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The paper derives the complete QCD×QED factorization theorem for B⁻→μ⁻ν̄(γ), and shows the photon-vetoed rate can be predicted at the percent level, including structure-dependent QED effects.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 09:15 UTC pith:FDV7AIWY

load-bearing objection A technically meticulous NLP SCET×QED factorization that is likely the new reference for B−→ℓν̄, but the percent-level accuracy claim rests on an all-orders subtraction assumption checked only at one loop, and the phenomenology was missing from the copy I saw. the 3 major comments →

arxiv 2601.14361 v2 pith:FDV7AIWY submitted 2026-01-20 hep-ph

The Simplest B Decay, Precisely

Claudia Cornella , Max Ferr\'e , Matthias K\"onig , Matthias Neubert This is my paper
classification hep-ph PACS 13.20.He
keywords B-meson leptonic decayQED correctionssoft-collinear effective theorynext-to-leading powerfactorizationresummation|V_ub|structure-dependent effects
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.

This paper establishes that the 'simplest' B-meson decay, B⁻→μ⁻ν̄, can be predicted at the percent level despite intricate electromagnetic corrections. It derives a factorization theorem that separates the many scales involved—weak, heavy-quark, hard-collinear, hadronic, lepton-mass, and photon-energy-cut—through a chain of effective field theories. The central result is an RG-improved amplitude for the photon-vetoed rate that includes the complete structure-dependent QED correction at next-to-leading power, with leading logarithms resummed to all orders. This matters because the channel is a clean probe of |V_ub| and of new scalar or pseudoscalar interactions, and future high-precision measurements are expected to reach 5–6% accuracy.

Core claim

Working from the effective weak Hamiltonian below the electroweak scale, the authors match onto soft-collinear effective theory at the hard scale, then onto a second variant at the hard-collinear scale, and finally onto heavy-hadron chiral perturbation theory below the QCD confinement scale. In doing so they construct the complete next-to-leading-power operator basis, introduce a Dirac reduction scheme that preserves d-dimensional identities and avoids power-enhanced evanescent operators, and use a refactorization-based subtraction scheme to systematically handle endpoint-divergent convolution integrals. The resulting factorization formula gives the virtual B⁻→μ⁻ν̄ amplitude with the complet

What carries the argument

The central machinery is a sequence of effective field theories: the weak-scale effective Hamiltonian, SCET-1 (soft-collinear effective theory with hard-collinear and soft modes) at the B-meson scale, SCET-2 below the hard-collinear scale, and heavy-hadron chiral perturbation theory augmented by boosted heavy-lepton effective theory below Λ_QCD. The load-bearing identity is the refactorization-based subtraction (RBS) scheme, which rewrites endpoint-divergent convolutions of hard and jet functions as plus-distribution integrals plus a subtraction term identified with the matrix element of a θ_T operator; exact refactorization conditions guarantee the subtraction term has the required form. To

Load-bearing premise

The argument rests on identifying the subtraction that removes endpoint-divergent convolutions with the matrix element of a step-function operator, and on assuming that identification holds to all orders in perturbation theory; if it fails beyond one loop, the factorization theorem no longer systematically removes the soft region and the percent-level claim collapses.

What would settle it

Compute the two-loop corrections to the refactorization condition and to the θ_T operator matrix element; if the endpoint behavior of the product of the hard and jet functions no longer matches the subtraction, the factorization theorem fails. On the experimental side, a future photon-vetoed B→μν rate measurement at roughly 5% accuracy that disagrees with the prediction by more than the quoted uncertainty would falsify the Standard Model treatment.

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

If this is right

  • The photon-vetoed B⁻→μ⁻ν̄ rate is known to about 1%, comparable to the expected future experimental accuracy, so the channel can deliver a clean determination of |V_ub|.
  • Structure-dependent QED corrections are no longer modeled but computed within a systematic EFT expansion, making percent-level comparisons meaningful.
  • The treatment of endpoint-divergent convolutions and the resummation of rapidity logarithms at next-to-leading power provides a template for other exclusive B-meson decays.
  • The indirect B→B*γ→ℓνγ channel and chiral-anomaly contributions must be included in analyses of the photon-vetoed rate.

Where Pith is reading between the lines

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

  • Editorial inference: the all-orders validity of the θ_T subtraction identification is the main structural assumption; a two-loop check would decisively test whether the factorization theorem holds at the claimed precision.
  • Editorial inference: in the presence of new physics with opposite-chirality couplings, QED corrections can be enhanced by a factor of order m_B/m_ℓ, so the SM baseline derived here sharpens the new-physics reach of the channel.
  • Editorial inference: the same EFT chain could be applied to the electron and tau channels, though the tau channel requires a different scale ordering and the electron channel is dominated by indirect contributions.
  • Editorial inference: the percent-level SM prediction provides a target for future lattice QCD calculations of QED corrections to B-meson decays, which do not yet exist.

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 / 4 minor

Summary. The paper develops a multi-scale QCD×QED factorization framework for the leptonic decay B^- → ℓ^- ν̄_ℓ(γ), with the muon channel as the main application. The construction proceeds through a chain of effective theories: LEFT → SCET-1 → SCET-2 → HHχPT⊗bHLET. The authors build the complete next-to-leading-power (NLP) operator basis in SCET-1 (26 physical plus evanescent operators), compute the hard and jet functions at one loop, treat endpoint-divergent convolution integrals with the refactorization-based subtraction (RBS) scheme, and resum leading QCD/QED logarithms. The final RG-improved virtual amplitude is given in Eqs. (4.99), (4.113), and (4.116). The low-energy section introduces a heavy-particle effective theory and HHχPT to describe real photon emission, including the indirect B→B^*γ→ℓν̄γ contribution. The abstract claims that the resulting photon-vetoed rate is accurate at the percent level, providing a clean determination of |V_ub| and a new-physics test.

Significance. If the central claim is correct, this is a landmark calculation: the first complete treatment of QED effects to an exclusive B-meson decay at next-to-leading power, with all-orders resummation of leading logarithms. The paper is unusually explicit and internally consistent: the κ-scheme dependence cancels between the LEFT matching and the SCET-1 hard functions, the subtraction scale Λ cancels at one loop (Eq. 4.115), and the RBS refactorization is checked at one loop. The technical apparatus — the complete SCET-1 operator basis, the 'SCET-friendly' Dirac reduction scheme, and the two-scale RG evolution of the hard function H_C1 — is a substantial contribution in its own right. The relationship (4.98) that eliminates the HQET parameter F_QCD in favor of the lattice-known f_B is a sensible and testable definitional step. However, the all-orders validity of the RBS subtraction/θ_T-operator identification is assumed rather than proved, and the numerical uncertainty budget needed to support the 'percent-level' claim is not accessible in the material under review.

major comments (3)
  1. [§4.4 (after Eq. 4.41) and §3.6 (Eq. 3.76)] The central all-orders claim relies on an assumption that is explicitly stated but verified only at one loop: 'we assume that the identification of the subtraction term with the matrix element of the θ_T operator holds to all orders of perturbation theory.' Likewise, the exact d-dimensional refactorization conditions (3.76) are asserted to hold to all orders but are checked only through the one-loop expression (3.77). Since the RG-improved amplitude (4.99)/(4.113)/(4.116) resums logarithms to all orders, a failure of these identifications beyond one loop would leave endpoint-divergent convolutions incompletely subtracted and invalidate the factorization formula at the claimed precision. A two-loop check of (3.76) and (4.41), or a symmetry/consistency argument that excludes uncontrolled Λ-dependent terms, is needed before the all-orders statement can be accepted. This is an internal-consi
  2. [§6 / Abstract] The headline 'accurate at the percent level' cannot be audited from the material provided: the numerical estimates and the uncertainty budget of Section 6 are not presented in the version under review (the text breaks off in §5.2). Even granting the formal factorization theorem, the size of the residual scale dependence (4.117) — estimated as O(αα_s ln, α^2 ln^2) — and the impact of the non-perturbative inputs (F_±, ω_- and the three-particle LCDA) must be quantified to support the percent-level statement. The scale-ambiguity estimate in footnote 16 appears to claim a partial cancellation of the O(αα_s ln^2) ambiguity without a detailed demonstration; a numerical scan over μ_0, Λ, and the LCDA parameters is required.
  3. [§4.9, Eq. (4.110) and (4.117)] The bootstrap of the unknown O(αα_s) homogeneous two-loop anomalous dimensions from the requirement of RG invariance is an ad hoc procedure. It assumes that the residual scale dependence has the specific form (4.110) and that no other O(αα_s) terms are present. The claim that the true scale ambiguity is reduced to O(αα_s ln) rests on this bootstrap and on the structure of (4.97); a direct two-loop calculation, or at least an independent estimate of the omitted O(αα_s) terms, would be needed to make the percent-level uncertainty claim credible. Without such a check, the numerical error estimate is itself a model-dependent input.
minor comments (4)
  1. [Eq. (4.101)] In the expression for T_C1, the three-particle LCDA appears as 'ϕB_3g(ω,ωgµ)' — a missing comma and the μ argument are typographical errors. The same function is written with the correct arguments in Eq. (4.83).
  2. [§3.3, Eq. (3.48)] The definition of the SCET-friendly reduction scheme is clear, but the text would benefit from an explicit statement that the scheme choice κ=0 is used consistently in all subsequent equations; the reader must otherwise infer this from the remark below (2.5).
  3. [Table 1] The dagger notation is explained, but the asterisk on 'Jet function' is only explained in a footnote. Consider moving these explanations into the table caption for clarity.
  4. [§3.7] The notation 'eUC(μ, mB)' in (3.97) and its distinction from 'UC(μ, mB)' in (3.92) is easy to miss; a sentence emphasizing that the tilde function is the ratio of the two evolution factors would help.

Circularity Check

0 steps flagged

No circularity found; the central derivation is self-contained, and the all-orders RBS assumption is a validity risk, not a circular reduction.

full rationale

I walked the derivation chain from the LEFT operator through SCET-1 → SCET-2 → HHχPT and the final factorization formula (4.99)/(4.113)/(4.116). The non-perturbative inputs — f_B from lattice, F_±(Λ,μ) from HQET parameterizations, ϕ_B^± and ϕ_B^{3g} from LCDA definitions — are not fitted to the target B→μν rate. The hard and jet functions are computed at one loop and RG-improved; their evolution equations are derived in the paper, not imported as an unexplained black box. The relation (4.98) between f_B and the hard functions is a standard HQET matching relation used to eliminate F_QCD, not a definition that builds the final prediction into the input. The endpoint-divergent convolutions are treated by the RBS scheme, and the subtraction terms are rewritten as matrix elements of the θ_T operator; the paper explicitly verifies this identification at one loop and then states the all-orders version as an assumption: 'we assume that the identification of the subtraction term with the matrix element of the θ_T operator holds to all orders of perturbation theory' (§4.4, after (4.41)). This is an unproven premise about higher orders, and the same is true for the refactorization conditions (3.76), which are checked only at one loop. That is a genuine limitation and a target for future two-loop verification, but it is not circular: the one-loop identification is not constructed by assuming the desired final answer. The Λ- and η-dependence cancel explicitly (4.115), which is a consistency check rather than a fit. Self-citations appear (e.g., 'first derived in [24]' and references to the RBS scheme), but the relevant ingredients are rederived here; they do not reduce the central claim to the cited authors' prior work. The κ-scheme dependence cancels, and no fitted parameter is renamed as a prediction. Accordingly, I find no step where the output equals an input by construction, and the circularity score is set to 0.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 1 invented entities

The derivation is anchored to external QCD inputs (f_B from lattice, LCDAs from QCD sum rules/models) and to internal consistency checks (κ- and Λ-cancellations). The genuinely paper-specific axioms are the all-orders RBS subtraction identification, the refactorization conditions, and the assumed validity of the QED-deformed hadronic parameter definitions. No free parameters are fitted to the decay rate itself; scale choices (μ_0 = 1.5 GeV, Λ) are declared and Λ cancels.

free parameters (3)
  • Common resummation scale μ_0 = 1.5 GeV (default)
    Choice of the common perturbative evaluation scale, varied by a factor √2 for scale uncertainty (§4.9, §6.2). Not fitted to data, but it sets the size of the resummed logarithms and enters the quoted accuracy.
  • RBS subtraction scale Λ (= η m_B)
    Introduced by hand in §4.4 to separate hard-collinear from soft momentum regions; shown to cancel at one loop (dR^virt/d ln Λ = 0). Not a fit parameter, but its all-orders treatment via the θ_T operator is assumed.
  • Logarithmic LCDA moment ω_-(μ_0)
    Non-perturbative input entering the type-C contribution, defined in (4.105) from the B-meson LCDA ϕ_B^-. Taken from external determinations, not fitted to the decay.
axioms (6)
  • ad hoc to paper The identification of RBS subtraction terms with the θ_T-operator matrix element holds to all orders of perturbation theory.
    §4.4, after (4.41): 'we assume that the identification of the subtraction term with the matrix element of the θ_T operator holds to all orders.' Load-bearing for the factorization theorem.
  • ad hoc to paper The exact d-dimensional refactorization conditions (3.76) — J H^C_1 K = H^A_1 S^C_1 and J H^C_2 K = −H^A_2/y S^C_2 — hold to all orders.
    §3.6; without them the endpoint-divergent convolutions cannot be systematically subtracted and reabsorbed into the type-A operators.
  • domain assumption SCET-1/SCET-2 mode separation and decoupling: hard-collinear, soft, and collinear fields with the stated power counting; soft×collinear interactions forbidden in SCET-2.
    §3.1 and §4.1–4.2, invoking the standard SCET framework [67–70, 80–82].
  • domain assumption Below Λ_QCD, heavy-hadron chiral perturbation theory with a point-like (B, B*) doublet, heavy-quark spin symmetry, and Low's theorem for ultrasoft photon emission.
    §5.1–5.2; needed for the direct/indirect real-photon contributions and for the matching of SCET-2 onto the low-energy theory.
  • standard math Dimensional regularization with the 'SCET-friendly' Dirac reduction scheme (κ=0) is a consistent evanescent-operator scheme; physical results are scheme independent.
    §3.3 and Appendix B; the paper shows κ-dependence cancels between K_EW and H_1^A.
  • ad hoc to paper The unknown O(αα_s) homogeneous two-loop anomalous dimensions are fixed by bootstrapping from RG invariance.
    §3.7 and §4.9: 'the corresponding corrections to the homogeneous terms... are not yet known' and are bootstrapped; residual scale dependence O(αα_s ln) is acknowledged in (4.117).
invented entities (1)
  • θ_T-subtracted heavy-light current operator Q^A_{1,θ} no independent evidence
    purpose: Removes hard-collinear momentum modes (n̄·(−l_s) > Λ) from the soft spectator-quark matrix element, cancelling the RBS endpoint subtractions.
    Defined in §4.4 (4.42) with a Taylor-expansion prescription; its matrix element defines the QED-modified decay constant F_-(Λ,μ). It is a technical construct with no independent falsifiable handle of its own.

pith-pipeline@v1.3.0-alltime-deepseek · 69655 in / 16213 out tokens · 169102 ms · 2026-08-03T09:15:13.429544+00:00 · methodology

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read the original abstract

We derive the QCD$\times$QED factorization theorem governing the leptonic decay $B^-\to\mu^-\bar\nu_\mu(\gamma)$ at all orders in $\alpha_s$ and $\alpha$. Electromagnetic corrections to this decay probe multiple scales, which we disentangle through a sequence of effective field theories (EFTs). The resulting state-of-the-art prediction for the photon-vetoed rate includes the complete structure-dependent component and is accurate at the percent level, establishing the theoretical framework required for future high-precision measurements of this channel, which will allow for a clean determination of $|V_{ub}|$ and powerful tests of new physics. Our work presents the first complete study of QED effects to an exclusive $B$-meson decay at next-to-leading power (NLP) in the heavy-quark expansion. Important milestones are (i) the construction of the complete NLP operator basis in soft-collinear effective theory (SCET); (ii) the proposal of a "SCET-friendly" reduction scheme for the Dirac structures of four-fermion operators in dimensional regularization, which avoids power-enhanced evanescent operators; (iii) the consistent refactorization of endpoint-divergent convolution integrals and the first complete resummation of "rapidity logarithms" arising at the boundary between the contributions involving soft and hard-collinear quarks; (iv) the systematic discussion of the EFT below the scale of QCD confinement and the non-perturbative matching of SCET onto this low-energy theory; (v) the decoupling of pseudoscalar mesons in the context of heavy-hadron chiral perturbation theory, so that they can be integrated out for processes in which they do not appear as external particles. We perform a phenomenological analysis of direct and indirect contributions to the decay rate and radiation-energy spectrum, highlighting the importance of the chiral anomaly.

Figures

Figures reproduced from arXiv: 2601.14361 by Claudia Cornella, Matthias K\"onig, Matthias Neubert, Max Ferr\'e.

Figure 1
Figure 1. Figure 1: Illustration of the appropriate effective theories below the electroweak scale to treat the B− → ℓ −ν¯ process for the cases ℓ = µ, τ, e. In the case of the τ lepton, two options are shown, in which the lepton is either treated as a hard-collinear or a hard particle. The dots in the electron case indicate the scales me and Ecut (me/mB), which are both much smaller than Ecut. relevant expansion parameters a… view at source ↗
Figure 2
Figure 2. Figure 2: Ingredients of the B− → µ − ν¯µ factorization formula and their respective natural scales. Solid arrows denote RG evolution to a common renormalization scale µ0. Dashed arrows indicate that the corresponding component functions are evaluated at µ = µ0 without resummation. function, which also depends on Ecut. As a consequence, they turn out to be relevant for the cases where ℓ = µ, e. Real photons emitted … view at source ↗
Figure 3
Figure 3. Figure 3: Tree-level and one-loop Feynman diagrams for the LEFT to SCET-1 matching for the type-A four-fermion operators. There is also a contribution from the wave-function renormalization of the b quark. u →  1 − 1 in¯ · ∂ n/¯ 2 [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Hard two-loop diagrams which can generate the operator OE 4 in SCET-1. The light quark, the charged lepton and the external gluons carry hard-collinear momenta. H D 6 (y1, y2) = − z y1 + O(αs), H D 7 (y1, y2) = zQb 1 − y1 − y2 + O(αs), H D 8 (y1, y2) = z 1 − y1 − y2 + O(αs). At O(αs), these coefficients receive contributions from yet unknown higher-order power cor￾rections to the heavy-light currents in SC… view at source ↗
Figure 5
Figure 5. Figure 5: Left: One-loop diagrams leading to a mixing of the operators OB i and OA i (with i = 1, 2). The lepton mass insertion is not shown explicitly. Right: One-loop diagram leading to a mixing of the operators OC i and OA i (with i = 1, 2). Here and in later figures, red lines refer to hard-collinear fields, gray lines to soft fields, and light blue lines to collinear or anti-collinear fields. The soft heavy-qua… view at source ↗
Figure 6
Figure 6. Figure 6: One-loop diagrams needed for the calculation of the diagonal renormalization factors Z AA ii . These graphs need to be supplemented by wave-function renormalization. this is an IR scale in SCET-1. We will see later how this puzzle is resolved. At O(α) there is also a mixing of the type-C operators with the type-B operators, but the corresponding contributions to our process of interest would only appear at… view at source ↗
Figure 7
Figure 7. Figure 7: Representative hard-collinear loop diagrams for the matching of type-B SCET-1 operators onto SCET-2. Red (dark blue) lines represent hard-collinear (soft) propagators and external states. Light blue lines represent collinear external states. The double line at the vertex represents the b-quark and neutrino fields. least two power-suppressed interactions, yielding SCET-2 operators which are suppressed by (a… view at source ↗
Figure 8
Figure 8. Figure 8: One-loop exchange of a hard-collinear photon, converting a hard-collinear to a soft quark. Red (dark blue) lines represent hard-collinear (soft) propagators and external states. Light blue lines represent collinear external states. fields along with (on-shell) collinear fields and soft fields, and then integrating out the off-shell collinear fields [81, 82]. Here we use symmetry arguments to constrain the … view at source ↗
Figure 9
Figure 9. Figure 9: Representative hard-collinear loop diagrams for the matching of the type-C SCET￾1 operators onto SCET-2. Red (dark blue) lines represent hard-collinear (soft) propagators and external states. Light blue lines represent collinear external states. The double line at the vertex represents the b-quark and neutrino fields. Type-A and type-B SCET-1 operators For these operators we find the non-zero matching rela… view at source ↗
Figure 10
Figure 10. Figure 10: Representative hard-collinear loop diagrams for the matching of type-D SCET-1 oper￾ators (upper row) and type-E SCET-1 operators (lower row) onto SCET-2. Red (dark blue) lines represent hard-collinear (soft) propagators and external states. Light blue lines represent collinear external states [PITH_FULL_IMAGE:figures/full_fig_p052_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Representative hard-collinear loop diagrams for the matching of type-F SCET-1 oper￾ators onto SCET-2. Red (dark blue) lines represent hard-collinear (soft) propagators and external states. Light blue lines represent collinear external states. of the operator contains a photon loop. Note that in each case the hard-collinear photon exchange between the up quark and the charged lepton is needed to transfer t… view at source ↗
Figure 12
Figure 12. Figure 12: Tree-level and one-loop diagrams contributing to the perturbative calculation of the matrix element in (4.41) in the region where Λ ≫ ΛQCD. The tree diagram vanishes in this region. Diagrams in which a gluon is emitted from the current (not shown) give vanishing contributions. We find that something analogous happens also in the present case, in which the soft matrix elements are non-perturbative hadronic… view at source ↗
Figure 13
Figure 13. Figure 13: Decay topologies describing the B− → ℓ − ν¯ℓγ process at energies far below Λc. The photon energy is restricted to be less than Ecut ≪ Λc in the B-meson rest frame. The crossed circle indicates the weak interaction vertex. In the third graph, the B meson transitions to an excited meson Xb via the emission of a photon. The black square indicates that the corresponding vertex is a power-suppressed interacti… view at source ↗
Figure 14
Figure 14. Figure 14: Feynman diagrams describing QCD corrections to the B− → ℓ −ν¯ℓ process at O(α) in the heavy-particle effective theory. For clarity, we label the lines by the corresponding EFT fields. A crossed circle indicates the weak interaction vertex, while a black square denotes a power-suppressed interaction. In the first and third graphs the photons are emitted from the Wilson lines contained in the effective weak… view at source ↗
Figure 15
Figure 15. Figure 15: Representative one-loop diagrams in HHχPT giving rise to higher-order corrections to the heavy-meson decay constant FQCD (first graph) and the BB∗γ coupling (second graph), and “genuine” one-loop corrections to the B− → ℓ − ν¯ℓγ decay amplitude (last two diagrams). Contri￾butions from local higher-order operators in HHχPT, which are needed as counterterms for these diagrams, exist but are not shown. A cro… view at source ↗
Figure 16
Figure 16. Figure 16: One-loop diagrams contributing to the wave-function renormalization of the pseudo￾scalar heavy-meson fields φ a v (additional tadpole graphs contribute to the self-energy but not to the wave-function renormalization). The label π c represents the pseudoscalar mesons π, K, η8, while ρ b v and φ b v represent B∗ and B mesons of different flavors. This result is identical to the corresponding expression obta… view at source ↗
Figure 17
Figure 17. Figure 17: Indirect decay topologies contributing to B− → ℓ − ν¯ℓ (γ) decay rate, which arise at leading order in the low-energy theory. In the first graph, a photon is emitted through the effective BB∗γ coupling. In the second graph an on-shell pion is first emitted, which subsequently decays into two photons. where q denotes the ultrasoft photon 4-momentum.22 The effective heavy-lepton spinor u(vℓ) is normalized t… view at source ↗
Figure 18
Figure 18. Figure 18: Dependence of the functions I(0, δB∗ /Ecut) (blue) and I(mπ/Ecut, δB∗ /Ecut) (red) on the photon energy cut. Both functions approach 1 from below in the limit Ecut → ∞. in agreement with [27]. Our HHχPT construction thus corresponds to the first-vector pole approximation for the form factors. Corrections to these relations have energy denominators in which δB∗ is replaced by δXb ∼ Λc and thus give contrib… view at source ↗
Figure 19
Figure 19. Figure 19: Left panel: The B− → µ − ν¯µ decay rate including QED corrections and resummation effects, normalized to the leading-order rate. The red curve shows the full rate including the indirect contributions from B → B∗γ and B → B∗π 0 transitions, while the blue curve shows only the direct contribution for comparison. Shaded areas indicate the combined uncertainties. The gray lines show the rate excluding the pio… view at source ↗
Figure 20
Figure 20. Figure 20: Different contributions to the radiation-energy spectrum Q(Erad) as a function of the radiation energy. The three curves show the direct contribution (blue), the indirect B∗γ contribution (green), and the full spectrum (red) including also the B∗π contribution. Shaded areas denote the uncertainty. The contribution from the BB∗π interaction is absent for values of Ecut below the neutral pion mass, but then… view at source ↗

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