Enhanced Quark-Loop Contribution to Pure Annihilation Nonleptonic B-meson decays in PQCD approach

Feng-Chen Chu; Hong-Yuan Shen; Yue-Long Shen; Zhi-Tian Zou

REVIEW 3 major objections 5 minor 52 references

Enhanced Quark-Loop Contribution to Pure Annihilation Nonleptonic B-meson decays in PQCD approach

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that quark-loop next-to-leading-order QCD corrections, though nearly invisible in branching ratios, become the dominant source of strong phases in pure annihilation B decays and sharply enhance both direct and…

desk verdict A real extension of the quark-loop mechanism to twist-3 and vector-vector final states, capped by a striking -40% direct CP asymmetry benchmark in Bd→φφ, but the paper asserts rather than demonstrates that other NLO corrections are negligible. read the letter →

arxiv 2504.15002 v1 pith:HLB44EWY submitted 2025-04-21 hep-ph

Hong-Yuan Shen , Feng-Chen Chu , Yue-Long Shen , Zhi-Tian Zou This is my paper · ORCID

classification hep-ph PACS 13.25.Hw12.38.Bx11.30.Er

keywords PQCDapproachpureannihilationBdecaysquark-loopcorrectionsCPasymmetriesstrongphasescharmlessnonleptonicchromo-magneticpenguinlight-conedistributionamplitudes

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 argues that a specific next-to-leading-order QCD effect—the quark-loop contraction of four-quark operators plus the chromo-magnetic penguin—dominates the NLO corrections to pure annihilation charmless B-meson decays in the PQCD approach. The NLO amplitudes carry large imaginary parts with signs opposite to the leading-order amplitudes, so the two cancel in magnitude and the branching ratios stay close to their leading-order values ($|A_{\rm LO+NLO}|/|A_{\rm LO}|\approx 1.02$). But that same cancellation creates sizable strong-phase differences, so the CP-violating observables change dramatically: for $\bar B_d\to\phi_L\phi_L$, the direct asymmetry goes from $0$ at leading order to about $-40\%$, and the mixing-induced asymmetry from $0$ to about $28\%$. The paper thereby turns pure annihilation decays, which are difficult for collinear factorization because of endpoint divergences, into a concrete probe of strong phases and CP violation.

What carries the argument

The machinery is the reduction of the quark-loop diagrams—the gluon attached to a quark loop contracted onto a four-quark operator, plus the chromo-magnetic penguin operator $O_{8g}$—to two effective operators with coefficients $C^{\rm eff}_{D,1}$ and $C^{\rm eff}_{D,2}$. The loop function $G(m)$, decomposed into quark and mass terms, controls the size and imaginary part of the correction; its antisymmetric structure $F_V(p_g^2,\tilde p_g^2)=-F_V(\tilde p_g^2,p_g^2)$ means the correction contributes to $PP$ and longitudinal $VV$ final states but not to $PV$ at leading power. The NLO hard kernels $h_{\rm nan}$ introduce new factorization formulas that are convolved with the $B$-meson and light-meson wave functions, including twist-2 and two-particle twist-3 light-cone distribution amplitudes; the endpoint-safe transverse-momentum-dependent framework of PQCD lets those convolutions be evaluated without parametric regularization.

What would settle it

Compute the triangle diagrams of Fig. 3 and the full QCD corrections to the Fig. 1 diagrams in the same PQCD framework: if their imaginary parts are comparable to or larger than the quark-loop contribution, the claimed dominance and the predicted strong phases collapse. On the experimental side, measuring $A_{\rm CP}^{\rm dir}(\bar B_d\to\phi_L\phi_L)$ near zero rather than the predicted about $-40\%$ would falsify the central claim.

Watch

Extended reading notes

Core claim

Within the PQCD factorization approach, the paper establishes that the quark-loop enhanced NLO contribution to pure-annihilation B decays is not a small perturbation of the branching ratios but is instead the dominant source of strong phases. The mechanism is a cancellation: the NLO amplitudes have large imaginary parts opposite in sign to the LO ones, so the modulus of the total amplitude and therefore the branching ratio barely change; for $\bar B_s\to\pi^+\pi^-$ and $\bar B_s\to\rho_L^+\rho_L^-$, $|A_{\rm LO+NLO}|/|A_{\rm LO}|\approx 1.02$. The same imaginary parts convert into strong-phase differences between tree and penguin amplitudes ($\sin\Delta\delta$ over $0.5$ for $\bar B_s\to\pi\pi$ and $\bar B_s\to\rho\rho$), and because CP asymmetries are proportional to $\sin\Delta\delta\,\sin\Delta\phi$ (or to imaginary parts of amplitudes), they are strongly enhanced. Concretely, the direct asymmetry in $\bar B_d\to\phi_L\phi_L$ moves from $0$ to $-39.7\%$, its mixing-induced asymmetry from $0$ to $27.8\%$, and the $\bar B_s\to\pi\pi/\rho\rho/\omega\omega$ direct asymmetries go to about $-4\%$ to $-6\%$; $\bar B_s\to\rho_L\omega_L$ is the exception, remaining unchanged because its two-gluon production violates isospin conservation.

Load-bearing premise

The calculation assumes that the quark-loop and chromo-magnetic-penguin diagrams are the only important next-to-leading-order corrections, and that the triangle diagrams of Fig. 3 plus the QCD corrections to the leading-order diagrams really are negligible; the paper gives this as a qualitative argument, without a numerical estimate of the discarded terms.

Editorial extensions

If this is right

Branching-ratio predictions for the pure annihilation channels remain essentially at their leading-order PQCD values and stay compatible with current experimental bounds, so the new ingredient does not disturb the earlier successful rate predictions.
Direct CP asymmetries become measurable in several modes: about $-5\%$ for $\bar B_s\to\pi\pi,\rho\rho,\omega\omega$, and about $-40\%$ for $\bar B_d\to\phi_L\phi_L$, creating concrete targets for current and future B-physics experiments.
Mixing-induced asymmetries receive the larger shift because they are proportional to the imaginary part of the amplitude; for example, $\bar B_s\to\pi\pi$ $A_{\rm CP}^{\rm mix}$ moves from $+35.9\%$ at LO to about $-4.2\%$, a qualitative sign change.
The longitudinal polarization fraction in $B\to VV$ modes stays near $1$, since the NLO correction only affects the longitudinal amplitude and the transverse amplitudes remain suppressed.
The $\bar B_s\to\rho_L\omega_L$ mode is predicted to keep its CP asymmetries unchanged, providing a control channel for the isospin-conservation argument behind the quark-loop contribution.

Reading between the lines

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

One consequence the paper leaves implicit: if the predicted large direct asymmetry in $\bar B_d\to\phi_L\phi_L$ survives a full NLO treatment, this mode could serve as a clean probe of the $b\to d$ penguin phase, although its tiny branching ratio means only a high-luminosity experiment could test it.
The paper's assertion that the Fig. 3 triangle diagrams are negligible is not backed by a numerical estimate; computing those diagrams in the same framework would be the direct test of whether the quoted strong phases are stable or only approximate.
The same quark-loop mechanism should also shift the CP asymmetries of other penguin-dominated B decays that proceed through annihilation-type topologies, and the size of the shift could be predicted by applying the same effective-operator calculation to those channels.
The cancellation pattern—large imaginary NLO parts with opposite sign—suggests that the real part of the correction is what must be controlled to keep branching ratios stable, so improvements in the light-cone distribution amplitudes would mainly sharpen the CP predictions rather than the rates.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This is a worthwhile follow-up to Ref. [47], extending the quark-loop enhancement from twist-2 only to twist-3 LCDAs and to longitudinally polarized vector-vector final states. The new numerical output is concrete and testable, especially A_CP^dir(Bd→φ_L φ_L) ≈ -40% where the leading order gives zero. The internal logic is coherent: the NLO amplitude is 35–40% of the LO one, its large imaginary part cancels in the branching ratios (|A_LO+NLO|/|A_LO| ≈ 1.02 in Table 2), and the CP asymmetries come from the strong phases that the loop generates. None of that is fitted to CP data; it follows from CKM inputs and independently sourced LCDA moments. That is genuine progress within the PQCD program, not a repackaging.

The soft spot is the one the stress-test correctly identifies. Section 3 dismisses the triangle diagrams and QCD corrections to the LO graphs as "less significant", citing CKM/Wilson enhancement and Ref. [47], but it gives no order-of-magnitude estimate in the present framework. For a difference-type observable like A_CP^dir, an omitted amplitude with an imaginary part of order 10–20% of the LO amplitude could shift the headline -40% by many points. The same Section also claims the NLO correction only affects longitudinal VV polarizations; that argument is made at leading power, and the text itself admits twist-3 internal gluons need not be transverse. No proof is given that the twist-3 NLO contribution to the transverse VV amplitudes vanishes. These concerns are real but not fatal: the qualitative claim that quark loops generate strong phases and can change CP asymmetries in annihilation modes likely survives, though the precise numbers should be treated as the central values of an incomplete NLO calculation.

Minor but visible: Table 1 quotes ω_b = 0.40 ± 0.5 GeV, which is almost certainly a typo and would inflate uncertainties if taken literally; and the text says experimental results are listed in Table 2 when they are in Table 3. Both are easy fixes.

The paper deserves a serious referee. A referee should push for a quantitative bound on the neglected NLO pieces, or at least a more careful argument for why they cannot compete with the charm-loop phase. For researchers working on PQCD annihilation decays or B→VV CP asymmetries, the B→φφ benchmark is a concrete target for LHCb and Belle II. I would send it to review and accept conditionally once the neglected-corrections question is addressed.

Referee Report

3 major / 5 minor

Summary. The paper computes next-to-leading-order quark-loop and chromo-magnetic-penguin contributions to pure-annihilation B-meson decays (Bs→ππ, ρρ, ρω, ωω; Bd→K+K−, K*+K*−, φφ) in the PQCD approach. The main finding is that the NLO amplitudes have large imaginary parts with signs opposite to the leading-order amplitudes, so |A_LO+A_NLO|/|A_LO|≈1.02 and the branching ratios barely change, while the resulting strong phases substantially modify the CP asymmetries. The central numerical results are in Tables 3 and 4, including A_CP^dir(Bd→φ_Lφ_L)≈−40% at NLO compared with 0 at LO. The paper excludes B→PV modes on the grounds that the quark-loop two-gluon form factor does not contribute to those channels.

Significance. If the result holds, it would establish that, within PQCD, the quark-loop and O8g diagrams provide the dominant source of strong phases in pure-annihilation decays, turning several CP asymmetries from unobservably small to measurable values (notably Bd→φφ) while leaving branching ratios essentially unchanged. The predictions are genuine output of the calculation rather than fits: the inputs are the standard PQCD parameters (ω_b, Gegenbauer moments, decay constants) and CKM elements from independent sources, and the amplitude ratios in Table 2 are internally consistent. The main weakness is that the dominance of the selected NLO diagrams is asserted rather than demonstrated, and the quoted uncertainties on A_mix make several of the claimed 'significant enhancements' statistically weak.

major comments (3)

[Section 3, after Eq. (11)] The central claim rests on the assertion that the quark-loop and chromo-magnetic-penguin diagrams of Fig. 2 dominate the NLO corrections, while the triangle diagrams of Fig. 3 and the QCD corrections to the Fig. 1 diagrams can be neglected. The text states this is true because of CKM/Wilson enhancement and cites Ref. [47], but no quantitative estimate is given within the PQCD framework. This is load-bearing because the headline observables in Table 4 are generated by the imaginary part of A_NLO; for example, A_CP^dir(Bd→φ_Lφ_L) changes from 0 at LO to −39.7% at NLO. An omitted amplitude whose imaginary part is 10–20% of A_LO would shift these predictions by tens of percent. Please provide an explicit estimate of the leading neglected terms, or state clearly that the predictions assume their smallness.
[Table 4 and Conclusions/Abstract] The abstract and conclusions claim that NLO QCD corrections 'significantly enhance' both A_CP^dir and A_CP^mix. The quoted uncertainties do not support this for A_mix: for Bs→π+π− the LO value 35.9+15.6−11.2 and NLO value −4.2+21.4−9.0 overlap within 1σ, and for Bd→K+K− the LO −47.0+15.7−18.8 and NLO −2.2+19.1−26.4 also overlap within 1σ. Even for Bd→φ_Lφ_L, A_mix=27.8+5.7−25.9 is consistent with 0 at the 1σ level. The only cleanly significant effect in Table 4 is A_CP^dir(Bd→φ_Lφ_L). The wording should be qualified, or the uncertainty propagation should be explained if the asymmetric errors are not intended as 1σ intervals.
[Table 2 and Table 4] The central mechanism is presented through the amplitude ratios for Bs→π+π− and Bs→ρ_Lρ_L, but no analogous decomposition is given for Bd→φ_Lφ_L, the channel with the largest claimed effect. Since A_CP^dir for φφ is zero at LO and becomes −39.7% at NLO, the reader cannot check whether the effect is driven by the same |A_NLO|/|A_LO| ≈ 0.36 cancellation pattern or by a different numerical balance. Please include the amplitude decomposition (or the relevant ratios and strong phases) for Bd→φφ, and ideally for all modes in Table 3.

minor comments (5)

[Section 4, paragraph before Table 3] The sentence 'In Table 2, we also list the experimental results' should read 'Table 3'; the experimental values appear in Table 3, not Table 2.
[Section 4, Eq. (21)] The B-meson shape parameters are quoted as ω_b = 0.40±0.5 GeV and ω_b = 0.50±0.5 GeV; as printed the errors exceed the central values. This is presumably a typo for ±0.05 GeV and should be corrected, since ω_b is a major source of the quoted uncertainties.
[Section 3, B→PV argument] The statement that the twist-3 ⟨P(p2)V(p3)|g*g*⟩ matrix element vanishes because the number of gamma matrices in the pseudo-scalar and vector LCDAs is odd is asserted without showing the trace. Since this is used to justify excluding B→PV modes, a one-line demonstration would help the reader verify the claim.
[Abstract and Conclusions] The phrase 'evidently revealed' is too emphatic for a calculation with the large uncertainties shown in Table 4; 'indicate' or 'suggest' would be more appropriate.
[Section 3, first paragraph] There is a spelling error in 'diffesr' (should be 'differs') in the discussion of the Lorentz structure of power-suppressed collinear quark fields.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the NLO CP-asymmetry enhancement follows from independently sourced CKM, Wilson coefficients, and LCDA inputs; self-citations supply published calculations rather than definitional reductions.

full rationale

The central claim—that the quark-loop NLO contribution supplies strong phases and enhances A_CP^dir and A_CP^mix—is derived from amplitudes built from the effective Hamiltonian (Eq. 1), the effective Wilson coefficients (Eq. 11), and the factorization formulae (Eqs. 16–20). The numerical predictions use CKM elements from the PDG, Wilson coefficients from Ref. [48], and LCDA parameters from the literature; no CP-asymmetry observable is fitted. The strong-phase difference Δδ in Table 2 is an output of the decomposition A ~ V_ub V*_uD A_u + V_tb V*_tD A_t, and the LO-vs-NLO comparison in Table 4 is a direct consequence of the computed amplitudes. The B-meson shape parameter ω_b is fixed to experimental data, but it is a standard nonperturbative input rather than a parameter fitted to the predicted CP asymmetries, so no 'fitted input called prediction' pattern occurs. The self-citations to Refs. [47] and [49] are to published calculations: Ref. [47] supplies the form-factor symmetry relations and the claimed suppression of the triangle diagrams, and Ref. [49] supplies the leading-order factorization formulae and LCDA conventions. These are independent supporting results with stated assumptions; the current paper extends them to twist-3 and vector final states rather than importing the target result. The paper's own caveat in Sec. 3 ('It is true that there are other NLO corrections... However, these are less significant...') flags a completeness/robustness concern about neglected diagrams, but that is an approximation judgment, not a definitional equivalence between the paper's equations and its inputs. No equation in the paper defines the predicted CP asymmetries in terms of the experimental CP asymmetries or reduces the NLO amplitude to a fitted parameter, so no circular step can be exhibited.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The calculation relies on standard PQCD machinery (Sudakov factors, TMD wave functions), non-perturbative LCDA parameters fitted to other data, and symmetry relations imported from Ref. [47] by overlapping authors. No new particles or forces are introduced, so the invented-entities list is empty.

free parameters (2)

omega_b (B meson wave-function shape parameter) = 0.40 GeV for B, 0.50 GeV for Bs; printed uncertainty 0.5 (likely a typo for 0.05)
Appears in Eq. (21) and controls the B meson distribution amplitude. The paper states it "has been fixed using the rich experimental data", so it is a fitted input from prior data.
Light-meson Gegenbauer moments and decay constants = Values in Table 1, e.g., a2_pi = 0.250, a2_rho = 0.180, aK_1 = 0.076
Non-perturbative inputs fitted to other data. The central CP asymmetry predictions depend on them, and their uncertainties propagate into the quoted errors.

assumptions (4)

domain assumption PQCD factorization with transverse-momentum-dependent wave functions properly regulates endpoint singularities in annihilation diagrams.
Section 2 states annihilation diagrams are considered factorizable in PQCD because transverse momenta of partons are retained; this is the standard assumption of the framework.
domain assumption The B-to-g*-g* transition form factors FV and FA obey the symmetry relations FV(p_g^2, p_tilde_g^2) = -FV(p_tilde_g^2, p_g^2) and FA symmetric, as derived in Ref. [47].
Section 3 uses these relations to conclude that PV decays receive no leading-power quark-loop contribution and to build the NLO amplitudes.
ad hoc to paper Neglected NLO corrections (triangle diagrams, vertex corrections) are numerically subdominant to the quark-loop and O8g contributions.
Section 3 asserts this without a quantitative estimate, citing Ref. [47] for the suppression of the triangle diagrams.
domain assumption Twist-3 LCDAs of the final-state light mesons do not introduce new endpoint divergences when included in the NLO convolution.
Section 3 states transverse momenta are neglected since "no additional endpoint divergence will be introduced"; this is a technical assumption of the computation.

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Pith. "Pith review of Enhanced Quark-Loop Contribution to Pure Annihilation Nonleptonic B-meson decays in PQCD approach." pith.science (2026). https://pith.science/paper/HLB44EWY

@misc{pith2026250415002,
  author       = {Pith},
  title        = {Pith review of: Enhanced Quark-Loop Contribution to Pure Annihilation Nonleptonic B-meson decays in PQCD approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HLB44EWY}},
  note         = {Machine review of arXiv:2504.15002}
}

abstract

In this study, we conduct an enhanced investigation of pure annihilation-type charmless hadronic B-decays within the framework of the PQCD approach. Specifically, we account for the quark-loop enhanced contribution to various decay processes at the next-to-leading order(NLO) in the strong coupling constant $\alpha_s$. The NLO amplitudes possess large imaginary parts that have signs opposite to those of the leading-order amplitudes. This cancellation effect implies that the NLO contribution does not significantly influence the branching ratios of the processes under consideration. However, the NLO effect explored in this work constitutes an important source of the strong phases in the weak - annihilation non-leptonic $\bar B_q$-meson decay amplitudes. As a result, it can have a substantial impact on CP - violating observables such as the direct CP asymmetry ${\cal A}_{\rm CP}^{\rm dir}$ and mixing induced CP asymmetry ${\cal A}_{\rm CP}^{\rm mix}$. The numerical result evidently revealed that NLO QCD correction significantly enhances both ${\cal A}_{\rm CP}^{\rm dir}$ and ${\cal A}_{\rm CP}^{\rm mix}$ of the pure annihilation type B decay processes.

Figures

Figures reproduced from arXiv: 2504.15002 by the authors.

**Figure 2.** The Feynman diagrams for next-to-leading order corrections to annihilation contri [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** The triangle diagrams for next-to-leading order corrections to annihilation contribution. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Works this paper leans on

52 extracted references · 17 canonical work pages

[47]

Enhanced Next-to-Leading-Order Corrections to Weak Annihilation $B$-Meson Decays

C.-D. Lü, Y.-L. Shen, C. Wang, and Y.-M. Wang, Nucl. Phys. B 990, 116175 (2023), arXiv:2202.08073 [hep-ph]

work page Pith review arXiv 2023
[1]

Beneke, G

M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999), arXiv:hep-ph/9905312

arXiv 1999
[2]

Beneke, G

M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, Nucl. Phys. B591, 313 (2000), arXiv:hep-ph/0006124

arXiv 2000
[3]

Nonleptonic B decays into two light mesons in soft-collinear effective theory

J. Chay and C. Kim, Nucl. Phys. B680, 302 (2004), arXiv:hep-ph/0301262

work page Pith review arXiv 2004
[4]

Beneke, A

M. Beneke, A. P. Chapovsky, M. Diehl, and T. Feldmann, Nucl. Phys. B643, 431 (2002), arXiv:hep-ph/0206152

arXiv 2002
[5]

C. W. Bauer, D. Pirjol, I. Z. Rothstein, and I. W. Stewart, Phys. Rev. D70, 054015 (2004), arXiv:hep-ph/0401188

arXiv 2004
[6]

C.W.Bauer, I.Z.Rothstein, andI.W.Stewart,Phys.Rev.D 74,034010(2006),arXiv:hep- ph/0510241

arXiv 2006
[7]

Y. Y. Keum, H.-N. Li, and A. I. Sanda, Phys. Rev. D 63, 054008 (2001), arXiv:hep- ph/0004173

arXiv 2001

Show all 52 references

[8]

C.-D. Lu, K. Ukai, and M.-Z. Yang, Phys. Rev. D63, 074009 (2001), arXiv:hep-ph/0004213

2001 arXiv
[9]

Li, C.-D

Y. Li, C.-D. Lu, Z.-J. Xiao, and X.-Q. Yu, Phys. Rev. D70, 034009 (2004), arXiv:hep- ph/0404028 . 18

2004
[10]

A. Ali, G. Kramer, Y. Li, C.-D. Lu, Y.-L. Shen, W. Wang, and Y.-M. Wang, Phys. Rev. D 76, 074018 (2007), arXiv:hep-ph/0703162

2007 arXiv
[11]

Li, Y.-L

H.-n. Li, Y.-L. Shen, Y.-M. Wang, and H. Zou, Phys. Rev. D 83, 054029 (2011), arXiv:1012.4098 [hep-ph]

2011 arXiv
[12]

Li, Y.-L

H.-n. Li, Y.-L. Shen, and Y.-M. Wang, Phys. Rev. D85, 074004 (2012), arXiv:1201.5066 [hep-ph]

2012 arXiv
[13]

Li, Y.-L

H.-N. Li, Y.-L. Shen, and Y.-M. Wang, JHEP02, 008 (2013), arXiv:1210.2978 [hep-ph]

2013 arXiv
[14]

Li, Y.-L

H.-N. Li, Y.-L. Shen, and Y.-M. Wang, JHEP01, 004 (2014), arXiv:1310.3672 [hep-ph]

2014 arXiv
[15]

Li and Y.-M

H.-n. Li and Y.-M. Wang, JHEP06, 013 (2015), arXiv:1410.7274 [hep-ph]

2015 arXiv
[16]

Z. L. Liu and M. Neubert, JHEP04, 033 (2020), arXiv:1912.08818 [hep-ph]

2020 arXiv
[17]

Z. L. Liu, B. Mecaj, M. Neubert, and X. Wang, Phys. Rev. D 104, 014004 (2021), arXiv:2009.04456 [hep-ph]

2021 arXiv
[18]

Z. L. Liu, B. Mecaj, M. Neubert, and X. Wang, JHEP01, 077 (2021), arXiv:2009.06779 [hep-ph]

2021 arXiv
[19]

Beneke, M

M. Beneke, M. Garny, S. Jaskiewicz, J. Strohm, R. Szafron, L. Vernazza, and J. Wang, JHEP 07, 144 (2022), arXiv:2205.04479 [hep-ph]

2022 arXiv
[20]

G. Bell, P. Böer, and T. Feldmann, JHEP09, 183 (2022), arXiv:2205.06021 [hep-ph]

2022 arXiv
[21]

Beneke, G

M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, Nucl. Phys. B606, 245 (2001), arXiv:hep-ph/0104110

2001 arXiv
[22]

Beneke and M

M. Beneke and M. Neubert, Nucl. Phys. B675, 333 (2003), arXiv:hep-ph/0308039

2003 arXiv
[23]

Beneke, J

M. Beneke, J. Rohrer, and D. Yang, Nucl. Phys. B774, 64 (2007), arXiv:hep-ph/0612290

2007 arXiv
[24]

Bartsch, G

M. Bartsch, G. Buchalla, and C. Kraus, (2008), arXiv:0810.0249 [hep-ph]

2008 arXiv
[25]

Cheng and K.-C

H.-Y. Cheng and K.-C. Yang, Phys. Rev. D78, 094001 (2008), [Erratum: Phys.Rev.D 79, 039903 (2009)], arXiv:0805.0329 [hep-ph]

2008 arXiv
[26]

Cheng and C.-K

H.-Y. Cheng and C.-K. Chua, Phys. Rev. D80, 114008 (2009), arXiv:0909.5229 [hep-ph]

2009 arXiv
[27]

Cheng and C.-K

H.-Y. Cheng and C.-K. Chua, Phys. Rev. D80, 114026 (2009), arXiv:0910.5237 [hep-ph]

2009 arXiv
[28]

Zhu, Phys

G. Zhu, Phys. Lett. B702, 408 (2011), arXiv:1106.4709 [hep-ph]

2011 arXiv
[29]

Wang and G

K. Wang and G. Zhu, Phys. Rev. D88, 014043 (2013), arXiv:1304.7438 [hep-ph]

2013 arXiv
[30]

Bobeth, M

C. Bobeth, M. Gorbahn, and S. Vickers, Eur. Phys. J. C75, 340 (2015), arXiv:1409.3252 [hep-ph]

2015 arXiv
[31]

Chang, J

Q. Chang, J. Sun, Y. Yang, and X. Li, Phys. Lett. B 740, 56 (2015), arXiv:1409.2995 [hep-ph]

2015 arXiv
[32]

Chang, X.-N

Q. Chang, X.-N. Li, J.-F. Sun, and Y.-L. Yang, J. Phys. G 43, 105004 (2016), arXiv:1610.02747 [hep-ph] . 19

2016 arXiv
[33]

Chang, X

Q. Chang, X. Li, X.-Q. Li, and J. Sun, Eur. Phys. J. C77, 415 (2017), arXiv:1706.06138 [hep-ph]

2017 arXiv
[34]

C.-D. Lu, Eur. Phys. J. C24, 121 (2002), arXiv:hep-ph/0112127

2002 arXiv
[35]

Lu and K

C.-D. Lu and K. Ukai, Eur. Phys. J. C28, 305 (2003), arXiv:hep-ph/0210206

2003 arXiv
[36]

Li and C.-D

Y. Li and C.-D. Lu, HEPNP27, 1062 (2003), arXiv:hep-ph/0305278

2003 arXiv
[37]

Li, C.-D

Y. Li, C.-D. Lu, and Z.-J. Xiao, J. Phys. G31, 273 (2005), arXiv:hep-ph/0308243

2005 arXiv
[38]

Song and C.-D

G.-L. Song and C.-D. Lu, Phys. Rev. D70, 034006 (2004), arXiv:hep-ph/0403233

2004 arXiv
[39]

Li and C.-D

Y. Li and C.-D. Lu, Commun. Theor. Phys.44, 659 (2005), arXiv:hep-ph/0502038

2005 arXiv
[40]

Lu, Y.-l

C.-D. Lu, Y.-l. Shen, and J. Zhu, Eur. Phys. J. C41, 311 (2005), arXiv:hep-ph/0501269

2005 arXiv
[41]

Zhu, Y.-L

J. Zhu, Y.-L. Shen, and C.-D. Lu, Phys. Rev. D72, 054015 (2005), arXiv:hep-ph/0504187

2005 arXiv
[42]

Li and C.-D

Y. Li and C.-D. Lu, Phys. Rev. D74, 097502 (2006), arXiv:hep-ph/0605220

2006 arXiv
[43]

Rui, Z.-T

Z. Rui, Z.-T. Zou, and C.-D. Lu, Phys. Rev. D86, 074008 (2012), arXiv:1112.1257 [hep-ph]

2012 arXiv
[44]

X.-Q. Yu, Y. Li, and C.-D. Lu, Phys. Rev. D73, 017501 (2006), arXiv:hep-ph/0511269

2006 arXiv
[45]

Yang, J.-F

Y.-L. Yang, J.-F. Sun, and N. Wang, Phys. Rev. D81, 074012 (2010), arXiv:1004.2825 [hep-ph]

2010 arXiv
[46]

Xiao, W.-F

Z.-J. Xiao, W.-F. Wang, and Y.-y. Fan, Phys. Rev. D85, 094003 (2012), arXiv:1111.6264 [hep-ph]

2012 arXiv
[48]

Buchalla, A

G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996), arXiv:hep-ph/9512380

1996 arXiv
[49]

J. Chai, S. Cheng, Y.-h. Ju, D.-C. Yan, C.-D. Lü, and Z.-J. Xiao, Chin. Phys. C46, 123103 (2022), arXiv:2207.04190 [hep-ph]

2022 arXiv
[50]

P. A. Zylaet al. (Particle Data Group), PTEP2020, 083C01 (2020)

2020
[51]

Nierste, in Helmholz International Summer School on Heavy Quark Physics (2009) pp

U. Nierste, in Helmholz International Summer School on Heavy Quark Physics (2009) pp. 1–38, arXiv:0904.1869 [hep-ph]

2009 arXiv
[52]

B physics at the Tevatron: Run II and beyond (2001) arXiv:hep-ph/0201071 . 20

2001 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [47]

Enhanced Next-to-Leading-Order Corrections to Weak Annihilation $B$-Meson Decays

C.-D. Lü, Y.-L. Shen, C. Wang, and Y.-M. Wang, Nucl. Phys. B 990, 116175 (2023), arXiv:2202.08073 [hep-ph]

work page Pith review arXiv 2023

[2] [1]

Beneke, G

M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999), arXiv:hep-ph/9905312

arXiv 1999

[3] [2]

Beneke, G

M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, Nucl. Phys. B591, 313 (2000), arXiv:hep-ph/0006124

arXiv 2000

[4] [3]

Nonleptonic B decays into two light mesons in soft-collinear effective theory

J. Chay and C. Kim, Nucl. Phys. B680, 302 (2004), arXiv:hep-ph/0301262

work page Pith review arXiv 2004

[5] [4]

Beneke, A

M. Beneke, A. P. Chapovsky, M. Diehl, and T. Feldmann, Nucl. Phys. B643, 431 (2002), arXiv:hep-ph/0206152

arXiv 2002

[6] [5]

C. W. Bauer, D. Pirjol, I. Z. Rothstein, and I. W. Stewart, Phys. Rev. D70, 054015 (2004), arXiv:hep-ph/0401188

arXiv 2004

[7] [6]

C.W.Bauer, I.Z.Rothstein, andI.W.Stewart,Phys.Rev.D 74,034010(2006),arXiv:hep- ph/0510241

arXiv 2006

[8] [7]

Y. Y. Keum, H.-N. Li, and A. I. Sanda, Phys. Rev. D 63, 054008 (2001), arXiv:hep- ph/0004173

arXiv 2001