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REVIEW 3 major objections 6 minor 42 references

Radiative B to tensor meson decays at NLO in SCET

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper computes NLO SCET branching ratios for radiative B decays to tensor mesons, finding B → K₂*γ close to experiment and CKM-suppressed rates near 10⁻⁷.

desk verdict A clean SCET extension to B->tensor-meson gamma, but the headline number leans on an imported soft form factor and an unexplained LCDA input. read the letter →

arxiv 2505.18624 v2 pith:LSJ2RPRT submitted 2025-05-24 hep-ph

Arslan Sikandar , M. Jamil Aslam This is my paper
classification hep-ph
keywords radiativeBdecaystensormesonssoft-collineareffectivetheorynext-to-leadingorderbranchingratiossoft-overlapfunctiontoK2*gammatwo-stepmatching
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

The paper aims to establish that radiative $B$-meson decays to tensor mesons (spin-2 final states) can be computed systematically in soft-collinear effective theory (SCET) at next-to-leading order in $\alpha_s$ and at leading power in $1/m_b$. The authors perform a two-step matching from QCD to SCET I and then to SCET II, which separates the amplitude into a soft-overlap (non-factorizable) part and a hard-spectator (factorizable) part, and they resum the large logarithms between the hard and intermediate scales. Their central numerical result is a branching ratio for $B \to K_2^*(1430)\gamma$ of $(16.7^{+6.36}_{-6.36} \pm 1.2)\times 10^{-6}$, which lies close to the measured average of $(12.4 \pm 2.4)\times 10^{-6}$, together with predictions of $(0.67^{+0.30}_{-0.23} \pm 0.06)\times 10^{-6}$ and $(0.18^{+0.23}_{-0.16} \pm 0.08)\times 10^{-6}$ for the CKM-suppressed channels $B \to a_2\gamma$ and $B \to f_2\gamma$. A curious reader should care because the calculation extends the SCET program to spin-2 final states and shows that the leading-power factorization, with the soft-overlap function extracted from light-cone sum rules, reproduces the measured $B \to K_2^*\gamma$ rate within uncertainties.

What carries the argument

The central object is the SCET factorization formula of Eq. (1), $\langle T\gamma|Q_i|B\rangle = C_i^I \zeta^{B\to T} + \int_0^\infty \frac{d\omega}{\omega}\phi_B(\omega)\int_0^1 du\, \phi_T(u)\, C_i^{II}(\omega,u)$, which splits the amplitude into a soft-overlap function $\zeta_T^\perp$ (non-factorizable) and a convolution of distribution amplitudes (factorizable). The two-step matching produces the Wilson coefficients $C^A$ and $C^{B'}$; the SCET I to SCET II step introduces the jet function $J_\perp$ and the evolution kernel $U_\perp$, which resum the large logarithms between the hard scale $m_b$ and the intermediate scale $\sqrt{2E\Lambda_{\rm QCD}}$. The soft-overlap function is fixed by Eq. (46), $V(q^2=0)=(1+m_T/m_B)(E_F/|\vec p_T|) C_{V1}^A \zeta_T^\perp$, using the light-cone-sum-rule value of the vector form factor.

What would settle it

A lattice QCD calculation of the $B \to K_2^*$ vector form factor $V(q^2=0)$ (or a direct determination of $\zeta_T^\perp$) that differs from the light-cone-sum-rule value by more than about 20% would shift the predicted $B \to K_2^*\gamma$ branching ratio by more than the experimental uncertainty, breaking the current agreement; likewise, a future measurement of $B \to a_2\gamma$ above roughly $2\times 10^{-7}$ would challenge the CKM-suppressed prediction.

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Extended reading notes

Core claim

The paper claims that the full QCD amplitude for $B \to T \gamma$ ($T = K_2^*(1430), a_2, f_2$) factorizes at leading power as a soft-overlap function term plus a convolution of the $B$-meson and tensor-meson light-cone distribution amplitudes, with the matching done in two steps (QCD to SCET I, then SCET I to SCET II). The non-factorizable soft-overlap function $\zeta_T^\perp$ is not computed from first principles but is extracted from the vector form factor $V(q^2=0)$ obtained in light-cone sum rules, through the heavy-quark-symmetry relation of Eq. (46). With this input, the predicted branching ratios are $(16.7^{+6.36}_{-6.36} \pm 1.2)\times 10^{-6}$ for $B \to K_2^*(1430)\gamma$, $(0.67^{+0.30}_{-0.23} \pm 0.06)\times 10^{-6}$ for $B \to a_2\gamma$, and $(0.18^{+0.23}_{-0.16} \pm 0.08)\times 10^{-6}$ for $B \to f_2\gamma$. The $B \to K_2^*\gamma$ value is closer to the experimental average than the earlier light-front quark model and heavy-quark effective theory estimates quoted in the paper, and the hard-spectator term contributes about 26% of the soft amplitude, with the resummation shifting the rate by 6.5%.

Load-bearing premise

The load-bearing premise is that the soft-overlap function $\zeta_T^\perp$, extracted from the light-cone-sum-rule vector form factor at $q^2=0$ through the heavy-quark-symmetry relation of Eq. (46), is the same non-perturbative object entering the radiative decay amplitude; the prediction is highly sensitive to this input, and its uncertainty dominates the error budget.

Editorial extensions

If this is right

  • If the prediction is correct, the measured $B \to K_2^*(1430)\gamma$ rate already constrains the soft-overlap function $\zeta_T^\perp$, so a sharper experimental average would directly reduce the dominant theory uncertainty.
  • The CKM-suppressed channels $B \to a_2\gamma$ and $B \to f_2\gamma$ are predicted at the $10^{-7}$ level; a measured rate significantly above roughly $2\times 10^{-7}$ would point to new physics in $b \to d\gamma$ transitions.
  • The same two-step SCET matching can be applied to other heavy-to-light transitions producing tensor mesons, such as semileptonic $B \to K_2^*\ell^+\ell^-$ decays, where $\zeta_T^\perp$ enters at $q^2=0$.
  • The hard-spectator term is about a quarter of the soft amplitude, so a leading-power calculation that omitted either the convolution term or the resummation would mis-estimate the branching ratio by tens of percent.

Reading between the lines

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

  • Because the soft-overlap function is a universal SCET quantity, a lattice QCD determination of $V(q^2=0)$ or of $\zeta_T^\perp$ would simultaneously sharpen all three predicted branching ratios and remove the largest theory error in this paper.
  • Ratios such as $\mathrm{BR}(B\to a_2\gamma)/\mathrm{BR}(B\to K_2^*\gamma)$ partially cancel the soft-overlap function and test the CKM ratio $|V_{td}/V_{ts}|^2$; that ratio would be a cleaner probe of the factorization than the absolute rates.
  • The same framework could be used for the isospin asymmetry in $B\to K_2^*\gamma$, where the soft-overlap function cancels to leading power, providing a direct test of the NLO matching and the resummation kernels.
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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

3 major / 6 minor

Summary. The paper presents a next-to-leading-order (NLO) soft-collinear effective theory (SCET) analysis of the radiative decays B -> T gamma, with T = K_2^*(1430), f_2(1270), and a_2(1320). The authors perform the two-step matching QCD -> SCET_I -> SCET_II, classify the relevant A-, B'-, and C-type operators, resum the large logarithms with the SCET_I anomalous dimension using a Jacobi-polynomial basis, and express the B -> T gamma amplitude as a soft-overlap term plus a hard-spectator convolution, Eq. (42). The soft-overlap function zeta_T^perp is extracted from the LCSR vector form factor V(0) through the heavy-quark-symmetry relation Eq. (46). The resulting branching ratios, Table 3, are (16.7^{+6.36}_{-6.36} +/- 1.2) x 10^-6 for B -> K_2^*(1430) gamma, to be compared with the measured average (12.4 +/- 2.4) x 10^-6, and (0.67^{+0.30}_{-0.23} +/- 0.06) x 10^-6 and (0.18^{+0.23}_{-0.16} +/- 0.08) x 10^-6 for B -> a_2 gamma and B -> f_2 gamma, respectively.

Significance. The formal machinery of the paper is sound: the matching procedure follows the Becher-Hill-Neubert framework and the authors' earlier SCET paper for axial-vector mesons, and the spin-2 matrix-element construction, Eqs. (37)-(39), is a non-trivial and plausible extension. If the numerical results hold, the paper provides the first NLO SCET branching-ratio predictions for radiative B-to-tensor-meson decays, with the B -> K_2^* gamma central value within about 1.4 sigma of experiment, and CKM-suppressed channels that may be testable in future experiments. However, the numerical headline is dominated by the non-perturbative soft-overlap function imported from LCSR, so the paper's lasting value is the extension of the SCET matching and resummation formalism to spin-2 final states rather than an ab initio prediction. The paper is commendably explicit about the resummation details (Fig. 5 with 80 basis functions) and about the 26%/6.5% decomposition of the hard-spectator and RG-improvement effects.

major comments (3)
  1. [Sec. 6, Eqs. (42), (46)-(48); Table 3] The headline branching ratio is dominated by the soft-amplitude term: the paper states that the hard-spectator contribution is only 26% of the soft-only value. The soft-overlap function zeta_T^perp is not computed but extracted from the LCSR vector form factor V(0) of Ref. [11] via the leading-power heavy-quark-symmetry relation Eq. (46). Only the quoted LCSR uncertainty of V(0) is propagated into zeta_T^perp; no estimate is provided for the systematic error of the HQS/leading-power relation itself for tensor mesons, or for the accuracy of identifying the SCET soft function with the full-QCD form factor at q^2 = 0. The consistency check quoted after Eq. (48) compares the vector- and tensor-current extractions, which share the same leading-power approximation, so it does not test the common assumption. Because an O(Lambda_QCD/m_b) violation of Eq. (46) would shift the central branching ratio by more than the quoted experimental uncertainty, the paper should either quantify this systematic or present the result as conditional on the LCSR+HQS input.
  2. [Table 2; Eq. (45)] The input a_1^perp = a_1^parallel = 5/3 is given without a source or explanation. With this value, the leading-twist LCDA phi_T^perp(u) = 6 u u-bar [1 + 3 a_1 (2u-1)] is negative on a substantial part of the domain (at u = 0.2 it equals about -1.92), and the stated normalization integral of phi^perp(u)/u from 0 to 1 is violated, evaluating to -2 instead of 1. Since the hard-spectator term in Eq. (42) is proportional to phi_T^perp, the computed 26% NLO hard-scattering correction is not reliable as it stands. The authors should either supply the correct Gegenbauer moments from Ref. [10] with a consistent normalization or justify the 5/3 value and test the sensitivity of the hard-spectator amplitude to this parameter.
  3. [Eq. (42), Table 3] The error budget in Table 3 contains only the zeta_T^perp uncertainty and the hadronic (inverse B-moment and decay-constant) uncertainty. For a paper whose central claim is an NLO calculation, the residual scale dependence of the RG-improved amplitude, in particular the variation of the hard scale mu_Q and the intermediate scale mu_i entering Eqs. (42) and (29), should be quoted; this also bears on the statement that the RG improvement changes the branching ratio by 6.5%.
minor comments (6)
  1. [Abstract] The abstract refers to a_2(1230), whereas the title, the running text, and Table 2 use a_2(1320); the mass label should be unified.
  2. [Eq. (40)] The integrand contains a typographical artifact, 'phi_B(omega, mu, )', with a stray comma after mu.
  3. [Sec. 2, Eq. (4)] The statement that the operator Q2 contributes only at alpha_s^2 needs clarification: in the standard current-current basis both Q1 and Q2 generate the charm penguin at one loop in the matching onto b -> s gamma. Please specify the operator basis (for instance the CMM basis or the basis of Ref. [42]) and state how the Q2 contribution is either absorbed into G1(x_c) in Eq. (43) or shown to be suppressed.
  4. [References] The reference list contains a stray bare '[4]' between the Faustov-Galkin entry and the Khodjamirian entry, which duplicates the number; please renumber the list.
  5. [Table 2] The table lists inputs without a column of sources; please add references for each input, and state the scale at which a_1^perp and a_1^parallel are evaluated, as is already done for the decay constants.
  6. [Sec. 1] The relation of the present calculation to the earlier LEET treatment of the same decays in Ref. [15] should be stated explicitly in the introduction, since the reader has to infer the new technical content.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the soft-overlap input is taken from an independent LCSR calculation, and the SCET hard-scattering amplitude is a genuine additional NLO contribution.

full rationale

The branching-ratio calculation is admittedly dominated by the soft-overlap function zeta_T^perp, but the paper does not fit zeta to the B -> K2* gamma branching-ratio data. Instead, zeta is extracted from the LCSR vector form factor V(q^2=0) of Aliev et al. [11] through the heavy-quark-symmetry relation in Eq. (46), which is an independent external input. The paper states this transparently: "The soft overlap function, zeta_T^perp, which is also the non-factorizable part and required as a nonperturbative input, is estimated from the form-factor values calculated in [11] by using Light-cone sum rules (LCSR)." This is a standard use of an external nonperturbative input, not a fitted parameter renamed as a prediction. The second term in Eq. (42) is a genuine NLO SCET hard-scattering contribution; the paper estimates it as 26% of the soft-only value, so it is not forced by the soft input. The tensor-meson LCDAs, inverse B-moment, and decay constants are likewise external inputs with stated uncertainties. The only same-author citation that enters the zeta extraction is [15] for the HQS relation, but that relation is written explicitly in Eqs. (46)-(47) as a conventional leading-power symmetry relation, and the paper also reports an internal 5% cross-check using a tensor-current determination. Any concern about the accuracy of the leading-power relation or about the dominance of the zeta input is a model-uncertainty/correctness issue, not a circularity by construction. The result is also compared against the experimental average and other independent calculations, so the paper is self-contained against external benchmarks.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The calculation relies on the standard SCET factorization theorem, a heavy-quark-symmetry relation connecting the vector form factor to the soft-overlap function, and several input parameters from prior LCSR and lattice determinations. No new theoretical entities are introduced.

free parameters (5)
  • ζ^⊥_{K2*}(0) = 0.295 ± 0.056
    Extracted from the LCSR vector form factor V(0) via Eq. (46); the soft term dominates the branching ratio.
  • ζ^⊥_{a2}(0) = 0.288 +0.06/-0.05
    Same extraction from LCSR for a2(1320).
  • ζ^⊥_{f2}(0) = 0.185 +0.12/-0.08
    Same extraction from LCSR for f2(1270).
  • λ_B^{-1} = 0.35 ± 0.10 GeV^-1
    Optimal value chosen from Beneke et al.; enters the hard-scattering contribution.
  • a_1^⊥ = a_1^∥ = 5/3
    Given in Table 2 without explanation; if taken literally, makes the LCDA negative over part of its support; likely a typo.
assumptions (4)
  • domain assumption The SCET factorization in Eq. (1) is valid at leading power in 1/m_b for B-to-tensor-meson transitions.
    The entire calculation is built on this factorization; standard in the field but not proved here.
  • domain assumption The soft-overlap function ζ^⊥_T extracted from the vector form factor V(q²=0) via Eq. (46) is the same object appearing in the radiative decay.
    Assumed heavy-quark symmetry relation from [15]; the branching ratio depends on this equation.
  • standard math The anomalous dimensions and evolution kernels of SCET I are taken from [22] without rederivation.
    The NLO running relies on these known results.
  • domain assumption Outgoing light quarks are massless (m_s,d < Λ_QCD), so no extra regulator is needed.
    Section 2, used to justify absence of soft-collinear mode.

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Cite this review

Pith. "Pith review of Radiative B to tensor meson decays at NLO in SCET." pith.science (2026). https://pith.science/paper/LSJ2RPRT

@misc{pith2026250518624,
  author       = {Pith},
  title        = {Pith review of: Radiative B to tensor meson decays at NLO in SCET},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LSJ2RPRT}},
  note         = {Machine review of arXiv:2505.18624}
}
abstract

The radiative $B$ to tensor $\left(K_2^*(1430),\; f_2(1270),\; a_2(1230)\right)$ meson decays are studied at next-to-leading order (NLO) in soft-collinear effective theory (SCET). The SCET allows the systematic treatment of factorizable and non-factorizable contributions along with the resummation of large perturbative logarithms. We performed a two step matching and determined the soft-overlap function $\zeta^\perp_{T}$ and branching ratios for these $B$ to tensor meson decays. In the case of $B\to K_2^*(1430)\gamma$, the numerical value of the branching ratio lies close to its experimental measurements. The estimated values of the branching ratios of CKM suppressed decays $B \to \left(a_2(1230),\; f_2(1270)\right)\gamma$ are significant small compared to that of $B\to K_2^*(1430)\gamma$, but still could be measured in some ongoing and future $B$ physics experiments.

Figures

Figures reproduced from arXiv: 2505.18624 by the authors.

Figure 1
Figure 1. QCD diagrams for Q7 operator matching onto J A ,J B′ 1 and J B′ 2 , respectively [8]. There could be a contribution from another set of diagrams with photon-emission from the spectator quark. The relevant SCETI operator for these diagrams (C −type matching) is a four￾quark operator. Together with its WC, we can write J C (s1, r, a) = ¯χhc(s1n¯)(1 + γ5) n/¯ 2 χhc(rn¯) ¯χhc¯ (an)(1 + γ5) n/ 2 h(0) C C k (u) = Z ds1 Z … view at source ↗
Figure 2
Figure 2. QCD diagrams for Q1 operator matching onto J A ,J B′ 1 and J B′ 2 , respectively [8] . (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. QCD diagrams for Q8 operator matching onto J A ,J B′ 1 and J B′ 2 , respectively [8]. The matching of Q1 operator with a charm quark loop (setting mu = 0) for A −type matching (Fig (2a)), and B′−type matching (Fig (2b,2c)) give the following WCs ∆1C A = αsCF 4π Gi(xc)∆7C A , ∆1C B′ 1 (u) = −∆ q 1C B′ 2 (u) = 2e 3 f  m¯ 2 c 4¯uEEγ  ∆7C B′ 1 , (20) where xc = ¯m2 c/m2 b and the functions G1(xc) and f  m2 c 4EEγu¯ … view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: SCETI diagram matching onto SCETII 4-quark operator. The dashed gluon is hard￾collinear[8]. the calculation of the one-loop anomalous dimension by keeping the UV divergent terms appearing in SCETI loop diagrams (c.f [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Evolution function with blue being Uk and orange being U⊥ using 80 basis functions. V⊥,k ) is used to construct the solution for Eq. (32). This solution is plotted in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

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Works this paper leans on

42 extracted references · 12 canonical work pages

  1. [11]

    T. M. Aliev, H. Dag, A. Kokulu and A. Ozpineci, Phys. Rev. D 100, no.9, 094005 (2019) doi:10.1103/PhysRevD.100.094005 [arXiv:1908.00847 [he p-ph]]

    arXiv 2019

  2. [10]

    K. C. Yang, Phys. Lett. B 695, 444-448 (2011) doi:10.1016/j.physletb.2010.11.053 [arXiv:1010.2944 [hep-ph]]

    work page Pith review arXiv 2011

  3. [1]

    Misiak, H

    M. Misiak, H. Asatrian, R. Boughezal, M. Czakon, T. Ewert h, A. Ferroglia, P. Fiedler, P. Gam- bino, C. Greub, U. Haisch, T. Huber, M. Kaminski, G. Ossola, M . Poradzinski, A. Rehman, T. Schutzmeier, M. Steinhauser and J. Virto, Phys. Rev. Lett . 114, no.22, 221801 (2015) doi:10.1103/PhysRevLett.114.221801 [arXiv:1503.01789 [hep-ph]]

    arXiv 2015

  4. [2]

    P. Ball, G. W. Jones and R. Zwicky, Phys. Rev. D 75, 054004 (2007) doi:10.1103/PhysRevD.75.054004 [arXiv:hep-ph/0612081 [hep-ph]]

    arXiv 2007

  5. [3]

    R. N. Faustov and V. O. Galkin, Mod. Phys. Lett. A 7, 2111-2117 (1992) doi:10.1142/S0217732392001853 [4]

    work page doi:10.1142/s0217732392001853 1992

  6. [4]

    Khodjamirian, T

    A. Khodjamirian, T. Mannel, A. A. Pivovarov and Y. M. Wang , JHEP 09 (2010), 089 doi:10.1007/JHEP09(2010)089 [arXiv:1006.4945 [hep-ph] ]

    arXiv 2010

  7. [5]

    Oshimo, Nucl

    N. Oshimo, Nucl. Phys. B 404, 20-41 (1993) doi:10.1016/0550-3213(93)90471-Z

    work page doi:10.1016/0550-3213(93)90471-z 1993

  8. [6]

    M. Jung, X. Q. Li and A. Pich, JHEP 10, 063 (2012) doi:10.1007/JHEP10(2012)063 [arXiv:1208.1251 [hep-ph]]

    arXiv 2012

Show all 42 references
  1. [7]

    Sikandar, M

    A. Sikandar, M. J. Aslam, I. Ahmed and S. Shafaq, Phys. Rev . D 100, no.5, 056013 (2019) doi:10.1103/PhysRevD.100.056013 [arXiv:1909.04384 [he p-ph]]. 14

    2019 arXiv

  2. [8]

    Sikandar, M

    A. Sikandar, M. J. Aslam, I. Ahmed and S. Shafaq, J. Phys. G 48 (2021) no.4, 045005 doi:10.1088/1361-6471/abdadd

    2021 doi

  3. [9]

    Wang, Phys

    W. Wang, Phys. Rev. D 83, 014008 (2011) doi:10.1103/PhysRevD.83.014008 [arXiv:1 008.5326 [hep-ph]]

    2011 doi

  4. [12]

    H. Y. Cheng, C. K. Chua and C. W. Hwang, Phys. Rev. D 69, 074025 (2004) doi:10.1103/PhysRevD.69.074025 [arXiv:hep-ph/0310359 [hep-ph]]

    2004 arXiv

  5. [13]

    H. Y. Cheng and C. K. Chua, Phys. Rev. D 81, 114006 (2010) [erratum: Phys. Rev. D 82, 059904 (2010)] doi:10.1103/PhysRevD.81.114006 [arXiv:0 909.4627 [hep-ph]]

    2010 doi

  6. [14]

    Veseli and M

    S. Veseli and M. G. Olsson, Phys. Lett. B 367, 309-316 (1996) doi:10.1016/0370-2693(95)01408- X [arXiv:hep-ph/9508255 [hep-ph]]

    1996 arXiv

  7. [15]

    Sikandar, M

    A. Sikandar, M. J. Aslam, I. Ahmed and S. Shafaq, J. Phys. G 49 (2022) no.10, 105002 doi:10.1088/1361-6471/ac8567

    2022 doi

  8. [16]

    R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022) doi:10.1093/ptep/ptac097

    2022 doi

  9. [17]

    Bauer, D

    C.W. Bauer, D. Pirjol, I.W. Stewart, Phys. Rev. D 67(200 3)071502, hep-ph/0206152

    arXiv

  10. [18]

    Beneke and T

    M. Beneke and T. Feldmann, Nucl. Phys. B 685, 249-296 (2004) doi:10.1016/j.nuclphysb.2004.02.033 [arXiv:hep-ph/0311335 [hep-ph]]

    2004 arXiv

  11. [19]

    B. O. Lange and M. Neubert, Nucl. Phys. B 690, 249-278 (2004) [erratum: Nucl. Phys. B 723, 201-202 (2005)] doi:10.1016/j.nuclphysb.2005.06.019 [ arXiv:hep-ph/0311345 [hep-ph]]

    2004 arXiv

  12. [20]

    Charles, A

    J. Charles, A. Le Yaouanc, L. Oliver, O. Pene and J. Rayna l, Phys. Rev. D 60, 014001 (1999) doi:10.1103/PhysRevD.60.014001 [arXiv:hep-ph/9812358 [hep-ph]]

    1999 arXiv

  13. [21]

    Becher, R

    T. Becher, R. J. Hill and M. Neubert, Phys. Rev. D 72, 094017 (2005) doi:10.1103/PhysRevD.72.094017 [hep-ph/0503263]

    2005 arXiv

  14. [22]

    R. J. Hill, T. Becher, S. J. Lee and M. Neubert, JHEP 0407, 081 (2004) doi:10.1088/1126- 6708/2004/07/081 [hep-ph/0404217]

    2004 arXiv

  15. [23]

    A. Ali, G. Kramer and G. h. Zhu, Eur. Phys. J. C 47 (2006), 625-641 doi:10.1140/epjc/s2006- 02596-4 [arXiv:hep-ph/0601034 [hep-ph]]

    2006 arXiv

  16. [24]

    Bobeth, M

    C. Bobeth, M. Misiak and J. Urban, Nucl. Phys. B 574, 291 (2000) doi:10.1016/S0550- 3213(00)00007-9 [hep-ph/9910220]

    2000 arXiv

  17. [25]

    Gorbahn, U

    M. Gorbahn, U. Haisch and M. Misiak, Phys. Rev. Lett. 95, 102004 (2005) doi:10.1103/PhysRevLett.95.102004 [arXiv:hep-ph/0504194 [hep-ph]]

    2005 arXiv

  18. [26]

    Czakon, U

    M. Czakon, U. Haisch and M. Misiak, JHEP 0703, 008 (2007) doi:10.1088/1126- 6708/2007/03/008 [hep-ph/0612329]. 15

    2007 arXiv

  19. [27]

    Blake, G

    T. Blake, G. Lanfranchi and D. M. Straub, Prog. Part. Nuc l. Phys. 92, 50-91 (2017) doi:10.1016/j.ppnp.2016.10.001 [arXiv:1606.00916 [hep -ph]]

    2017 arXiv

  20. [28]

    Ebert, R

    D. Ebert, R. N. Faustov and V. O. Galkin, Phys. Rev. D 64 (2001), 094022 doi:10.1103/PhysRevD.64.094022 [arXiv:hep-ph/0107065 [hep-ph]]

    2001 arXiv

  21. [29]

    Becher, R

    T. Becher, R. J. Hill and M. Neubert, Phys. Rev. D 69, 054017 (2004) doi:10.1103/PhysRevD.69.054017 [arXiv:hep-ph/0308122 [hep-ph]]

    2004 arXiv

  22. [30]

    Beneke, Y

    M. Beneke, Y. Kiyo and D. S. Yang, Nucl. Phys. B 692, 232 (2004) doi:10.1016/j.nuclphysb.2004.05.018 [hep-ph/0402241]

    2004 arXiv

  23. [31]

    Beneke, Nucl

    M. Beneke, Nucl. Part. Phys. Proc. 261-262, 311-337 (2015) doi:10.1016/j.nuclphysbps.2015.03.021 [arXiv:1501.07374 [hep-ph]]

    2015 arXiv

  24. [32]

    C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys . Rev. D 63, 114020 (2001) doi:10.1103/PhysRevD.63.114020 [hep-ph/0011336]

    2001 arXiv

  25. [33]

    Greub, T

    C. Greub, T. Hurth and D. Wyler, Phys. Rev. D 54, 3350 (1996) doi:10.1103/PhysRevD.54.3350 [hep-ph/9603404]

    1996 arXiv

  26. [34]

    Chay and C

    J. Chay and C. Kim, Phys. Rev. D 68, 034013 (2003) doi:10.1103/PhysRevD.68.034013 [arXiv:hep-ph/0305033 [hep-ph]]

    2003 arXiv

  27. [35]

    Descotes-Genon and C

    S. Descotes-Genon and C. Sachrajda, Nucl. Phys. B 693, 103-133 (2004) doi:10.1016/j.nuclphysb.2004.06.018 [arXiv:hep-ph/0403277 [hep-ph]]

    2004 arXiv

  28. [36]

    Bosch, R

    S. Bosch, R. Hill, B. Lange and M. Neubert, Phys. Rev. D 67, 094014 (2003) doi:10.1103/PhysRevD.67.094014 [arXiv:hep-ph/0301123 [hep-ph]]

    2003 arXiv

  29. [37]

    Aoki et al

    S. Aoki et al. [Flavour Lattice Averaging Group], Eur. Phys. J. C 80, no.2, 113 (2020) doi:10.1140/epjc/s10052-019-7354-7 [arXiv:1902.08191 [hep-lat]]

    2020 arXiv

  30. [38]

    Beneke and J

    M. Beneke and J. Rohrwild, Eur. Phys. J. C 71, 1818 (2011) doi:10.1140/epjc/s10052-011- 1818-8 [arXiv:1110.3228 [hep-ph]]

    2011 arXiv

  31. [39]

    Y. M. Wang, JHEP 09, 159 (2016) doi:10.1007/JHEP09(2016)159 [arXiv:1606.03 080 [hep-ph]]

    2016 doi

  32. [40]

    Y. M. Wang and Y. L. Shen, Nucl. Phys. B 898 (2015), 563-604 doi:10.1016/j.nuclphysb.2015.07.016 [arXiv:1506.00667 [hep-ph]]

    2015 arXiv

  33. [41]

    J. Gao, C. D. L¨ u, Y. L. Shen, Y. M. Wang and Y. B. Wei, Phys. Rev. D 101 (2020) no.7, 074035 doi:10.1103/PhysRevD.101.074035 [arXiv:1907.11 092 [hep-ph]]

    2020 doi

  34. [42]

    S. W. Bosch and G. Buchalla, Nucl. Phys. B 621, 459 (2002) doi:10.1016/S0550-3213(01)00580- 6 [hep-ph/0106081]. 16

    2002 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.