REVIEW
Exclusive radiative and electroweak b->d and b->s penguin decays at NLO
High Energy Physics - Phenomenology · Submitted 28 Dec 2004 · 36 references
Pith's verdict: A careful, honest NLO extension to b→d penguins; the new amplitudes and B→ρℓℓ rates are the real content, and the CKM bound is weaker than the text implies.
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Referee report
The paper
The paper claims
The paper extends the QCD-factorization program for exclusive rare B decays from $b$ to $s$ transitions down to $b$ to $d$ transitions, supplying next-to-leading-order Standard Model reference values for $B to rho gamma$, $B to omega gamma$
The reading
What carries the argument
The machinery is QCD factorization for exclusive decays: the hadronic matrix element is written as a form-factor term $xi_a C_a^{(i)}$ plus a hard-scattering term involving the $B$ meson light-cone distribution amplitude and the light-meson distribution amplitude, with the transverse and longitudinal amplitudes separated. The objects that carry the argument are the coefficient functions $C_7^{(i)}$ and $C_{9,a}^{(i)}(q^2)$, which package the short-distance physics and are split into a top-quark sector $i = t$ and an up-quark sector $i = u$ weighted by the CKM ratio $lambda_u^{(d)}/lambda_t^{(d)} = -R_{ut} e^{i alpha}$. The new content is the up-sector amplitude $T_a^{(u)}$, whose two-loop correction from light-quark loops $C^{(1)}$ turns out to be as large as the one-loop function $Y^{(u)}(q^2)$ and largely cancels its imaginary part, and the power-suppressed annihilation amplitude $Delta T_perp^{(u)}$ that is enhanced by the large Wilson coefficient $C_2$ and dominates the charged-mode transverse amplitude. Because the up-sector amplitudes enter with a different weak phase, the interference terms produce the isospin and CP asymmetries and shift the forward-backward zero differently for $B$ and $bar B$.
Extended reading notes
Core claim
The central claim is that exclusive radiative and electroweak penguin decays of $B$ mesons to $K^*$, $rho$ and $omega$ are now computable at next-to-leading order in $alpha_s$ for both $b to s$ and $b to d$ at once, within QCD factorization at leading power in $1/m_b$, with weak annihilation included at leading order and the numerically relevant $1/m_b$ annihilation and hard-spectator corrections added. Concretely the paper delivers: $mathrm{Br}(B^0 to rho^0 gamma) = (5.0 pm 0.5) times 10^{-7}$ and $mathrm{Br}(B^+ to rho^+ gamma) = (10.3^{+1.5}_{-1.2}) times 10^{-7}$ for a reference tensor form factor $T_1^rho(0) = 0.21$; partially integrated rates $int_1^6 dq^2, dmathrm{Br}(B^0 to rho^0 ell^+ ell^-)/dq^2 = 4.2^{+0.6}_{-0.4} times 10^{-9}$ and $9.6^{+1.5}_{-1.1} times 10^{-9}$ for the charged mode; direct CP asymmetries $A_{CP}(rho^0 gamma) = (-10.4^{+1.6}_{-2.4}|_{mathrm{CKM}}{}^{+3.0}_{-3.6}|_{mathrm{had}})%$ and $A_{CP}(rho^+ gamma) = (-10.7^{+1.5}_{-2.0}{}^{+2.6}_{-3.7})%$; isospin asymmetries $Delta(rho gamma) = (-4.6^{+5.4}_{-4.2}{}^{+5.8}_{-5.6})%$ and $Delta(K^* gamma) = 0.28, T_1^{K^*}(0), (5.8^{+3.3}_{-2.9})%$; and forward-backward zero positions around $q^2 = 3.6$ to $4.5 , mathrm{GeV}^2$ for the four $B to rho ell^+ ell^-$ modes, distinguishable between $B$ and $bar B$ because the up-quark amplitude now carries a large absorptive part. The authors' own framing of why this matters is that $b to d$ penguins are the cleanest place to look for new flavour-changing neutral currents, and that the $B to rho gamma$ observables give a determination of the apex of the unitarity triangle independent of the standard fit.
Load-bearing premise
The $b to d$ branching fractions and the $|V_{td}/V_{ts}|$ bound rest on carrying the $B to K^*$ tensor form factor, fixed to $T_1^{K^*}(0) = 0.28$ from $B to K^* gamma$ data, over to $B to rho$ through a ratio assumed to lie between $0.75$ and $1.05$.
Editorial extensions
If this is right
- The existing experimental upper limit on $B^0 to rho^0 gamma$ translates into $|V_{td}/V_{ts}| < 0.21$ for form-factor ratio $F > 0.7$, a bound that already cuts into the range $0.204^{+0.029}_{-0.046}$ obtained from the standard unitarity-triangle fit.
- A measurement of $mathrm{Br}(B^0 to rho^0 gamma)$ near the current limit, combined with the $B to K^* gamma$ rate, fixes $|V_{td}/V_{ts}|$ up to the form-factor ratio $F = T_1^rho(0)/T_1^{K^*}(0)$.
- Simultaneous measurement of the branching fraction, isospin asymmetry and direct CP asymmetry in $B to rho gamma$ determines $(bar rho, bar eta)$ from a circle intersection that is nearly orthogonal to the standard fit, giving a genuinely independent cross-check.
- The location of the forward-backward asymmetry zero in $B to rho ell^+ ell^-$ measures the real part of the $b to d$ coefficient $C_9$, and the $B$ versus $bar B$ splitting of that zero measures the phase of the up-sector amplitude.
- Once the CKM elements and the $b to s$ coefficients $C_9, C_{10}$ are known, the $B to rho ell^+ ell^-$ observables become constraints on the $b to d$ coefficients $C_9^d, C_{10}^d$, which new-physics scenarios with non-minimal flavour violation can shift in magnitude and phase.
Reading between the lines
- The factor-of-two gap between the Belle partially integrated $B to K^* ell^+ ell^-$ rate and the prediction points, if it persists, to a tension between the tensor form factor preferred by $B to K^* gamma$ and the longitudinal form factor $A_0$ preferred by the dilepton rate, which would be a concrete handle on where the heavy-quark expansion is failing.
- Because the charged-mode predictions lean on the annihilation amplitude and the $B to gamma^*$ form factors $xi_perp^{(B gamma^*)}, xi_parallel^{(B gamma^*)}$, a future lattice or sum-rule determination of those form factors at $q^2 approx 0$ would convert the largest hadronic error bar in this paper into a genuine test.
- The same formalism applied with $delta_{qd} to delta_{qs}$ would generate the corresponding spectator-charge terms for $B_s to phi ell^+ ell^-$, which the appendix flags in passing but does not compute.
- If the direct CP asymmetry in $B to rho gamma$ were measured far from the predicted $-10%$, the circle in the $(bar rho, bar eta)$ plane would move rather than shrink, so the asymmetry is a sharper discriminant of new phases in $b to d gamma$ than of the CKM angle itself.
Original abstract
We provide Standard Model expectations for the rare radiative decays B->K^* gamma, B->rho gamma and B-> omega gamma, and the electroweak penguin decays B->K^* l^+ l^- and B->rho l^+ l^- at the next-to-leading order (NLO), extending our previous results to b->d transitions. We consider branching fractions, isospin asymmetries and direct CP asymmetries. For the electroweak penguin decays, the lepton-invariant mass spectrum and forward-backward asymmetry is also included. Radiative and electroweak penguin transitions in b->d are mainly interesting in the search for new flavour-changing neutral current interactions, but in addition the B->rho gamma decays provide constraints on the CKM parameters (\bar\rho,\bar\eta). The potential impact of these constraints is discussed.
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 provide Standard Model expectations for the rare radiative decays B->K^* gamma, B->rho gamma and B-> omega gamma, and the electroweak penguin decays B->K^* l^+ l^- and B->rho l^+ l^- at the next-to-leading order (NLO), extending our previous results to b->d transitions. We consider branching fractions, isospin asymmetries and direct CP asymmetries. For the electroweak penguin decays, the lepton-invariant mass spectrum and forward-backward asymmetry is also included. Radiative and electroweak penguin transitions in b->d are mainly interesting in the search for new flavour-changing neutral current interactions, but in addition the B->rho gamma decays provide constraints on the CKM parameters (\bar\rho,\bar\eta). The potential impact of these constraints is discussed.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We provide Standard Model expectations for the rare radiative decays B->K^* gamma, B->rho gamma and B-> omega gamma, and the electroweak penguin decays B->K^* l^+ l^- and B->rho l^+ l^- at the next-to-leading order (NLO), extending our previous results to b->d transitions. | contribution 1.00 | 0.93 | 0.24 | asserted 0.99 |
| s1 | We consider branching fractions, isospin asymmetries and direct CP asymmetries. | contribution 0.97 | 0.79 | 0.05 | asserted 0.74 |
| s2 | For the electroweak penguin decays, the lepton-invariant mass spectrum and forward-backward asymmetry is also included. | contribution 0.98 | 0.83 | 0.04 | asserted 0.99 |
| s3 | Radiative and electroweak penguin transitions in b->d are mainly interesting in the search for new flavour-changing neutral current interactions, but in addition the B->rho gamma decays provide constraints on the CKM parameters (\bar\rho,\bar\eta). | interpretation 0.52 | 0.12 | 0.18 | asserted 0.97 |
| s4 | The potential impact of these constraints is discussed. | interpretation 0.82 | 0.51 | 0.06 | unclear 0.78 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:hep-ph/0412400 by the authors.
References
How later papers used it
99 later papers cite it
3 later papers by other authors are listed below: 3 used its method or tool.
Uses its method or tool 3 of 26 papers with this label
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Computed B to K(*) l+ l- and B_s to phi l+ l- decays using QCD factorization from this paper.
Constraints on Lepton Universality Violation from Rare $B$ Decays (2022) · 1.00 · paraphrased: the licence does not allow quoting
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Obtained its spectator scattering terms from equations 20 and 22 of another work and equations 49 and 50 of this paper.
Predictions of Angular Observables for $\bar{B}_s\to K^{\ast}\ell\ell$ and $\bar{B}\to \rho\ell\ell$ in Standard Model (2018) · 0.85 · paraphrased: the licence does not allow quoting
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Employed a factorization scheme within QCDF relating the xi_perp,parallel to the V, A_1,2 as in this paper.
The Benefits of B ---> K* l+ l- Decays at Low Recoil (2010) · 0.99 · paraphrased: the licence does not allow quoting
How the literature uses it
98 arXiv papers cite this in 206 sentences
Papers citing it · 133 sentences without a confident label
Builds on it 2 papers
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Radiative corrections at first order of the strong coupling $α_s$ are included by adapting the results in [ref] to the corresponding $c →u$ transitions; in particular, this includes non-factorizable spectator scattering effects.
$D \to \rho \,\ell^+\ell^-$ Decays in the QCD Factorization Approach · Introduction · 0.94
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The results of [ref] were soon generalized to isospin asymmetries [ref] and the $B→ργ, ρℓℓ$ decays [ref] .
Soft-collinear factorization in B decays · Radiative and electroweak penguin decays · 0.91
Uses its method or tool 25 papers
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In order to calculate the universal form factors we use the QCDf scheme [ref] , where they are expressed as [equation] The $q^2$ dependence of the form factors $V, A_1, A_2$ can be parameterized as [equation] The values of the parameters involved in the calculation of form factors are taken from [ref] .
Study of the rare semileptonic decays $B_d^0 \to K^* l^+ l^-$ in scalar leptoquark model · Observables in the large recoil · 0.91
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We use the framework of QCD factorization (QCDF) valid in the low-$q^2$ region [ref] and exploit the symmetries of QCD in the large recoil limit of heavy-to-light transitions [ref] .
Angular Distributions of B -> K ll Decays · Introduction · 1.00
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The theory predictions of all the relevant semileptonic and radiative processes at large recoil are based on the QCDF results [ref] .
Bayesian Fit of Exclusive $b \to s \bar\ell\ell$ Decays: The Standard Model Operator Basis · Numerical Input · 0.84
Uses its data or values 5 papers
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The QCD form factors for the $B̅ →K̅^* l^+ l^-$ processes in the low $q^2$ region are taken from [ref] .
Impact of vector leptoquark on $\bar B \to \bar K^* l^+ l^-$ anomalies · $B̅ →K̅^* l^+ l^-$ processes · 0.99
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In this case we have used the results of Ref. [ref] for the hadronic form factors and their corresponding errors, taking into account additional theoretical uncertainty due to neglected additional long distance effects [ref] .
Constraints on New Physics in MFV models: a model-independent analysis of Delta F=1 processes · Numerical analysis · 0.97
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In Eq. ([ref]), $b̅^V_u,d$ are the corresponding ratios for the charge conjugate modes, and $R =|L_V|^2/|L_V|^2$. $L_V$ can be written as [equation] where the transition form factor for $B →V$, $ζ_⊥^V$, is defined as [equation] and we will use $ζ_⊥^K^* = 0.36±0.07$ and $ζ_⊥^ρ = 0.27±0.05$ [ref] for the numerical analysis.
Nonperturbative Charming Penguin Contributions to Isospin Asymmetries in Radiative B decays · Introduction · 0.80
Compares with it 12 papers
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Then we enlarge the theoretical errors by the differences between our SM predictions and known QCD factorization and SCET results [ref] .
Constraints on New Physics in MFV models: a model-independent analysis of Delta F=1 processes · $B →(K^*,K) ℓ^+ ℓ^-$, $B →(K^*,K) νν̅$ · 0.98
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[ref]), is in good agreement with that obtained in [ref] .
Study of the $b \to d \ell\ell$ transitions in the Standard Model and test of New Physics sensitivities · $B→ρℓℓ$ · 1.00
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We compare the impact of non-local form factors obtained in QCD factorisation (QCDf) [ref] with those determined using the more recent $z$-expansion approach [ref] .
Overview of $B \to K^{(*)}\ell \ell$ Theoretical Calculations and Uncertainties · Impact of Non-Local Form Factors · 1.00
Cites it as support 1 paper
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This also guarantees the applicability of the QCDF approach for these rare processes [ref] .
$\bar{B}_{s,d}^{0} \to J/\psi \mu^{+}\mu^{-}$ Decays in QCD Factorization · Branching ratios and dimuon invariant mass distributions · 0.97
Cites it as background 12 papers
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In the framework of QCDF, one can separate long-distance and short-distance contributions via a simultaneous expansion in the power of strong coupling constant and $Λ_QCD/m_Q$, and the factorization approach has been applied to various radiative heavy meson decays [ref] , and other processes such as factorization of the correlation function in light-cone sum rules [ref] .
Factorization of radiative leptonic D-meson decay with sub-leading power corrections · Introduction · 0.81
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Beyond these terms, there are also contributions from four-quark and chromomagnetic dipole operators, which require the calculation of hadronic matrix elements of the $T$-product of these local operators and the electromagnetic current [ref] .
Non-local matrix elements in $B_{(s)}\to \{K^{(*)},\phi\}\ell^+\ell^-$ · Introduction · 0.93
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As these are beyond the scope of this writing I just refer to selected papers [ref] and recent reviews [ref] .
Climbing NLO and NNLO Summits of Weak Decays: 1988-2023 · Exclusive Radiative Decays · 1.00
Discussion
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