Next-to-next-to-leading QCD corrections to the $\mathbf{B^+}$-$\mathbf{B_d^0}$, $\mathbf{D^+}$-$\mathbf{D^0}$, and $\mathbf{D_s^+}$-$\mathbf{D^0}$ lifetime ratios

Francesco Moretti; Matthias Steinhauser; Pascal Reeck; Ulrich Nierste

arxiv: 2604.24841 · v1 · submitted 2026-04-27 · ✦ hep-ph

Next-to-next-to-leading QCD corrections to the B^+-B_d⁰, D^+-D⁰, and D_s^+-D⁰ lifetime ratios

Francesco Moretti , Ulrich Nierste , Pascal Reeck , Matthias Steinhauser This is my paper

Pith reviewed 2026-05-08 02:52 UTC · model grok-4.3

classification ✦ hep-ph

keywords mathbftermsalphacontributionselementsfindhadronicheavy

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The pith

Three-loop perturbative corrections to B and D meson lifetime ratios are calculated, producing values that agree with experiment when using HQET sum rules or lattice inputs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Heavy mesons containing a bottom or charm quark decay at rates that differ slightly between charged and neutral versions. These differences arise mainly from effects suppressed by three powers of the heavy quark mass. The authors calculate the next-to-next-to-leading perturbative QCD pieces of these effects, which involve three-loop diagrams. They do this under exact isospin or V-spin symmetry. The resulting coefficients are then multiplied by hadronic matrix elements taken from separate calculations using heavy quark effective theory sum rules or lattice QCD. The final numbers for the lifetime ratios are 1.072 for B plus over B zero, 2.344 for D plus over D zero, and 1.289 for D_s plus over D zero. These match measured values within uncertainties. The agreement lets the authors estimate the size of still-missing higher-order terms.

Core claim

Combining our perturbative coefficients with hadronic matrix elements calculated from Heavy Quark Effective Theory sum rules, we find τ(B⁺)/τ(B_d⁰)=1.072±0.024. Using hadronic matrix elements from a recent lattice QCD calculation we find τ(D⁺)/τ(D⁰)=2.344±0.170 and τ(D_s⁺)/τ(D⁰)=1.289±0.042. We find good agreement of our predictions with experimental data.

Load-bearing premise

The hadronic matrix elements from HQET sum rules and lattice QCD are accurate enough that the 1/m_Q^4 and V-spin breaking contributions remain smaller than the quoted uncertainties; the calculation is performed in the exact isospin and V-spin symmetric limit.

read the original abstract

The total decay widths of heavy mesons can be systematically calculated in terms of an expansion in the two parameters $1/m_Q$ and $\alpha_s(m_Q)$, where $Q=c,b$ denotes the heavy quark. The dominant contributions to meson lifetime splittings stem from terms which are suppressed by $1/m_Q^3$ with respect to the leading universal contribution to the total decay width. We calculate three-loop contributions of order $\alpha_s^2/m_q^3$ to the lifetime ratios $\tau(B^+)/\tau(B_d^0)$, $\tau(D^+)/\tau(D^0)$, and $\tau(D_s^+)/\tau(D^0)$ in the limit of exact isospin and V-spin symmetry, respectively. Furthermore, we present new $\alpha_s/m_q^3$ corrections to the Cabibbo-suppressed terms in $\tau(B^+)/\tau(B_d^0)$. Combining our perturbative coefficients with hadronic matrix elements calculated from Heavy Quark Effective Theory sum rules, we find $\tau(B^+)/\tau(B_d^0)= {1.072 \pm 0.024 }$. Using hadronic matrix elements from a recent lattice QCD calculation we find $\tau(D^+)/\tau(D^0)={2.344 \pm 0.170} $ and $\tau(D_s^+)/\tau(D^0)={1.289 \pm 0.042}$. We find good agreement of our predictions with experimental data, which constitutes a successful probe of the calculations of hadronic matrix elements and permits estimates of the unknown $1/m_Q^4$ contributions as well as the V-spin breaking terms in $\tau(D_s^+)/\tau(D^0)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers new three-loop perturbative coefficients for B and D meson lifetime ratios that look technically solid, but the headline predictions still rest on external hadronic matrix elements whose uncertainties may not fully cover omitted higher-order terms.

read the letter

The core advance here is the calculation of the three-loop α_s²/m_q³ pieces for the lifetime ratios in the isospin and V-spin symmetric limits, plus the new α_s/m_q³ corrections to the Cabibbo-suppressed contributions in the B sector. Those coefficients were not available before, and the authors combine them with existing HQET sum-rule and lattice matrix elements to produce numerical predictions that sit close to the measured values: 1.072 ± 0.024 for B⁺/B_d⁰, 2.344 ± 0.170 for D⁺/D⁰, and 1.289 ± 0.042 for D_s⁺/D⁰. That is useful concrete output for the heavy-flavor community. The perturbative work itself appears careful and fills a clear gap in the literature. The agreement with data then serves as a consistency check on the matrix elements and gives a rough handle on the size of missing 1/m_Q⁴ pieces. The main limitation is that the quoted errors are driven almost entirely by the external hadronic inputs. For the charm sector the 1/m_c expansion parameter is not small, V-spin breaking is known to matter in other observables, and those effects are left out of the present calculation. The B-sector result is on firmer ground because the expansion converges better and isospin breaking is smaller. Readers who need the new coefficients for their own work will find them directly usable; the paper is aimed at people doing precision calculations in HQET or lattice QCD for lifetimes, mixing, or rare decays. It is technically grounded enough to deserve a serious referee who can check the diagram evaluation and the matching to the matrix elements.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculation rests on the heavy-quark expansion in 1/m_Q and α_s, exact isospin and V-spin symmetry, and the accuracy of external hadronic matrix elements; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Heavy quark expansion in powers of 1/m_Q and α_s(m_Q) is valid for bottom and charm mesons
Invoked in the first sentence of the abstract as the systematic framework for total decay widths.
domain assumption Exact isospin symmetry for B mesons and exact V-spin symmetry for D mesons
Explicitly stated as the limit in which the three-loop contributions are computed.

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