REVIEW
Fat penguins and imaginary penguins in perturbative QCD
High Energy Physics - Phenomenology · Submitted 1 Apr 2000 · 23 references
Pith's verdict: An explicit, checkable PQCD calculation of B→Kπ whose headline 50% penguin enhancement is really a statement about the Eq. (9) scale choice, not an independent dynamical measurement.
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Referee report
The paper
The paper claims
The paper argues that penguin amplitudes in $B$ meson decays are about 50% larger than the standard factorization approximation assumes, because the strong-interaction scale inside the decay is lower than usually chosen.
The reading
What carries the argument
The carrying mechanism is perturbative QCD factorization for exclusive heavy-meson decays: the amplitude is written as a convolution of $B$, $K$ and $pi$ wave functions with a hard kernel $H([x],[b],M_B)$, with double logarithms resummed into Sudakov factors $exp[-S_B - S_pi]$ and $exp[-S_K - S_pi]$ that suppress the end-point region and keep the exchanged gluon hard. The scale-setting rule $t = max(sqrt{x},M_B, 1/b_i, 1/b_j)$ is what converts the running of $C_4$ and $C_6$ into the numerical enhancement, and the same framework makes annihilation diagrams formally the same order in $1/M_B$ as factorizable ones, which is what allows them to generate a large imaginary part.
Extended reading notes
Core claim
The central claim is a dynamical enhancement of penguin amplitudes in $B to Kpi$ decays. The mechanism is the running of the Wilson coefficients $C_4$ and $C_6$, which depend steeply on the renormalization scale, combined with the prescription that the hard scale be taken as the maximum of $sqrt{x},M_B$ and the inverse transverse extents $1/b$ of the participating mesons. Because a large fraction of the integrand sits at $t < m_b/2$, the penguin coefficients are evaluated where they are larger, giving $(F^{Pe6})_{PQCD}/(F^{Pe6})_{FA} = 1.6$, $(F^{Pe4})_{PQCD}/(F^{Pe4})_{FA} = 1.4$, and $(F^{e})_{PQCD}/(F^{e})_{FA} = 1.0$. A second claim is that the penguin annihilation amplitude $F^{Pa6}$ is not negligible: in Table I it has real part $0.30$ and imaginary part $2.58$ in units of $10^{-2}$, i.e. a strong phase near $90^circ$, arising from cuts on the intermediate $sbar{d}$ state. The authors argue this is allowed because the $O_6$ operator's helicity structure lets a left-handed quark and left-handed antiquark emerge back to back, so the usual helicity-suppression argument for annihilation does not apply. They check perturbativity by showing that about 97% of the $B to pi$ form factor and roughly 90% (80%) of $F^{Pe6}$ come from the region $alpha_s(t)/pi < 0.3$ (0.2).
Load-bearing premise
Everything hinges on taking the hard scale $t$ as the maximum of the internal-particle virtuality and the inverse transverse sizes, since the quoted 50% enhancement is literally the difference between evaluating the Wilson coefficients at that scale and freezing them at $M_B/2$.
Editorial extensions
If this is right
- The $B to Kpi$ and $B to pipi$ branching ratios can be fitted with a unitarity angle $phi_3 sim 90^circ$ using one consistent set of parameters, which the authors say is difficult in the plain factorization approximation.
- Penguin matrix elements extracted from data under the factorization approximation are systematically underestimated by roughly a third to a half.
- Annihilation diagrams must be retained in any CP-asymmetry prediction for $B to Kpi$, since they supply strong phases of order $90^circ$ rather than negligible ones.
- The strong phases are computable in this framework to roughly 20%, which the authors translate into about 30% uncertainty on predicted CP asymmetries.
- The relative weight of annihilation versus factorizable contributions should shrink as $M_B to infty$, giving a concrete asymptotic check on the picture.
Reading between the lines
- If the enhancement is really a scale-choice effect rather than a dynamical one, then the same prescription applied to other channels with steeply running coefficients — charmless $B to pipi$, $B to Keta$, $B to K K$ — should show a correlated pattern of shifts, which is a sharper test than any single branching ratio.
- The near-$90^circ$ annihilation phase suggests measurable direct CP asymmetries in $B^pm to K^pm pi^0$ and $B^0 to K^pm pi^mp$ that are larger than factorization-based expectations; comparing the two charge-conjugate rates is the natural place to look.
- Because the enhancement rests on integrand weight below $m_b/2$, an independent determination of the effective scale from, say, the $B to pi$ form factor's shape at large recoil would either corroborate or undercut the claim without needing new decay data.
- The paper's perturbativity check is a statement about the fraction of the integrand in a small-$alpha_s$ region, not a bound on the neglected higher-order terms; a genuine next-to-leading calculation would be the way to convert that fraction into an error bar.
Original abstract
We evaluate $B\to K\pi$ decay amplitudes in perturbative QCD picture. It is found that penguin contributions are dynamically enhanced by nearly 50% compared to those assumed in the factorization approximation. It is also shown that annihilation diagrams are not negligible, and give large strong phases. Our results for branching ratios of $B\to K\pi$ decays for a representative parameter set are consistent with data. We evaluate $B\to K\pi$ decay amplitudes in perturbative QCD picture. It is found that penguin contributions are dynamically enhanced by nearly 50% compared to those assumed in the factorization approximation. It is also shown that annihilation diagrams are not negligible, and give large strong phases. Our results for branching ratios of $B\to K\pi$ decays for a representative parameter set are consistent with data.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We evaluate $B\to K\pi$ decay amplitudes in perturbative QCD picture. It is found that penguin contributions are dynamically enhanced by nearly 50% compared to those assumed in the factorization approximation. It is also shown that annihilation diagrams are not negligible, and give large strong phases. Our results for branching ratios of $B\to K\pi$ decays for a representative parameter set are consistent with data. We evaluate $B\to K\pi$ decay amplitudes in perturbative QCD picture. It is found that penguin contributions are dynamically enhanced by nearly 50% compared to those assumed in the factorization approximation. It is also shown that annihilation diagrams are not negligible, and give large strong phases. Our results for branching ratios of $B\to K\pi$ decays for a representative parameter set are consistent with data.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We evaluate $B\to K\pi$ decay amplitudes in perturbative QCD picture. | contribution 0.93 | 0.95 | 0.05 | asserted 0.98 |
| s1 | It is found that penguin contributions are dynamically enhanced by nearly 50% compared to those assumed in the factorization approximation. | result 1.00 | 0.93 | 0.44 | asserted 1.00 |
| s2 | It is also shown that annihilation diagrams are not negligible, and give large strong phases. | result 1.00 | 0.92 | 0.14 | asserted 1.00 |
| s3 | Our results for branching ratios of $B\to K\pi$ decays for a representative parameter set are consistent with data. | result 0.98 | 0.93 | 0.44 | hedged possibility 0.94 |
| s4 | We evaluate $B\to K\pi$ decay amplitudes in perturbative QCD picture. | contribution 0.88 | 0.92 | 0.02 | asserted 0.99 |
| s5 | It is found that penguin contributions are dynamically enhanced by nearly 50% compared to those assumed in the factorization approximation. | result 1.00 | 0.90 | 0.35 | asserted 1.00 |
| s6 | It is also shown that annihilation diagrams are not negligible, and give large strong phases. | result 1.00 | 0.91 | 0.11 | asserted 1.00 |
| s7 | Our results for branching ratios of $B\to K\pi$ decays for a representative parameter set are consistent with data. | result 0.98 | 0.92 | 0.37 | hedged possibility 0.90 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:hep-ph/0004004 by the authors.
References
How later papers used it
86 later papers cite it
4 later papers by other authors are listed below: 2 used its method or tool, 1 used its data or values and 1 cited it as support.
Uses its method or tool 2 of 12 papers with this label
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Used the perturbative QCD approach based on kT factorization, citing this paper, to compute the B to T form factors.
B to tensor meson form factors in the perturbative QCD approach (2010) · 0.98 · paraphrased: the licence does not allow quoting
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Calculated the form factors in the PQCD approach of this paper near q2=0, where the K* meson recoils very fast, and determined their values at some points.
Study of $B\to K^{(*)} \ell^+\ell^-$ Decays in the Family Non-universal $Z'$ Models (2009) · 1.00 · paraphrased: the licence does not allow quoting
Uses its data or values 1 of 5 papers with this label
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The sources of the first two errors are from those for the $B$ meson wave function: $ω_B=(0.40±0.05)$ and $f_B=(0.19±0.02)GeV$ for $B$ mesons [ref] .
Investigate the glueball-like particle $X(2370)$ in $B$ meson decays (2025) · Transition form factors · 1.00
Cites it as support 1 of 4 papers with this label
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Said that the large value of phi_A^i lends some support to this paper's pQCD claim that annihilation amplitudes can provide a large strong phase.
Spectator Scattering and Annihilation Contributions as a Solution to the ${\pi}K$ and ${\pi}{\pi}$ Puzzles within QCD Factorization Approach (2014) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
72 arXiv papers cite this in 104 sentences
Papers citing it · 52 sentences without a confident label
Uses its method or tool 11 papers
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As an estimate we will adopt the perturbative QCD (PQCD) approach [ref] based on $k_T$ factorization to compute the transition form factor in which the leading-order Feynman diagram for the $B→G lν̅$ decays is displayed in Fig [ref].
Investigate the glueball-like particle $X(2370)$ in $B$ meson decays · Transition form factors · 1.00
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In this paper, we shall study the pure annihilation type charmed decays $B^0→D_s^-K_2^*+$ and $B_s→D̅a_2$ in the PQCD approach, which is based on the $k_T$ factorization [ref] .
The pure annihilation type decays $B^0\rightarrow D_{s}^{-}K_{2}^{*+}$ and $B_{s}\rightarrow \bar{D}a_{2}$ in perturbative QCD approach · Introduction · 0.96
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The motif of this work is to handle the first sector with the computation of the $B→T$ form factors, and in particular we will use the perturbative QCD (PQCD) approach [ref] which is based on the $k_T$ factorization.
B to tensor meson form factors in the perturbative QCD approach · Introduction · 0.98
Uses its data or values 6 papers
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Noted that $ϕ_B$ is normalized by the decay constant $f_B$, [equation] As aforementioned, the parameters $ω_b=0.40±0.08$ for $B^0$ meson and $ω_b=0.50±0.10$ for $B^0_s$ meson are almost best fits from the well measured results of the $B_d,s→Kπ$, $ππ$ decays [ref] , including the branching fractions and the $CP$ asymmetries.
Effects from Hadronic Structure of Photon on $B\to\phi\gamma$ and $B_s\to(\rho^0,\omega)\gamma$ Decays · Framework and Inputs · 0.84
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We will take $ω_B=(0.4 ± 0.04)$ GeV and $f_B=(0.21 ± 0.02)$ GeV for B meson [ref] .
The $B(B_{s})\rightarrow D_{(s)}(\bar{D}_{(s)}) T$ and $D_{(s)}^{*}(\bar{D}_{(s)}^{*})T$ decays in perturbative QCD approach · FORMALISM AND WAVE FUNCTION · 1.00
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The sources of the first two errors are from those for the $B$ meson wave function: $ω_B=(0.40±0.05)$ and $f_B=(0.19±0.02)GeV$ for $B$ mesons [ref] .
Investigate the glueball-like particle $X(2370)$ in $B$ meson decays · Transition form factors · 1.00
Compares with it 2 papers
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Compared with the theoretical predictions for these quantities in Refs. [ref] , based on the QCD factorization (QCDF) [ref] , the perturbative QCD (PQCD) [ref] , and the soft-collinear effective theory (SCET) [ref] , respectively, one would find the experimental measurement of this branching ratio agrees with the SM predictions with SCET [ref] , but lower than the predictions with QCDF and PQCD [ref] .
Studying of Bs --> K(*)-pi+, K(*)-rho+ decays within supersymmetry · Introduction · 0.97
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Compared with the theoretical predictions for these quantities in Refs. [ref] , [ref] and [ref] , based on the QCD factorization approach (QCDF) [ref] , the perturbative QCD (PQCD) [ref] , and the soft-collinear effective theory (SCET) [ref] , respectively, one would find the experimental measurements of branching ratios agree with the SM predictions within their large theoretical uncertainties.
B_s\to K^{(*)-}K^{(*)+}, K^{(*)-}\pi^{+}, K^{(*)-}\rho^{+} decays in R-parity violating supersymmetry · Introduction · 0.90
Cites it as support 3 papers
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It has been tested in the study of charmless $B$ meson decays successfully [ref] , especially for the direct CP asymmetries [ref] .
The two-body B_c {\to} D^{(*)}_{(s)}P, D^{(*)}_{(s)}V decays in the Perturbative QCD Approach · Introduction · 0.96
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The importance of annihilation diagrams are already tested in the predictions of direct CP asymmetries in $B^0 →π^+π^-$, $K^+π^-$ decays [ref] and in the explanation of $B→ϕK^*$ polarization problem [ref] .
Radiative charmless B_{(s)}\to V \gamma and B_{(s)}\to A \gamma decays in pQCD approach · Introduction · 0.99
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The large value of $ϕ_A^i$, corresponding to a large imaginary part of the enhanced complex corrections, also lends some support to the pQCD claim that the annihilation amplitudes can provide a large strong phase [ref] .
Spectator Scattering and Annihilation Contributions as a Solution to the ${\pi}K$ and ${\pi}{\pi}$ Puzzles within QCD Factorization Approach · Scenario I · 1.00
Cites it as background 28 papers
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In the past ten years, many efforts have been promoted to estimate the non-factorizabale contributions quantitatively, such as the QCD fatorization approach (QCDF) [ref] , the perturbative QCD approach (PQCD) [ref] and the soft collinear effective theory (SCET) [ref] .
Comments on $\left|\frac{V_{ub}}{V_{cb}}\right|$ and $|V_{ub}|$ from Non-leptonic $B$ Decays within the Perturbative QCD Approach · Introduction · 0.88
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Indeed, several quantum chromodynamics (QCD)-based factorization approaches, such as pQCD [ref] , QCD factorization (QCDF) [ref] and soft collinear effective theory (SCET) [ref] make use of the large $B$ mass and give predictions on the facrorization amplitudes, $A^fac$.
Revisiting Final State Interaction in Charmless $B_q\to P P$ Decays · Introduction · 0.81
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Recently three commonly-accepted approaches are developed to study exclusive $B$ decays: QCD factorization (QCDF) [ref] , soft collinear effective theory (SCET) [ref] and perturbative QCD approach [ref] .
Radiative charmless B_{(s)}\to V \gamma and B_{(s)}\to A \gamma decays in pQCD approach · Introduction · 0.93
Discussion
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