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Two-loop QCD penguin contribution to the width difference in $B_s-\bar{B}_s$ mixing
High Energy Physics - Phenomenology · Submitted 10 Jun 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our new corrections are comparable in size to the current experimental error and slightly increase $\Delta\Gamma_s$.
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The paper's own sentences, arranged. 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.
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The abstract, sentence by sentence
We consider two-loop QCD corrections to the element $\Gamma_{12}^q$ of the decay matrix in $B_q-\bar{B}_q$ mixing, $q=d,s$, in the leading power of the Heavy Quark Expansion. The calculated contributions involve one current-current and one penguin operator and constitute the next step towards a theory prediction for the width difference $\Delta\Gamma_s$ matching the precise experimental data. We present compact analytic results for all matching coefficients in an expansion in $m_c/m_b$ up to second order. Our new corrections are comparable in size to the current experimental error and slightly increase $\Delta\Gamma_s$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider two-loop QCD corrections to the element $\Gamma_{12}^q$ of the decay matrix in $B_q-\bar{B}_q$ mixing, $q=d,s$, in the leading power of the Heavy Quark Expansion. | contribution 0.97 | 0.74 | 0.05 | asserted 0.79 |
| s1 | The calculated contributions involve one current-current and one penguin operator and constitute the next step towards a theory prediction for the width difference $\Delta\Gamma_s$ matching the precise experimental data. | interpretation 0.46 | 0.86 | 0.17 | asserted 0.94 |
| s2 | We present compact analytic results for all matching coefficients in an expansion in $m_c/m_b$ up to second order. | contribution 0.60 | 0.95 | 0.10 | asserted 1.00 |
| s3 | Our new corrections are comparable in size to the current experimental error and slightly increase $\Delta\Gamma_s$. | result 0.85 | 0.93 | 0.77 | asserted 1.00 |
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The paper
Original abstract
We consider two-loop QCD corrections to the element $\Gamma_{12}^q$ of the decay matrix in $B_q-\bar{B}_q$ mixing, $q=d,s$, in the leading power of the Heavy Quark Expansion. The calculated contributions involve one current-current and one penguin operator and constitute the next step towards a theory prediction for the width difference $\Delta\Gamma_s$ matching the precise experimental data. We present compact analytic results for all matching coefficients in an expansion in $m_c/m_b$ up to second order. Our new corrections are comparable in size to the current experimental error and slightly increase $\Delta\Gamma_s$.
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References
How the literature uses it
13 arXiv papers cite this in 35 sentences
Papers citing it · 24 sentences without a confident label
Uses its method or tool 2 papers
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The update on the theoretical calculation in Refs. [ref] makes use of the fact that the numerical results converge quickly in an expansion in $z≡m_c^2/m_b^2$ and the NNLO QCD corrections are evaluated to $𝒪(z)$ by expanding naively in $m_c$.
NNLO QCD corrections to $\Delta \Gamma_s$ in the $B_s-\overline{B}_s$ system · Technical details on the calculation · 0.80
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As in Ref. [ref] we use the CMM basis [ref] for the $|ΔB|=1$ operators and calculate the two-loop QCD corrections as an expansion in [equation] up to linear order.
The width difference in \bbm\ at order $\alpha_s$ and beyond · Introduction · 0.97
Uses its data or values 1 paper
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The penguin contributions are expanded up to the linear term in $z$ and can be found in Refs. [ref] .
Current-current operator contribution to the decay matrix in $B$-meson mixing at next-to-next-to-leading order of QCD · Matching of $ΔB=1$ and $ΔB=2$ theory · 1.00
Cites it as background 5 papers
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These include NLO QCD corrections to the contributions with current-current and chromomagnetic penguin operators [ref] , the corresponding NNLO corrections (and NLO corrections involving four-quark penguin operators) enhanced by the number $N_f$ of active quark flavours [ref] as well as NLO results with one current-current and one penguin operator [ref] or two penguin operators [ref] .
The width difference in the $B-\bar{B}$ system at next-to-next-to-leading order of QCD · Introduction · 0.81
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The complete set of penguin operators has only been added recently [ref] , where terms of order $z^0$ and $z^1$ have been computed in an expansion for $z→0$.
$B$ meson mixing at NNLO: technical aspects · Introduction and notation · 0.81
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The four-quark penguin operators $Q_3,…,Q_6$ and the chromomagnetic penguin operator $Q_8$ are not needed for the calculation presented in this paper; their definition can be found, e.g. in Refs. [ref] .
Current-current operator contribution to the decay matrix in $B$-meson mixing at next-to-next-to-leading order of QCD · $Δ𝐁 =1$ theory · 1.00
What it cites · 28 sentences without a confident label
Uses the method or tool of 13 papers
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For the NNLO calculation it is instrumental to use the CMM operator basis [ref] as we did in this paper.
|Delta F| = 1 Nonleptonic Effective Hamiltonian in a Simpler Scheme · Conclusions · 0.98
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This non-trivial computational task was done using [ref] , where the special hints described in the manual have been used.
FORM version 4.0 · Projector methodology · 1.00
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For the reduction to master integrals we use FIRE [ref] and LiteRed [ref] .
Presenting LiteRed: a tool for the Loop InTEgrals REDuction · Calculation · 1.00
Uses data or values from 6 papers
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In addition to the quark masses and $α_s$ given above we have the following input parameters [ref] [equation]
$B$- and $D$-meson leptonic decay constants from four-flavor lattice QCD · Analytical and numerical results · 1.00
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In addition to the quark masses and $α_s$ given above we have the following input parameters [ref] [equation]
Neutral B-meson mixing from full lattice QCD at the physical point · Analytical and numerical results · 1.00
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For the input values we use $α_s(M_Z)=0.1179$ [ref] and the $MS$ quark masses $m_c(3 )=0.993$ GeV [ref] and $m_b(m_b)=4.163$ GeV [ref] which leads $m_c(m_b)=0.929$ GeV and $z̅≈0.0497$.
Addendum to arXiv:0907.2110: "Charm and Bottom Quark Masses: An Update" · Analytical and numerical results · 1.00
Compares with 7 papers
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The two-loop terms proportional to $N_L$, $N_V$ and $N_H$ have recently been computed in Ref. [ref] and we find complete agreement after expanding the exact expression up to linear order in $z$ and transforming to the operator basis used in [ref] .
Penguin contribution to width difference and CP asymmetry in $B_q$-$\bar B_q$ mixing at order $\alpha_s^2 N_f$ · Analytical and numerical results · 1.00
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As a byproduct we reproduce the NLO result for the contribution with two copies of $C_1,2$ of Refs. [ref] (expanded in $z$) after transforming the $ΔB=1$ Wilson coefficients to the CMM basis, which is a powerful check of our calculational set-up.
CP asymmetry in flavour-specific B decays beyond leading logarithms · Introduction · 1.00
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As a byproduct we reproduce the NLO result for the contribution with two copies of $C_1,2$ of Refs. [ref] (expanded in $z$) after transforming the $ΔB=1$ Wilson coefficients to the CMM basis, which is a powerful check of our calculational set-up.
Lifetime Differences and CP Violation Parameters of Neutral B Mesons at the Next-to-Leading Order in QCD · Introduction · 1.00
Cites as support 2 papers
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The benefit of using $z̅$ instead of $z$ for the quality of the prediction has been demonstrated in Refs. [ref] and we refrain from using $z$ in our numerics.
The B^+ B_d^0 Lifetime Difference Beyond Leading Logarithms · Analytical and numerical results · 0.99
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The benefit of using $z̅$ instead of $z$ for the quality of the prediction has been demonstrated in Refs. [ref] and we refrain from using $z$ in our numerics.
Theoretical update of Bs-Bs-bar mixing · Analytical and numerical results · 0.99
Cites as background 10 papers
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Next-to-leading logarithmic order (NLO) QCD corrections at leading power in $1/m_b$ have been computed in Refs. [ref] .
Lifetime Differences and CP Violation Parameters of Neutral B Mesons at the Next-to-Leading Order in QCD · Introduction · 0.95
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Next-to-leading logarithmic order (NLO) QCD corrections at leading power in $1/m_b$ have been computed in Refs. [ref] .
CP asymmetry in flavour-specific B decays beyond leading logarithms · Introduction · 0.95
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Next-to-leading logarithmic order (NLO) QCD corrections at leading power in $1/m_b$ have been computed in Refs. [ref] .
Next-to-Leading Order QCD Corrections to the Lifetime Difference of $B_s$ Mesons · Introduction · 0.95
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