Perturbative corrections to semileptonic b decay distributions

G. Ridolfi; N. Uraltsev; P. Gambino; V. Aquila

REVIEW

Perturbative corrections to semileptonic b decay distributions

V. Aquila, P. Gambino, G. Ridolfi, N. Uraltsev

High Energy Physics - Phenomenology · Submitted 8 Mar 2005 · 32 references

Pith's verdict: A solid, well-cross-checked one-loop plus BLM calculation for semileptonic b decay; the tau-sector structure functions W4 and W5 are the one genuinely unverified piece, and the authors say so themselves.

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arXiv IDhep-ph/0503083 (version 2)

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CitationG. Ridolfi, N. Uraltsev, P. Gambino, V. Aquila. "Perturbative corrections to semileptonic b decay distributions." arXiv:hep-ph/0503083 (2005). https://pith.science/paper/LUB6JODO

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The thing to know: this is the reference calculation for the O(alpha_s) and BLM corrections to the semileptonic b -> q l nu structure functions, and it holds up. The soft-divergence handling is the technical core and it is done properly. The log(lambda) in Eq. (2.29) cancels against Eq. (2.46), and the finite remainder S(q^2) in Eq. (2.48) is defined cleanly through a generalized plus distribution (Eq. 2.26). That is what makes the numerics stable, and it is a real improvement on what was in the literature before. What is new beyond ref. [17]: the subtraction scheme itself, the BLM representation for the triple differential rate (Eq. 3.15), and the L(2), L(3), H(2), H(3) coefficients in Tables 1 to 4, which the authors state were not previously reported. The cross-checks are the right ones: total width against [12, 21, 22], leptonic moments against [15, 23], hadronic moments against [13, 16]. Appendix C's analytic moments with a lepton-energy cut are a useful bonus. Now the soft spot, and the stress-test note is right about it. The five structure functions include W4 and W5, which only matter for tau. For W1 to W3 the calculation is anchored by the massless-lepton cross-checks. For W4 and W5 it is not: every table entry is massless-lepton, and the only independent calculation (ref. [17]) disagrees on the virtual pieces. The authors report the disagreement openly and assert their result is right, but they do not show which integral differs. That is a correctness question in a sector the paper itself flags, and it is the one place where I would want an independent check before trusting B -> Xc tau nu numbers from this code. It does not undermine the Vcb-relevant massless-lepton results, which is where the paper's real weight sits. Two smaller things: the code is on request only, so the tables are not regenerable from the paper alone, and no numerical error estimates accompany them. And the BLM truncation is by construction the large-beta0 ladder with the non-BLM remainder unbounded, so the higher-order columns are indicative rather than predictive. The authors are upfront about the second; the first is a practical annoyance. Who it is for: anyone computing semileptonic moments, fitting Vcb, or needing the tau case. Worth a serious referee. The tau-sector discrepancy should be raised explicitly in review, but it is a fixable gap, not a reason to reject.

Referee report

Major objections (3)

[Section 2, paragraph following Eq. (2.48)] The paper reports a direct disagreement with ref. [17] on the virtual contributions to W3, W4 and W5 (in the notation of [17]) and does not resolve it: 'the virtual contributions to W3, W4, W5 ... are different from our results. The discrepancy, however, is such that our V1, V2, V3 are reproduced correctly. Therefore, a difference should be observed when computing the decay rate into massive leptons only.' This is precisely the sector that motivates computing all five structure functions (Section 1 and Section 5 both state that W4 and W5 are relevant for B -> Xc tau nu), and it is the one sector where the paper's own numerical checks are blind: every entry of Tables 1-4 is for massless leptons, for which Eq. (2.4) shows W4 and W5 drop out. The tau-relevant part of the calculation therefore rests on a single evaluation with no independent confirmation, while the only independent calculation disagrees. Three concrete requests would settle this: (a) state whether the W4 and W5 virtual pieces were obtained by both of the two methods the paper says it used ('We calculated the structure functions in both ways', Section 2), and whether those two methods agree; (b) identify which term in Eq. (2.40) or Eq. (B.12) is the source of the difference with [17], as was done for V1-V3; (c) provide at least one tau-sensitive numerical result, for example the tau energy spectrum or the tau-mass dependence of a moment, so that the massive-lepton sector has a check that does not rely on the massless-lepton tables.
[Section 3, Eq. (3.3) and Tables 1-4] The BLM truncation is acknowledged in Eq. (3.3), a_{n+2} = a^{BLM}_{n+2} beta_0^{n+1} + O(beta_0^n), so the approximation itself is not at issue. What remains unresolved is how the third and fourth columns of Tables 1-4 are presented: L(2), L(3), H(2) and H(3) are given as reference numbers with no indication of the size of the dropped O(beta_0^n) term, and Section 5's claim to be 'far more general than those present in the literature' is a statement about coverage of the beta_0 ladder rather than about accuracy of the full series. Please label these columns explicitly as BLM-only in the table captions, and, where a comparison is possible, quote the known exact two-loop values at the zero-recoil point [26] next to the corresponding BLM entry so the reader can gauge the non-BLM remainder. If no such comparison exists for the moments, say so explicitly rather than leaving the reader to infer that the higher-order columns are predictive.
[Section 3, Eq. (3.15)] Eq. (3.15) requires the real-emission one-loop differential rate da^R_1/dq^2 du at finite gluon mass lambda, that is, the lambda-dependent structure functions R_i. The paper states that these were computed analytically ('we have performed analytically the calculation of the structure functions with a finite gluon mass'), but Eqs. (2.32)-(2.38) print only the lambda -> 0 limit. As a result the BLM entries of Tables 1-4 cannot be checked from the manuscript alone, and the general formula of Eq. (3.15) is not usable without the code. Please include the finite-lambda structure functions in an appendix, or state precisely in what form and where they are available, so that the BLM numbers are reproducible from the paper.

Minor objections (7)

[Tables 1 and 3] The n = 0 entries of Table 1 and the (i,j) = (0,0) entries of Table 3 both represent the total rate and should be identical, but L(1)_0 = -1.777588 while H(1)_00 = -1.777615, a difference of about 3 x 10^{-5}, whereas the tree-level and BLM entries agree exactly. The same pattern appears in reverse between Table 2 (L(0)_0 = 0.814810) and Table 4 (H(0)_00 = 0.814811). Please state the numerical precision of the code and reconcile these entries, since they are the reader's only handle on the accuracy of the integration.
[Section 4 and Appendix C] The analytic moments M10, M20 and M11 of Appendix C are given for arbitrary cut xi, but no numerical evaluation is shown that connects them to Tables 1-4, which correspond to xi = 0 and xi = 2(1 GeV)/m_b. One explicit evaluation of, say, M10 at xi = 0 reproducing the corresponding entry of Table 3 would let the reader verify the Appendix C expressions independently of the code.
[Eqs. (2.32)-(2.38)] The R_i contain explicit 1/lambda_b and 1/lambda_b^{3/2} factors multiplied by log tau, and lambda_b vanishes at the upper endpoint u_+. Please state that the apparent singularities at lambda_b = 0 are integrable against the phase-space measure and describe how the numerical integration handles them, since this is exactly the region where a naive implementation of the plus distribution would fail.
[Section 3, final paragraph] The conversion to short-distance masses is described as 'elementary' but is not given, and all tables are in the pole scheme with m_b = 4.6 GeV and m_c = 1.15 GeV, which the authors state do not correspond to the physical pole masses. Since the stated goal is a tool for precision phenomenology, please give the conversion explicitly, or add at least one table entry in a short-distance scheme such as the kinetic mass.
[Reference [2]] The first item of ref. [2] reads 'Delphi, babar hadr exp results' and is not a usable citation; please supply the proper references for the Delphi and BaBar hadronic-moment measurements.
[Eq. (2.4)] The sign of the W3 term in Eq. (2.4) depends on the convention for epsilon^{0123}. Please state the convention explicitly, since this term is one of those that matters for the massive-lepton case.
[Abstract and Section 1] The abstract refers to b -> q l nu while the body computes b -> c l nu with rho = m_c^2/m_b^2, the b -> u case being the rho -> 0 limit. A one-line clarification in the abstract would avoid the impression that the massless-charm case is treated throughout.

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The paper

The paper claims

The paper derives the complete one-loop QCD corrections to all five structure functions that govern the differential rate of semileptonic bottom-quark decay, with the soft singularity subtracted so the result can be integrated numerically,.

The reading

Inclusive semileptonic $B$ decays are now measured well enough that the theoretical uncertainty from QCD corrections is becoming the limiting factor in extracting $|V_{cb}|$ and the heavy-quark masses. The paper's aim is to put the perturbative side of that accounting on a complete footing: it computes the order-$O(α_s)$ corrections to all five hadronic structure functions $W_i$ entering the triple differential rate for $b → qℓν̄$, not just the three that survive when the leptons are treated as massless. The technical obstacle is that the gluon-emission contribution diverges logarithmically as the gluon mass goes to zero near the edge of phase space, and the paper's central practical move is to isolate that divergence and absorb it into a generalized plus distribution, leaving a finite remainder that can be integrated over any kinematic region. Building on the same gluon-mass regulator, the authors then obtain the leading large-$β_0$ (BLM) tower $O(α_s^n β_0^{n-1})$ for the differential rate, and they tabulate leptonic and hadronic moments with and without a lepton-energy cut. If the results hold, a single set of analytic formulae plus a numerical integrator covers arbitrary experimental cuts, including the $τ$ channel where all five structure functions matter.

What carries the argument

The load-bearing device is the treatment of the soft singularity. The real-emission contribution is written as $D(q̂^2,û,λ̂^2)$, which in the massless-gluon limit becomes $D_0(q̂^2,û)/û$ and diverges logarithmically at $û = 0$; the authors add and subtract a term $A(q̂^2,λ)δ(û)$, where $A$ is the integral of the endpoint approximation $D_s$ over the allowed range, so that the bracketed remainder is regular and the divergence sits entirely in $A$. The singular piece of $A$, $A ∼ (2 - a^{-1}log((1+a)/(1-a)))log λ̂$ with $a = √{λ_b^0}/ω̂$, cancels exactly against the corresponding soft logarithm $V_0(q̂^2,λ)$ from the virtual diagrams. What survives is a plus distribution $(1/û)_+$ defined by $∫ dû (1/û)_+ f(û) = ∫ dû [f(û)-f(0)]/û$, which is what makes the result integrable with arbitrary cuts. The same finite-gluon-mass machinery then feeds the BLM resummation, since the $λ^2$ dependence of the one-loop coefficient is exactly what the large-$β_0$ weight in Eq. (3.5) requires.

Extended reading notes

Core claim

The paper's claim is that the full order-$α_s$ structure functions for $b → cℓν̄$ can be written compactly and analytically, in the form of Eq. (2.47): a tree-level term proportional to $δ(û)$, a virtual-plus-soft correction proportional to $δ(û)$ and built from the finite combination $S(q̂^2) = lim_{λ→0}[A(q̂^2,λ) - V_0(q̂^2,λ)]$, and a real-emission piece split into a plus-distribution term $T_i D_0 (1/û)_+$ plus regular functions $R_i$. The authors state that they agree with all the real-emission contributions of the earlier calculation of Ref. [17] but disagree with its virtual contributions to $W_3$, $W_4$ and $W_5$, while their $V_1, V_2, V_3$ are reproduced correctly; the discrepancy would therefore show up only in the decay rate into massive leptons. On top of the one-loop result, Section 3 gives a convergent integral representation, Eq. (3.15), for the BLM coefficients of the triple differential rate, obtained by integrating the finite-gluon-mass calculation against a weight in $λ^2$.

Load-bearing premise

The higher-order numbers in the tables are the large-$β_0$ part of the series only, and the paper never estimates the size of the terms it drops, so the third and fourth columns are indicative of the running-coupling ladder rather than a bound on the full perturbative correction.

Editorial extensions

If this is right

Any moment of the semileptonic decay distribution — leptonic energy, hadronic invariant mass, hadronic energy — can be computed at order $α_s$ with arbitrary experimental cuts from one set of formulae, rather than case by case.
Decays into $τ$ leptons become accessible at the same order, because all five structure functions are given, not just the three that survive for massless leptons.
The BLM tower $O(α_s^n β_0^{n-1})$ can be evaluated for the differential rate and hence for truncated moments, extending earlier BLM results that existed only for the total width and the lepton energy spectrum.
The tabulated numbers in Tables 1 to 4 give reference values for the first few leptonic and hadronic moments at tree level, one loop, and the second and third BLM orders, in the pole-mass scheme with $m_b = 4.6$ GeV and $m_c = 1.15$ GeV.
Because the authors' virtual contributions to $W_3$, $W_4$ and $W_5$ differ from Ref. [17] while $V_1, V_2, V_3$ agree, the massive-lepton channel is the place where one of the two calculations must fail.

Reading between the lines

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

The disagreement with Ref. [17] is a concrete, resolvable discrepancy rather than a matter of convention: an independent evaluation of the $b → cτν̄$ rate at order $α_s$ would decide it, and the paper supplies enough intermediate structure ($V_i$, $I_x$, $I_y$, $I_{xy}$) for that check to be made without redoing the loop integrals.
The plus-distribution construction is generic: the same add-and-subtract of an endpoint integral could be applied to other heavy-quark spectra where a soft logarithm sits at a kinematic edge, such as $b → u$ or radiative decays, and the paper notes that the $b → u$ structure functions follow as the $m_c → 0$ limit.
The higher-order columns of the tables inherit the usual status of BLM results: they track the piece of the series that runs with the number of light fermions, so their use in fitting $|V_{cb}|$ carries an unquantified remainder that only a full two-loop calculation at the relevant kinematic points could bound.

Original abstract

We compute O(alpha_s) and O(alpha_s^n beta_0^{n-1}) (BLM) corrections to the five structure functions relevant for b->q l nu decays and apply the results to the moments of a few distributions of phenomenological importance. We present compact analytic one-loop formulae for the structure functions, with proper subtraction of the soft divergence.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We compute O(alpha_s) and O(alpha_s^n beta_0^{n-1}) (BLM) corrections to the five structure functions relevant for b->q l nu decays and apply the results to the moments of a few distributions of phenomenological importance. We present compact analytic one-loop formulae for the structure functions, with proper subtraction of the soft divergence.

background contribution method result interpretation main claim

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s0	We compute O(alpha_s) and O(alpha_s^n beta_0^{n-1}) (BLM) corrections to the five structure functions relevant for b->q l nu decays and apply the results to the moments of a few distributions of phenomenological importance.	contribution 0.99	0.96	0.44	asserted 1.00
s1	We present compact analytic one-loop formulae for the structure functions, with proper subtraction of the soft divergence.	contribution 0.52	0.95	0.18	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.92.

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References

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hep-ph 2011-02 accept novelty 2.0 of 10

A 35-year chronicle of next-to-leading and next-to-next-to-leading order QCD corrections to weak decays, with reference tables, an eight-class taxonomy of correction structures, and personal anecdotes.

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47 later papers cite it

Evidence: substantial. 33 of the citing papers share no author with it, and 6 of those carry a confident label from Pith's citation classifier.

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1 paper by the same authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 4 papers with this label · 1 by the same authors

By the same authors

Performed the calculation of BLM corrections using the massive gluon technique and integration over the gluon mass, a standard procedure described in this paper.

Inclusive semileptonic B decays and the determination of |V_ub| (2007) (same authors) · 0.90 · paraphrased: the licence does not allow quoting

How the literature uses it

44 arXiv papers cite this in 76 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 50 sentences without a confident label

Uses its method or tool 4 papers

In the analysis of [ref] the theoretical prediction for the moments $M_HQE$ are calculated perturbatively in a Heavy-Quark Expansion (HQE) in the kinetic-mass scheme up to $𝒪(1/m_b^3)$ with perturbative contributions [ref] resulting in a dependence on six parameters: the running masses of the b- and c-quarks, $m_b(μ)$ and $m_c(μ)$, the parameters $$ and $$ at $𝒪(1/m_b^2)$ in the HQE, and, at $𝒪(1/m_b^3)$, the parameters $$ and $$.

Limit on a Right-Handed Admixture to the Weak $b \to c$ Current from Semileptonic Decays · Fit Setup · 0.82
The $O(^2 β_0)$ corrections to the hadronic structure functions and therefore to the triple differential distribution have been computed in Ref. [ref] , which gives explicit numerical results for the leptonic and hadronic form factors, as well as a code implementing the calculation.

The $q^2$ moments in inclusive semileptonic $B$ decays · BLM corrections · 0.86
The original code underwent a subsequent upgrade to include the full $O(β_0 ^2)$, computed numerically following [ref] and implemented through precise interpolation formulas, and in this form has been employed in recent HFAG analyses.

B semileptonic moments at NNLO · Hadronic mass moments at NNLO · 0.95

Uses its data or values 3 papers

They read W^(1)_i=w_i^(0) { S_i δ(û) -2 (1 - E_0 I_1) [1/û]_+ +θ(û)/(ρ+ û) } + R_i θ(û) , where $S_i= S+Δ_i$ and S = 2E_0 (I_2,0-I_4,0) -1 -1-ρ-6q̂^2/4q̂^2 lnρ-(1 - ρ)^2 - 6 q̂^2 (1 + ρ) + 5 (q̂^2)^2 /4q̂^2 I_1,0 ; Δ_1 = -ρ/E_0 I_1,0; Δ_2= 1-ρ/4q̂^2 lnρ+((1-ρ)^2/4q̂^2-1+ρ/4)I_1,0; Δ_3=0, and the functions $R_i$ are given in Eqs. (2.32-2.34) of Ref. [ref] .[The variables $ω̂$, $λ_b$, and $τ$ of Ref. [ref] correspond to $-2 E_0$, $λ$ and $(1-t)/(1+t)$, respectively.] The integrals $I_1$, $I_1,0$, $I_2,0$, and $I_4,0$ are given in Eqs. (A.6-8) of [ref] and the plus distribution is defined by its action on a generic test function $f(û)$:[ Ref. [ref] uses $û_+$ as upper limit, and the two definitions can be easily related, see [ref] .] ∫f(û) [1/û]_+dû = ∫_0^1 f(û)-f(0)/û dû.

Perturbative corrections to power suppressed effects in $\bar B\to X_u\ell\nu$ · Notation and $O()$ corrections · 0.81
They read W^(1)_i=w_i^(0) { S_i δ(û) -2 (1 - E_0 I_1) [1/û]_+ +θ(û)/(ρ+ û) } + R_i θ(û) , where $S_i= S+Δ_i$ and S = 2E_0 (I_2,0-I_4,0) -1 -1-ρ-6q̂^2/4q̂^2 lnρ-(1 - ρ)^2 - 6 q̂^2 (1 + ρ) + 5 (q̂^2)^2 /4q̂^2 I_1,0 ; Δ_1 = -ρ/E_0 I_1,0; Δ_2= 1-ρ/4q̂^2 lnρ+((1-ρ)^2/4q̂^2-1+ρ/4)I_1,0; Δ_3=0, and the functions $R_i$ are given in Eqs. (2.32-2.34) of Ref. [ref] .[The variables $ω̂$, $λ_b$, and $τ$ of Ref. [ref] correspond to $-2 E_0$, $λ$ and $(1-t)/(1+t)$, respectively.] The integrals $I_1$, $I_1,0$, $I_2,0$, and $I_4,0$ are given below in Eqs. ([ref],[ref],[ref]) and the plus distribution is defined by [1/û]_+ =lim_ε→0 [ lnε δ(û) + 1/û θ(û - ε) θ(1-û) ]

Kinetic operator effects in B-> X_c l nu at O(alpha_s) · Leading Order results · 0.83
Gambino for providing the results of Refs. [ref] in electronic form.

Kolya: an open-source package for inclusive semileptonic $B$ decays · Acknowledgments · 0.91

Compares with it 5 papers

Starting from the $O()$ contributions to the form factors $W_i$ given in [ref] , we find excellent numerical agreement with [ref] , namely [equation] where $𝒞_0^NLO,F$ is given in Eq. (50) of [ref] .

The $q^2$ moments in inclusive semileptonic $B$ decays · The calculation · 0.95
Ref. [ref] uses $û_+$ as upper limit, and the two definitions are related by the simple expression [1/û]_+,[10] =[1/û]_+ - lnû_+ δ(û)- θ(û - û_+) θ(1-û)/û.

Kinetic operator effects in B-> X_c l nu at O(alpha_s) · Leading Order results · 0.84
We compare the lattice results with the corresponding OPE prediction (red curve) including $O(1/m^3)$ [ref] and $O(α_s)$ [ref] terms with $α_s=0.27$.

Inclusive semi-leptonic decays from lattice QCD · Introduction · 0.81

Cites it as support 1 paper

However, we point out that also in the $MS$ mass scheme, the $α_s$ corrections to the semileptonic decay rates have a bad convergence as well if one uses the normalization point $μ=m_Q$ [ref] .

The Heavy Quark Expansion for Inclusive Semileptonic Charm Decays Revisited · Charm-Quark Mass and QCD Corrections · 0.98

Cites it as background 12 papers

The Wilson coefficient $C^Γ_Q̅Q$ has been calculated to the $𝒪(α_s^3)$ order [ref] , where the highest order calculation has been done recently [ref] .

Determination of $|V_{cb}|$ using N$^3$LO perturbative corrections to $\Gamma(B \to X_c \ell \nu)$ and 1S masses · Introduction · 0.85
The leading terms in these double expansions are given by the free $b$ quark decays, while the $O(α_s,α_s^2 β_0)$ perturbative corrections [ref] and the $O (Λ^2_QCD/m_b^2, Λ^3_QCD/m_b^3)$ non-perturbative corrections [ref] have been known for a long time.

Perturbative corrections to power suppressed effects in semileptonic B decays · Introduction · 0.88
In the free-quark approximation, corrections up to $O(α_s^2)$ are available [ref] together with the leading $β_0$ terms at higher orders [ref] , where $β_0$ is the one-loop coefficient of the QCD beta function.

Third order corrections to the semi-leptonic \boldmath{$b\to c$} and the muon decays · Introduction · 0.91

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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