REVIEW
Semileptonic B decays to excited charmed mesons
High Energy Physics - Phenomenology · Submitted 30 May 1997 · 36 references
Pith's verdict: A careful, honest HQET paper whose 'model-independent' slope prediction is real but numerically soft, and whose flagged continuity assumption is safe in practice.
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Referee report
The paper
The paper claims
Heavy-quark corrections fix the slope of a B decay without a model
The reading
What carries the argument
The argument runs on the trace formalism of heavy quark effective theory. Meson fields are packaged into $4 times 4$ matrices $H_v$ for the ground-state ${1 over 2}^-$ doublet and $F^{mu}_v$ for the ${3 over 2}^+$ doublet, so that matrix elements of any Dirac current become traces over spinor indices. The leading-order matrix element is $tau(w)$ times a trace of $v_sigma bar F^sigma_{v'} Gamma H_v$, which vanishes at $w=1$ because $v_sigma bar F^sigma_v=0$. The $1/m_Q$ corrections come from two sources: matching the weak current (insertions of $-i$ left-derivative over $2m_c$ and $i$ right-derivative over $2m_b$, parameterized by functions $tau_j^{(Q)}(w)$) and corrections to the states from the kinetic and chromomagnetic terms in the effective Lagrangian. The load-bearing step is that the zero-recoil piece of the current correction is controlled by a single coefficient $tau_4$, which the equation of motion and the translation identity tie to $(barLambda'-barLambda)tau(1)$. Chromomagnetic contributions drop out at zero recoil because $v_alpha(1+not v)sigma_{alpha beta}(1+not v)=0$, and kinetic-energy contributions merely renormalize $tau$. The same trace technology, applied with the ${1 over 2}^+$ field $K_v$, gives the analogous zero-recoil relations for $D_0^*$ and $D_1^*$, where the smallness of $epsilon_c-3epsilon_b$ makes the enhancement much weaker.
Extended reading notes
Core claim
The central claim is that for $B$ decays into the four lightest excited charmed mesons ($D_1$, $D_2^*$, $D_0^*$, $D_1^*$), the order-$Lambda_{QCD}/m_Q$ corrections to the weak-current matrix elements at zero recoil are computable in terms of the leading Isgur-Wise functions and hadron mass splittings, not new unknown parameters. In the infinite-mass limit these matrix elements vanish because the initial and final mesons sit in different heavy-quark spin multiplets. At order $1/m_Q$ the current-matching corrections survive at zero recoil through a single tensor structure, and the heavy-quark equation of motion plus a translation identity reduce it to $tau_4^{(b)}(1)=-tau_4^{(c)}(1)=(barLambda'-barLambda)tau(1)$, with $barLambda'-barLambda$ already known from $B$ and $D$ mass splittings. Because one form factor, $f_{V1}$, dominates the $D_1$ rate near zero recoil, the ratio of the leading and next-to-leading terms in the $(w-1)$ expansion is fixed by kinematics alone, giving the model-independent slope $rho^2_{D_1}=-[(1+r_1)/(1-r_1)][2m_c/(barLambda'-barLambda)]+O(1)$ for the helicity-zero $D_1$ differential rate. The same machinery yields the leading $1/m_Q$ coefficients for $B$ to $D_2^*$, $D_0^*$ and $D_1^*$, a predicted ratio $R$ of the $D_2^*$ to $D_1$ branching ratios that drops from about 1.65 in the infinite-mass limit to 0.5-0.7, a predicted helicity-zero fraction in $B$ to $D_1 e$ neutrino of roughly 0.8, and an extraction $tau(1)[6.0 times 10^{-3}/Br(B to D_1 e nu)]^{1/2}=0.71$ from the measured branching ratio.
Load-bearing premise
The zero-recoil relation that ties the subleading coefficient to the mass splitting assumes the subleading Isgur-Wise functions $tau_j^{(Q)}(w)$ are continuous at $w=1$, a point the authors flag explicitly after Eq. (2.15).
Editorial extensions
If this is right
- The helicity-zero $D_1$ differential rate has a slope at zero recoil that is fixed by $m_{D_1}/m_B$ and the measured $barLambda'-barLambda$ splitting, so a measurement of that slope is a direct test of the heavy-quark expansion rather than of a form-factor model.
- The ratio $R=Br(B to D_2^* e nu)/Br(B to D_1 e nu)$ is predicted to be well below its infinite-mass value of about 1.65, in the range 0.4 to 0.7, which is consistent with the tight ALEPH upper bound on the $D_2^*$ channel.
- The measured $B$ to $D_1 e$ neutrino branching ratio of about $0.6$ percent implies $tau(1)$ near 0.71, roughly a factor of three enhancement over what the infinite-mass limit would require, showing that $1/m_Q$ corrections dominate these channels.
- Via factorization, the same differential rates at maximal recoil give predictions for the $B$ to $D_1 pi$ and $B$ to $D_2^* pi$ branching ratios; the observed ratio of about 0.2 for $D_1 pi$ over $D_1 e$ neutrino favours the assumed slope $hat tau'$ near $-1.5$.
- Including the excited states strengthens the Bjorken lower bound on the slope of the ground-state Isgur-Wise function to $rho^2>0.75$, and tightens bounds on $lambda_1$ and on the zero-recoil $B$ to $D^*$ axial form factor relevant to $|V_{cb}|$.
Reading between the lines
- If the predicted slope of the helicity-zero $D_1$ rate were confirmed, the same zero-recoil technique would give a clean handle on the subleading Isgur-Wise normalization $tau(1)$ without needing a form-factor model, which is a route to $|V_{cb}|$ that is logically independent of the ground-state $B$ to $D^*$ extraction.
- The continuity assumption at $w=1$ is the kind of premise that lattice calculations could check directly: a determination of $tau_4^{(c)}(w)$ on a few nearby knots would either confirm the extrapolation or reveal a discontinuity that would break the mass-splitting relation.
- The tension the authors note between the sum of exclusive channels (about 8.2 percent of $B$ decays) and the measured inclusive semileptonic rate suggests that the missing strength sits in broad ${1 over 2}^+$ states or non-resonant $D^{(*)}pi$ pairs; measuring the $D_0^*$ and $D_1^*$ line shapes would discriminate between those two options.
- Because the enhancement of the $D_1$ channel hinges on $barLambda'-barLambda$, a better determination of the excited $B$ multiplet mass, for instance from $B_{s1}$ and $B_{s2}^*$ masses, would sharpen the numerical prediction without any new theory input.
Original abstract
Exclusive semileptonic B decays into excited charmed mesons are investigated at order $\Lambda_{QCD}/m_Q$ in the heavy quark effective theory. Differential decay rates for each helicity state of the four lightest excited $D$ mesons ($D_1$, $D_2^*$, $D_0^*$, and $D_1^*$) are examined. At zero recoil, $\Lambda_{QCD}/m_Q$ corrections to the matrix elements of the weak currents can be written in terms of the leading Isgur-Wise functions for the corresponding transition and meson mass splittings. A model independent prediction is found for the slope parameter of the decay rate into helicity zero $D_1$ at zero recoil. The differential decay rates are predicted, including $\Lambda_{QCD}/m_Q$ corrections with some model dependence away from zero recoil and including order $\alpha_s$ corrections. Ratios of various exclusive branching ratios are computed. Matrix elements of the weak currents between $B$ mesons and other excited charmed mesons are discussed at zero recoil to order $\Lambda_{QCD}/m_Q$. These amplitudes vanish at leading order, and can be written at order $\Lambda_{QCD}/m_Q$ in terms of local matrix elements. Applications to $B$ decay sum rules and factorization are presented.
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
Exclusive semileptonic B decays into excited charmed mesons are investigated at order $\Lambda_{QCD}/m_Q$ in the heavy quark effective theory. Differential decay rates for each helicity state of the four lightest excited $D$ mesons ($D_1$, $D_2^*$, $D_0^*$, and $D_1^*$) are examined. At zero recoil, $\Lambda_{QCD}/m_Q$ corrections to the matrix elements of the weak currents can be written in terms of the leading Isgur-Wise functions for the corresponding transition and meson mass splittings. A model independent prediction is found for the slope parameter of the decay rate into helicity zero $D_1$ at zero recoil. The differential decay rates are predicted, including $\Lambda_{QCD}/m_Q$ corrections with some model dependence away from zero recoil and including order $\alpha_s$ corrections. Ratios of various exclusive branching ratios are computed. Matrix elements of the weak currents between $B$ mesons and other excited charmed mesons are discussed at zero recoil to order $\Lambda_{QCD}/m_Q$. These amplitudes vanish at leading order, and can be written at order $\Lambda_{QCD}/m_Q$ in terms of local matrix elements. Applications to $B$ decay sum rules and factorization are presented.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Exclusive semileptonic B decays into excited charmed mesons are investigated at order $\Lambda_{QCD}/m_Q$ in the heavy quark effective theory. | contribution 0.95 | 0.85 | 0.08 | asserted 0.94 |
| s1 | Differential decay rates for each helicity state of the four lightest excited $D$ mesons ($D_1$, $D_2^*$, $D_0^*$, and $D_1^*$) are examined. | contribution 0.95 | 0.86 | 0.06 | asserted 0.98 |
| s2 | At zero recoil, $\Lambda_{QCD}/m_Q$ corrections to the matrix elements of the weak currents can be written in terms of the leading Isgur-Wise functions for the corresponding transition and meson mass splittings. | result 0.82 | 0.76 | 0.16 | asserted 0.99 |
| s3 | A model independent prediction is found for the slope parameter of the decay rate into helicity zero $D_1$ at zero recoil. | result 1.00 | 0.93 | 0.27 | asserted 1.00 |
| s4 | The differential decay rates are predicted, including $\Lambda_{QCD}/m_Q$ corrections with some model dependence away from zero recoil and including order $\alpha_s$ corrections. | result 0.65 | 0.91 | 0.25 | asserted 1.00 |
| s5 | Ratios of various exclusive branching ratios are computed. | result 0.76 | 0.91 | 0.08 | asserted 0.99 |
| s6 | Matrix elements of the weak currents between $B$ mesons and other excited charmed mesons are discussed at zero recoil to order $\Lambda_{QCD}/m_Q$. | contribution 0.95 | 0.81 | 0.09 | asserted 0.90 |
| s7 | These amplitudes vanish at leading order, and can be written at order $\Lambda_{QCD}/m_Q$ in terms of local matrix elements. | result 0.93 | 0.71 | 0.09 | asserted 1.00 |
| s8 | Applications to $B$ decay sum rules and factorization are presented. | contribution 0.81 | 0.84 | 0.09 | asserted 0.98 |
What was examined, and what was not
- arXiv listing abstract v2
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- P abstract
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- T body sentences
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Figures
Figures reproduced from arXiv:hep-ph/9705467 by the authors.
References
How the literature uses it
37 arXiv papers cite this in 69 sentences
Papers citing it · 46 sentences without a confident label
Contests it 2 papers
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Notably, the measurement of Ref. [ref] is in conflict with the measured branching fractions of Refs. [ref] and with the expectation of $ℬ(B →D_1^* ℓν̅_ℓ)$ being of similar size to $ℬ(B →D_0^* ℓν̅_ℓ)$ [ref] .
Measurements of $q^2$ Moments of Inclusive $B \rightarrow X_c \ell^+ \nu_{\ell}$ Decays with Hadronic Tagging · Data Set and Simulated Samples · contest 0.82
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Notably, the measurement of Ref. [ref] is in conflict with the measured branching fractions of Refs. [ref] and with the expectation of $ℬ(B →D_1^* ℓν̅_ℓ)$ being of similar size than $ℬ(B →D_0 ℓν̅_ℓ)$ [ref] .
Measurements of Partial Branching Fractions of Inclusive $B \to X_u \, \ell^+\, \nu_{\ell}$ Decays with Hadronic Tagging · Data Set and Simulated Samples · contest 0.83
Builds on it 1 paper
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Analogously to the sum rule formulated in the mesonic case for the axial-vector current (Eqns (114) and (5.6) of [ref] and [ref] respectively), we have for the form factor $G_1$ at the order $O(1/m_Q^2)$:
On the form factors of semileptonic baryon decays in Heavy Quark Effective Theory · HQET sum rule for the correction $δ^(G_1)_1/m_Q^2$ to the form factor $G_1(1)$ · 0.90
Uses its method or tool 8 papers
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To overcome this difficulty, we use a method, which amounts to compute the operator matrix element based on an expression of the derivative of that matrix element in terms of the recoil four-velocity of the final meson [ref] .
Lattice calculation of the Isgur-Wise functions tau_{1/2} and tau_{3/2} with dynamical quarks · Decay form factors in the heavy quark limit · 0.97
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Following the “Approximation $A$” of [ref] , which means $O(w-1) ∼O(ϵ_c,b)$ and their higher orders are truncated, we use f_V_1(w) = 1/√(6)[(1-w^2)τ(w) - 4ϵ_c(w+1)(wΛ̅' - Λ̅)τ(w) ] and wf_V_1(w) + (w^2 - 1)f_V_3(w) = w/√(6)[(1-w^2)τ(w) - 4ϵ_c(w+1)(wΛ̅' - Λ̅)] + w^2 - 1/√(6)(w-2)τ(w) to extract the Isgur-Wise form factor $τ(w)$.
Study of intermediate states in the inclusive semileptonic $B \rightarrow X_c l \nu$ decay structure function · Results · 1.00
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We follow the approximation A used in [ref] to express $A_V_0 V_0^D_s1$ and $A_V_∥V_∥^D_s1$ (we stress that $A_V_0 V_∥^D_s1$ is not a free parameter in our analysis due to the equality) using the zero-recoil values of the Isgur-Wise form factors $τ(w) ≡√(3) τ_3/2(w)$ and its first derivative $τ(w) = τ_0 + τ^'(w-1) + …$ as [equation] Similarly, the contributions from $D_s0^*$ to $C_A_0 A_0$ and $C_A_∥A_∥$ can also be approximated using the zero-recoil values of the Isgur-Wise form factors $ζ(w) ≡2 τ_1/2(w)$ and its first derivative $ζ(w) = ζ_0 + ζ^'(w-1) + …$ as [equation] The basic idea of approximation A is to treat $w-1$ as the same order as $1/m_Q$ and retain only the second order in $𝒪(w-1)∼𝒪(1/m_Q)$.
Study on the $P$-wave form factors contributing to $ B_s $ to $D_s$ inclusive semileptonic decays from lattice simulations · Numerical results · 1.00
Uses its data or values 2 papers
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To estimate the systematic uncertainties in these results, we consider contributions from the following sources: uncertainties in the background component normalizations, uncertainty in the MC tracking efficiency, errors in the world average of $ℬ(D^*+→D^0π^+)$ and $ℬ(D^0→K^-π^+)$ as well as in the $ℬ(B→D^**ℓν)$ components [ref] , uncertainties in the shape of the $w$ distribution of $B→D^**ℓν$ events based on the LLSW model [ref] , uncertainties in the $B^0$ lifetime [ref] , and the uncertainties in the total number of $B^0$ mesons in the data sample.
Measurement of the form factors of the decay B0 -> D*- ell+ nu and determination of the CKM matrix element |Vcb| · Results of the fits and investigation of the systematic uncertainties in the subsamples · 0.99
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Parameter values used for the transition current $⟨D|_0^* | A^μ| B |$⟩ taken from the pole parametrisation in [ref] .$
Comparing different ansatzes to describe electroweak radiative corrections to exclusive semileptonic B meson decays into (pseudo)scalar final state mesons using Monte-Carlo techniques · Form factors for $B →D^*_0 ℓ ν$ · 0.97
Compares with it 2 papers
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(The $R_3/2 = 1.67$ central value is very close to the `$B_∞$' result, 1.65, in Table II in Ref. [ref] .) The severe tension for $R_3/2$, about $10σ$, is evidence for the presence of large deviations from the heavy quark limit.
Model independent analysis of semileptonic $B$ decays to $D^{**}$ for arbitrary new physics · Fits in the heavy quark limit · 0.96
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Comparison of the zero-recoil form factor for $B→D^**$ with the heavy-quark effective theory expectation [ref] has been attempted [ref] .
Next challenges in semi-leptonic B decays · Next challenge III: $B→D^**$ · 0.91
Cites it as background 4 papers
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From a theoretical perspective, semileptonic $B_(s)$ decays to the excited charmed mesons $D_0(2300)$, $D_s0(2317)$, $D_1(2420)$, $D_1'(2430)$, $D_s1(2460)$, and $D_s1'(2536)$ have been extensively studied using various approaches, including the covariant light-front quark model (CLFQM) [ref] , the Isgur–Scora–Grinstein–Wise (ISGW) model [ref] , QCD sum rules [ref] , light-cone sum rules (LCSRs) [ref] , and heavy quark effective theory (HQET) [ref] .
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At the zeroth order of HQET, the transitions from the $B_s$ to the $D_s$ mesons are described by three sets of form factors $ξ^(n)(w)$, $τ_1/2^(n)(w)$ and $τ_3/2^(n)(w)$ for $S$, $P_1/2$ and $P_3/2$ channels, respectively [ref] .
Study on the $P$-wave form factors contributing to $ B_s $ to $D_s$ inclusive semileptonic decays from lattice simulations · Decompositions of $C_J_μJ_ν$ into different final states · 0.86
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The decays of $B$ mesons into orbitally excited $D$ mesons have been investigated in the context of heavy quark symmetry in [ref] , including also the $1/m_c$ corrections.
$1/m$ Corrections for Orbitally Excited Heavy Mesons and the $1/2$ - $3/2$ Puzzle · Effects on Semileptonic Decays · 0.96
Discussion
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