper · ORCID
classification hep-ph
keywords Lambda_bbaryondecayslight-conesumrulessoft-collineareffectivetheoryheavy-to-lightformfactorsrareflavor-changingneutralcurrentsforward-backwardasymmetrydistributionamplitudes
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 sets out to establish that the rare flavor-changing decays $\Lambda_b\to\Lambda\ell^+\ell^-$ and $\Lambda_b\to\Lambda\gamma$ are, at large recoil, governed by a single 'soft' form factor $\xi_\Lambda(n\cdot p)$ rather than by an irreducible set of hadronic form factors. It derives this form factor from light-cone sum rules inside soft-collinear effective theory, computing the perturbative jet function at next-to-leading order and matching $\mathrm{SCET}_I$ onto heavy quark effective theory. It then argues that the competing hard-scattering (B-type) and non-factorizable (C-type) contributions are numerically negligible, so the decays are dominated by soft physics. The concrete payoff is a set of predictions: $\xi_\Lambda^{\mathrm{NLO}}(q^2=0)=0.182\pm0.05$, $B(\Lambda_b\to\Lambda\gamma)=(1.22^{+0.66}_{-0.67})\times10^{-5}$, and $q^2$-dependent rates and angular observables for the dilepton channel that are consistent with current data within sizable uncertainties.

What carries the argument

The load-bearing object is the A-type soft form factor $\xi_\Lambda(n\cdot p)$, defined by $\langle\Lambda(p,s')|\bar\xi_s(0)\Gamma_A^i h(0)|\Lambda_b(v,s)\rangle=\xi_\Lambda(n\cdot p)\,\bar u_\Lambda(p,s')\Gamma_A^i u_{\Lambda_b}(v,s)$, and it is what remains of the $\Lambda_b\to\Lambda$ matrix elements after the large-recoil symmetry reduction. The paper evaluates it by a light-cone sum rule: the final-state $\Lambda$ is replaced by the interpolating current $J_\Lambda$, the vacuum-to-$\Lambda_b$ correlation function with a $\mathrm{SCET}_I$ A-type operator is expanded on the light cone, and after $\mathrm{SCET}_I\to$ HQET matching the one-loop jet function is obtained. B-type form factors are extracted from a correlation function with a hard-collinear gluon field, which vanishes at leading power by transverse rotational invariance, and C-type form factors come from a $\gamma^*$-to-$\Lambda_b$ correlation function evaluated at tree level. All numerical results rest on the exponential model of the $\Lambda_b$ distribution amplitude, $\phi_4(\omega_1,\omega_2)=\omega_0^{-2}e^{-(\omega_1+\omega_2)/\omega_0}$, with $\omega_0=430^{+70}_{-50}\,\mathrm{MeV}$ fixed by matching to the external input $f_+(0)=0.18\pm0.04$.

What would settle it

Measure $B(\Lambda_b\to\Lambda\gamma)$ with total uncertainty below about $20\%$ and compare it with the predicted band $(1.22^{+0.66}_{-0.67})\times10^{-5}$, or compute the second moments of the $\Lambda_b$ distribution amplitude on the lattice: a clear miss in either test would falsify the exponential-LCDA-plus-single-soft-form-factor picture.

Watch

Extended reading notes

Core claim

The central claim is that the $\Lambda_b\to\Lambda$ transition at large recoil reduces to one function: the A-type soft form factor $\xi_\Lambda(n\cdot p)$, which survives the combined heavy-quark and large-recoil symmetries that would otherwise leave ten form factors. The paper builds a light-cone sum rule from a vacuum-to-$\Lambda_b$ correlation function in which the final-state $\Lambda$ is represented by an interpolating current and the $\mathrm{SCET}_I$ current is inserted perturbatively, then performs the $\mathrm{SCET}_I\to$ HQET matching at one loop to obtain $\xi_\Lambda^{\mathrm{NLO}}(q^2=0)=0.182^{+0.050}_{-0.041}$. It finds that the leading-power hard-collinear contribution begins at $O(\alpha_s^2)$ and is numerically small, that the B-type form factor vanishes at leading power in the relevant correlation function, and that the C-type non-factorizable corrections shift the hard coefficients by roughly $1\%$ (under $2\%$ for $\Lambda_b\to\Lambda\gamma$). From this single soft form factor the paper computes the differential branching fraction, forward-backward asymmetry, and dilepton longitudinal polarization fraction for $\Lambda_b\to\Lambda\ell^+\ell^-$, together with $B(\Lambda_b\to\Lambda\gamma)=(1.22^{+0.66}_{-0.67})\times10^{-5}$.

Load-bearing premise

Everything rests on the assumed exponential shape of the $\Lambda_b$ light-quark distribution amplitude: a single parameter $\omega_0$, fixed by matching to the external number $f_+(0)=0.18\pm0.04$, and no dependence on the momentum split $u$ between the two light quarks; if the true amplitude has a different shape or $u$-dependence, the predicted $q^2$ dependence of $\xi_\Lambda$ and all derived rates change.

Editorial extensions

If this is right

  • In the window $1\,\mathrm{GeV}^2<q^2<7\,\mathrm{GeV}^2$, all hadronic uncertainty in both $\Lambda_b\to\Lambda\ell^+\ell^-$ and $\Lambda_b\to\Lambda\gamma$ is concentrated in $\xi_\Lambda(n\cdot p)$, so improved data on either mode sharpen the same nonperturbative function.
  • The NLO sum rule lowers the soft form factor at $q^2=0$ from $0.222^{+0.057}_{-0.058}$ (LO) to $0.182^{+0.050}_{-0.041}$ (NLO), and the predicted energy dependence of $\xi_\Lambda$ scales between $1/E_\Lambda^2$ and $1/E_\Lambda^3$.
  • The paper predicts $B(\Lambda_b\to\Lambda\gamma)=(1.22^{+0.66}_{-0.67})\times10^{-5}$, above the current central value $(7.1\pm1.7)\times10^{-6}$ from experiment but consistent within the quoted uncertainties.
  • For $\Lambda_b\to\Lambda\ell^+\ell^-$, the differential branching fraction, forward-backward asymmetry, and longitudinal polarization fraction are predicted in the factorization limit, with the two angular observables carrying substantially smaller theoretical errors because uncertainties cancel in the ratios.
  • The non-factorizable C-type corrections shift the effective hard coefficients by about $1\%$ over most of the dilepton window, except near $q^2\simeq3.2\,\mathrm{GeV}^2$ where the relative shift reaches $\sim50\%$ because $Q_7$ and $Q_9$ cancel.

Reading between the lines

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

  • If the single-soft-form-factor picture survives, $\Lambda_b\to\Lambda$ becomes a clean indirect probe of the $\Lambda_b$ light-cone distribution amplitude: low-recoil lattice results and these large-recoil sum rules could be combined to extract $\omega_0$ and test the assumed $u$-independence of $\phi_4$.
  • The $u$-independence assumption is directly testable: a lattice calculation of the second $\omega_1-\omega_2$ moment, or of the first $u$-dependent correction to $\phi_4$, would show whether the predicted $q^2$ dependence of $\xi_\Lambda$ is trustworthy.
  • The $\sim50\%$ enhancement at $q^2\sim3.2\,\mathrm{GeV}^2$ marks the region where $Q_7$ and $Q_9$ cancel in $C_1^A$; completing the one-loop matching and RG evolution of the C-type operators would be needed before precision statements can be made in exactly that window.
  • A natural extension is to apply the same SCET sum-rule construction to other heavy-baryon transitions such as $\Lambda_b\to p$: if the suppression of B- and C-type terms is a general baryon feature, the framework would simplify those modes too.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper constructs light-cone sum rules (LCSRs) within soft-collinear effective theory (SCET) for the effective form factors governing Λ_b → Λ ℓ^+ℓ^- and Λ_b → Λ γ decays. The authors classify SCET_I operators into A-, B-, and C-type, compute the A-type soft form factor ξ_Λ(n·p) at next-to-leading order (NLO) in the matching from SCET_I to HQET, argue that B-type hard-scattering contributions are power-suppressed, and evaluate C-type non-factorizable contributions at tree level, including weak-annihilation, chromomagnetic, and quark-loop topologies. They then use the soft form factor to compute the differential branching fraction, forward-backward asymmetry, and dilepton longitudinal polarization fraction for Λ_b → Λ ℓ^+ℓ^- in the factorization limit, and the branching fraction B(Λ_b → Λγ) = (1.22^{+0.66}_{-0.67}) × 10^{-5}. The numerical analysis normalizes the single exponential LCDA parameter ω0 so that the NLO sum rule reproduces f_+(0) = 0.18 ± 0.04 from earlier LCSR work.

Significance. If the result holds, the paper provides a systematic SCET-based framework for heavy-baryon rare decays, with a complete one-loop jet-function calculation for the A-type soft form factor, explicit verification of factorization-scale cancellation, and new sum rules for C-type non-factorizable form factors, including a corrected sign relative to Ref. [21]. The predicted q^2-dependent observables and the Λ_b → Λγ branching fraction are falsifiable against existing LHCb and CDF data. However, the central quantitative claim is conditional on the exponential, u-independent model of the Λ_b light-cone distribution amplitude, and on the explicit neglect of several computed contributions whose size is not fully quantified. The paper is transparent about many of these gaps, which helps the reader assess the robustness of the results.

major comments (4)
  1. [Sec. 3.1.5 and Sec. 4.1 (Eqs. (52), (53), (82), (88))] The central numerical result ξ_NLO(q^2=0) = 0.182^{+0.050}_{-0.041} is not an independent prediction: the exponential u-independent LCDA model of Eqs. (94)-(95) is normalized by requiring the NLO sum rule to reproduce the external value f_+(0) = 0.18 ± 0.04 from Refs. [13,15]. Since the same model controls the q^2 dependence in Eq. (53), the uncertainty band and hence the branching ratio in Eq. (88) inherit the specific shape assumption. Appendix A (Eqs. (96)-(109)) shows that the full projector contains five independent generalized distribution-amplitude combinations with different exponential dependences, while the sum rule only uses the u-averaged combination of Eq. (39). The paper supplies no independent check of the u-independence or of the exponential shape, so the quoted uncertainty is a conditional uncertainty rather than a full model uncertainty.
  2. [Sec. 3.1.3 and Sec. 5 (Eqs. (81), (88), (123))] The leading-power contribution ξ_LP(q^2=0) = -0.012^{+0.009}_{-0.023} from Ref. [37] is quoted, but it is not stated whether it is added to the soft form factor in the numerical predictions. The phenomenological analysis in Sec. 5 uses ξΛ from the soft LCSR alone, so the branching ratio in Eq. (88) appears to omit a known ~7% amplitude-level contribution. The authors should either add ξ_LP to the soft form factor or explicitly justify and quantify the omission.
  3. [Sec. 3.3 and the text after Eq. (15)] The paper explicitly states that vertex corrections to the weak-annihilation graph are neglected and that the C-type matching remains incomplete at O(α_s). Nevertheless, the numerical smallness of the C-type non-factorizable contributions, quantified by |ΔC_A^γ|/|C_A^γ| < 2% in Eq. (84) and by the O(1%) estimate for |ΔC_A^1|/|C_A^1|, is used to neglect them in the phenomenological analysis. Because the matching is incomplete at the order claimed, the numerical suppression is established only for the leading and partial NLO terms, and no estimate of the omitted vertex-correction uncertainty is provided. This gap should be closed or the conclusion should be softened.
  4. [Sec. 3.2] The B-type form factor Δξ^B_Λ is not actually computed by the new sum rule. The correlation function is argued to vanish at leading power by rotational invariance, and the numerical size Δξ_B/ξ_Λ ≈ -0.8% is taken from Ref. [14] rather than derived here. Thus the statement in the introduction and conclusion that various types of form factors are determined by the LCSRs overstates what is obtained in this work; the B-type contribution should be explicitly labeled as an external input or the corresponding sum rule should be provided.
minor comments (4)
  1. [Appendix B, Eq. (119)] The definition of λ contains unbalanced parentheses; the second factor should read ((M_{Λ_b} - m_Λ)^2 - q^2) to close the square root.
  2. [Fig. 7 caption] The caption refers to 'sum rules for the B → π form factors' in a paper on Λ_b decays; the comparison with the B → π spectral function should be explained more clearly in the caption or the main text.
  3. [Sec. 2.1, Eq. (11)] The notation 'barred' coefficients ¯C_1-6 is used without an explicit definition of the relation to the standard BBL basis Wilson coefficients; a brief definition or reference would improve readability.
  4. [Sec. 3.3, text after Eq. (71)] The sign difference relative to Eq. (3.32) of Ref. [21] is noted only in a footnote; the authors should state explicitly which sign convention is used in Eq. (71) so that the comparison is unambiguous.

Circularity Check

2 steps flagged · score 6.0 of 10

The A-type form-factor normalization and the Λb→Λγ branching fraction are not independent predictions: ω0 is fixed by requiring the NLO sum rule to reproduce f_+(0)=0.18±0.04 from the same group's earlier LCSR, and B(Λb→Λγ) is proportional to |ξΛ(0)|^2.

  1. fitted input called prediction [Section 4.1 'Input parameters', Eq. (95) and the text following Eq. (82)]
    "Using the matching procedure from Ref. [15], we take ξNLO Λ (q2 = 0) ≃ f + Λb→Λ(0) = 0.18 ± 0.04 as input to determine ω0 = 430 +70 −50 MeV."

    The only shape/normalization parameter of the exponential LCDA (Eq. (95)) is fixed by requiring the NLO sum rule (52)–(53) to reproduce f_+(0). Hence the subsequently quoted 'prediction' ξNLO Λ (0) = 0.182+0.050−0.041 in Eq. (82) is the fitted calibration point, not an independent sum-rule result. Its uncertainty band mainly propagates the 0.04 error on the training input and does not include model-shape uncertainty. The input f_+(0) itself is taken from Ref. [15], a prior LCSR calculation by overlapping authors (Y.-L. Shen).

  2. fitted input called prediction [Section 5, Eq. (123) and Eq. (88)]
    "B(Λb → Λγ) = τΛb 4πM 3 Λb (M 4 Λb − m4 Λ) |C A γ (µ)| 2 |ξΛ(q2 = 0)| 2 . ... B(Λb → Λγ) = (1.22+0.66 −0.67) × 10−5"

    Because ξΛ(q2 = 0) was set equal to f_+(0) = 0.18 ± 0.04 via the ω0 calibration, this branching fraction is proportional to the square of the same calibrated input. The headline rate therefore does not test the sum-rule framework; it is a re-expression of the adopted normalization multiplied by known Wilson coefficients and phase-space factors.

full rationale

The derivation is not globally circular: the NLO SCETI→HQET matching, the construction of the A/B/C-type correlation functions and sum rules, the Borel/duality-threshold analysis, and the shape/ratio observables (AFB, fL, R1(EΛ), relative sizes of B- and C-type contributions) are independent calculations that do not reduce to the input. However, the absolute normalization of the central object is calibrated. In Section 4.1, the single LCDA parameter ω0 is chosen by imposing ξNLO(q2=0) = f_+(0) = 0.18 ± 0.04, taken from Ref. [15]; Eq. (82) then reports ξNLO(q2=0) = 0.182, which is the fitted point, and Eq. (88) combined with Eq. (123) gives B(Λb→Λγ) ∝ |ξΛ(0)|^2, so the radiative branching fraction is a restatement of that same input rather than a from-scratch SM prediction. The quoted error bands mainly propagate the 0.04 uncertainty of f_+(0); the exponential, u-independent LCDA shape (Eqs. 94–95) is an adopted ansatz whose shape uncertainty is not included, as the paper itself acknowledges when it says the uncertainties are 'dominated by the poorly constrained Λb LCDAs'. This is partial circularity: the absolute predictions are calibrated, while the shape and relative contributions retain independent content.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new particles or forces are introduced; the A-, B-, C-type operators are known SCET constructs. The free parameters are the LCDA shape parameter and sum-rule parameters, all standard in LCSR practice.

free parameters (3)
  • ω0 (Λ_b LCDA inverse width) = 430^{+70}_{-50} MeV
    Fitted so that the NLO sum rule reproduces f_+(0) = 0.18 ± 0.04 from Ref. [15].
  • Borel mass M^2 = 1.6 ± 0.4 GeV^2
    Chosen by hand within a stability window defined by continuum contribution below 40% and R_ωM below 40%.
  • Continuum threshold s0 = 2.56 ± 0.10 GeV^2
    Chosen to match the values adopted in Refs. [14,15,44] and to make the sum rule stable.
assumptions (5)
  • domain assumption The light-cone OPE for the vacuum-to-Λ_b and γ*-to-Λ_b correlation functions is valid in the region 1 GeV^2 < q^2 < 7 GeV^2 and at large recoil.
    Used throughout Section 3 to justify the sum-rule dispersion relations.
  • ad hoc to paper The Λ_b LCDA can be modeled by the exponential ansatz ψ_v ∝ exp(-(x1+x2)/ω0), and ϕ4 is independent of the momentum fraction u.
    Eqs. (94)-(95) and the comment below Eq. (53).
  • domain assumption The Wandzura-Wilczek approximation for the subleading LCDAs holds.
    Eq. (92) used in the one-loop A-type correlation function.
  • ad hoc to paper At O(α_s) only SCET_I operators with a single gluon field are needed, and vertex corrections to the weak annihilation graph are neglected.
    Section 2.1, explicit statement that the matching is incomplete at O(α_s).
  • domain assumption Isospin symmetry for u and d spectator quarks, with light-quark masses neglected.
    Used for the B- and C-type correlation functions, Sections 3.2 and 3.3.

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Cite this review

Pith. "Pith review of SCET sum rules for $\Lambda_b \to \Lambda \ell^+\ell^-$, $\Lambda \gamma$ decays." pith.science (2026). https://pith.science/paper/C2OKULS7

@misc{pith2026250621419,
  author       = {Pith},
  title        = {Pith review of: SCET sum rules for $\Lambda_b \to \Lambda \ell^+\ell^-$, $\Lambda \gamma$ decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C2OKULS7}},
  note         = {Machine review of arXiv:2506.21419}
}
abstract

We construct light-cone sum rules for various types of effective form factors in the $\Lambda_b \to \Lambda \ell^+\ell^-$ and $\Lambda_b \to \Lambda \gamma$ decays by analyzing vacuum-to-$\Lambda_b$ (or $\gamma^\ast$-to-$\Lambda_b$) correlation functions with the light $\Lambda$-baryon interpolating current. These form factors, defined via hadronic matrix elements within soft-collinear effective theory (SCET), enter the next-to-leading-power QCD factorization formulas for large-recoil transitions. Implementing the perturbative matching from $\text{SCET}_\text{I}$ to heavy quark effective theory, we determine the hard-collinear functions at next-to-leading-order accuracy. Based on light-cone sum rule predictions for the $\Lambda_b \to \Lambda$ form factors, we compute the $q^2$-dependent differential branching fraction, forward-backward asymmetry and dilepton longitudinal polarization fraction for $\Lambda_b \to \Lambda \ell^+\ell^-$ decay, as well as the branching fraction for $\Lambda_b \to \Lambda \gamma$ decay.

Figures

Figures reproduced from arXiv: 2506.21419 by the authors.

Figure 1
Figure 1. Feynman diagrams where the (virtual) photon, as denoted by the crossed circle, is emitted [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Diagrammatic representation of the correlation function Π [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Diagrammatic representation of the non-vanishing one-loop contribution to Π [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Diagram (a) shows the leading-power one-loop representation of Π [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Diagrammatic representation of the leading-power Π [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Diagrammatic representation of the leading-power Π [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Spectral functions correspond to sum rules for the [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: Dependence of RωM (left) and Rcontinuum (right) on the Borel mass. Solid, dashed and dotted curves correspond to the threshold s0 = 2.56 GeV2 , 2.66 GeV2 , 2.46 GeV2 , respectively, while all the other input parameters are fixed at their central values. 20 [PITH_FULL_…
Figure 9
Figure 9. Figure 9: It is evident that the sum rule for ξΛ(q 2 = 0) exhibits exceptionally mild dependence on the Borel mass parameter, owing to a strong cancellation of systematic uncertainties between the LCSR prediction for ξΛ(q 2 = 0) and the QCD sum rule for the coupling fΛ. Also the…
Figure 9
Figure 9. Figure 9: Dependence of ξΛ(q 2 = 0) on the Borel parameter (top left), on the threshold parameter (top right) and on the factorization scale (bottom left). Solid, dashed and dotted-dashed curves are obtained from the sum rules with s0 = 2.56 GeV2 , 2.66 GeV2 , 2.46 GeV2 (top lef…
Figure 10
Figure 10. Figure 10: Dependence of the ratio R1(EΛ) on the Λ-baryon energy EΛ. The blue (left panel) and the black (right panel) curves are obtained from the LO and NLO sum rule predictions, respectively. The two green curves refer to a pure 1/E2 Λ and a pure 1/E3 Λ dependence. The domina…
Figure 11
Figure 11. Figure 11: The differential branching fraction, the leptonic forward-backward asymmetry and the [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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    hep-ph 2026-07 conditional novelty 5.0 of 10

    The Lambda_Q baryon LCDA at mass m_Q equals (m_Q/m_Q^0)^2 times the LCDA at m_Q^0 evaluated at rescaled fractions x_i*m_Q/m_Q^0, times an exponentiated anomalous dimension, plus renormalon-model power corrections.

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