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Cited by 22
- 20 June 2025
- Published by Springer Nature in Journal of High Energy Physics
One-loop renormalization group equations in generic effective field theories. Part I. Bosonic operators
by Mikołaj Misiak, Ignacy Nałȩcz
We consider a generic class of effective quantum field theories with arbitrary gauge groups and scalar matter fields. In such theories, we derive the one-loop Renormalization Group Equations (RGEs) for the physical dimension-six operators. The present paper is the first one in a series that is going to cover theories with spin- $$ \frac{1}{2} $$ $\frac{1}{2}$ matter fields, too, including the phenomenologically most relevant SMEFT and LEFT cases. Our present approach provides tools for deriving the yet unknown two-loop RGEs in the SMEFT.
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- 20 June 2025
- Published by Springer Nature in Journal of High Energy Physics
Top Cited Papers
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One-loop renormalization group equations in generic effective field theories. Part I. Bosonic operators
by Mikołaj Misiak, Ignacy Nałȩcz
We consider a generic class of effective quantum field theories with arbitrary gauge groups and scalar matter fields. In such theories, we derive the one-loop Renormalization Group Equations (RGEs) for the physical dimension-six operators. The present paper is the first one in a series that is going to cover theories with spin- $$ \frac{1}{2} $$ $\frac{1}{2}$ matter fields, too, including the phenomenologically most relevant SMEFT and LEFT cases. Our present approach provides tools for deriving the yet unknown two-loop RGEs in the SMEFT.
Cited by 22
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Cited by 2
- 5 February 2025
- Published by Springer Nature in Journal of High Energy Physics
Complete $ \mathcal{O}\left({\alpha}_s^2\right) $ corrections to the leptonic invariant mass spectrum in $ b\to {X}_cl{\overline{\nu}}_l $ decay
by Mateusz Czaja, Mikołaj Misiak, Abdur Rehman
In the determination of the Cabibbo-Kobayashi-Maskawa matrix element |V _cb| from inclusive semileptonic B-meson decays, moments of the leptonic invariant mass spectrum constitute valuable observables. To evaluate them with sufficient precision, perturbative $$ \mathcal{O}\left({\alpha}_s^2\right) $$ $O (α_{s}^{2})$ corrections to the analogous spectrum in the partonic $$ b\to {X}_cl{\overline{\nu}}_l $$ $b \to X_{c} l {\bar{ν}}_{l}$ decay are necessary. In the present paper, we compute such perturbative corrections in a complete manner, including contributions from the triple-charm channel, namely from the $$ cc\overline{c}l{\overline{\nu}}_l $$ $cc \bar{c} l {\bar{ν}}_{l}$ final states. We present our results in terms of numerical fits in both the single- and triple-charm cases. We confirm the recently found results for the single-charm correction, and analyze the triple-charm channel impact on centralized moments of the spectrum.
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- 5 February 2025
- Published by Springer Nature in Journal of High Energy Physics
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Complete $ \mathcal{O}\left({\alpha}_s^2\right) $ corrections to the leptonic invariant mass spectrum in $ b\to {X}_cl{\overline{\nu}}_l $ decay
by Mateusz Czaja, Mikołaj Misiak, Abdur Rehman
In the determination of the Cabibbo-Kobayashi-Maskawa matrix element |V _cb| from inclusive semileptonic B-meson decays, moments of the leptonic invariant mass spectrum constitute valuable observables. To evaluate them with sufficient precision, perturbative $$ \mathcal{O}\left({\alpha}_s^2\right) $$ $O (α_{s}^{2})$ corrections to the analogous spectrum in the partonic $$ b\to {X}_cl{\overline{\nu}}_l $$ $b \to X_{c} l {\bar{ν}}_{l}$ decay are necessary. In the present paper, we compute such perturbative corrections in a complete manner, including contributions from the triple-charm channel, namely from the $$ cc\overline{c}l{\overline{\nu}}_l $$ $cc \bar{c} l {\bar{ν}}_{l}$ final states. We present our results in terms of numerical fits in both the single- and triple-charm cases. We confirm the recently found results for the single-charm correction, and analyze the triple-charm channel impact on centralized moments of the spectrum.
Cited by 2
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- 19 November 2024
- Published by arXiv in arXiv
Complete $\mathcal{O}(α_s^2)$ Corrections to the Leptonic Invariant Mass Spectrum in $b\to X_c l\barν_l$ Decay
by Mateusz Czaja, Mikołaj Misiak, Abdur Rehman
In the determination of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ from inclusive semileptonic $B$-meson decays, moments of the leptonic invariant mass spectrum constitute valuable observables. To evaluate them with sufficient precision, perturbative $\mathcal{O}(α_s^2)$ corrections to the analogous spectrum in the partonic $b\to X_c l\barν_l$ decay are necessary. In the present paper, we compute such perturbative corrections in a complete manner, including contributions from the triple-charm channel, namely from the $cc\bar{c}l\barν_l$ final states. We present our results in terms of numerical fits in both the single- and triple-charm cases. We confirm the recently found results for the single-charm correction, and analyze the triple-charm channel impact on centralized moments of the spectrum.
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- 19 November 2024
- Published by arXiv in arXiv
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Complete $\mathcal{O}(α_s^2)$ Corrections to the Leptonic Invariant Mass Spectrum in $b\to X_c l\barν_l$ Decay
by Mateusz Czaja, Mikołaj Misiak, Abdur Rehman
In the determination of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ from inclusive semileptonic $B$-meson decays, moments of the leptonic invariant mass spectrum constitute valuable observables. To evaluate them with sufficient precision, perturbative $\mathcal{O}(α_s^2)$ corrections to the analogous spectrum in the partonic $b\to X_c l\barν_l$ decay are necessary. In the present paper, we compute such perturbative corrections in a complete manner, including contributions from the triple-charm channel, namely from the $cc\bar{c}l\barν_l$ final states. We present our results in terms of numerical fits in both the single- and triple-charm cases. We confirm the recently found results for the single-charm correction, and analyze the triple-charm channel impact on centralized moments of the spectrum.
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Cited by 17
- 18 July 2024
- Published by MDPI AG in Symmetry
Current Status of the Standard Model Prediction for the B_s → μ⁺μ⁻ Branching Ratio
by Mateusz Czaja, Mikołaj Misiak
The rare decay $B_{s} \to μ^{+} μ^{-}$ provides an important constraint on possible deviations from the Standard Model in b-s-ℓ-ℓ interactions. The present weighted average of its branching ratio measurements amounts to $(3.34 \pm 0.27) \times 10^{- 9}$ , which remains in good agreement with the theoretical prediction of $(3.64 \pm 0.12) \times 10^{- 9}$ within the Standard Model. In the present paper, we review calculations that have contributed to this prediction and discuss the associated uncertainties.
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- 18 July 2024
- Published by MDPI AG in Symmetry
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Current Status of the Standard Model Prediction for the B_s → μ⁺μ⁻ Branching Ratio
by Mateusz Czaja, Mikołaj Misiak
The rare decay $B_{s} \to μ^{+} μ^{-}$ provides an important constraint on possible deviations from the Standard Model in b-s-ℓ-ℓ interactions. The present weighted average of its branching ratio measurements amounts to $(3.34 \pm 0.27) \times 10^{- 9}$ , which remains in good agreement with the theoretical prediction of $(3.64 \pm 0.12) \times 10^{- 9}$ within the Standard Model. In the present paper, we review calculations that have contributed to this prediction and discuss the associated uncertainties.
Cited by 17
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Cited by 5
- 23 April 2024
- Published by Elsevier in Nuclear Physics B
Erratum to “Two-loop QCD anomalous dimensions of flavour-changing four-quark operators within and beyond the standard model” [Nucl. Phys. B 586 (2000) 397]
by Andrzej J. Buras, Mikołaj Misiak, Jörg Urban
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- 23 April 2024
- Published by Elsevier in Nuclear Physics B
Erratum to “Two-loop QCD anomalous dimensions of flavour-changing four-quark operators within and beyond the standard model” [Nucl. Phys. B 586 (2000) 397]
by Andrzej J. Buras, Mikołaj Misiak, Jörg Urban
Cited by 5
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Cited by 8
- 6 December 2023
- Published by Springer Nature in The European Physical Journal C
The $Q_{1,2}$–$Q_7$ interference contributions to $b \rightarrow s \gamma $ at ${\mathcal O}(\alpha _{\mathrm s}^2)$ for the physical value of $m_c$
by M. Czaja, M. Czakon, T. Huber, M. Misiak, M. Niggetiedt, A. Rehman, K. Schönwald, M. Steinhauser
The $$\bar{B}\rightarrow X_s\gamma $$ $\bar{B} \to X_{s} γ$ branching ratio is currently measured with around $$5\%$$ $5 %$ accuracy. Further improvement is expected from Belle II. To match such a precision on the theoretical side, evaluation of $${\mathcal O}(\alpha _{\mathrm s}^2)$$ $O (α_{s}^{2})$ corrections to the partonic decay $$b \rightarrow X_s^\textrm{part}\gamma $$ $b \to X_{s}^{part} γ$ are necessary, which includes the $$b \rightarrow s \gamma $$ $b \to s γ$ , $$b \rightarrow s g\gamma $$ $b \to s g γ$ , $$b \rightarrow s gg\gamma $$ $b \to s g g γ$ , $$b \rightarrow sq\bar{q}\gamma $$ $b \to s q \bar{q} γ$ decay channels. Here, we evaluate the unrenormalized contribution to $$b \rightarrow s \gamma $$ $b \to s γ$ that stems from the interference of the photonic dipole operator $$Q_7$$ $Q_{7}$ and the current–current operators $$Q_1$$ $Q_{1}$ and $$Q_2$$ $Q_{2}$ . Our results, obtained in the cut propagator approach at the 4-loop level, agree with those found in parallel by Fael et al. who have applied the amplitude approach at the 3-loop level. Partial results for the same quantities recently determined by Greub et al. agree with our findings, too.
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- 6 December 2023
- Published by Springer Nature in The European Physical Journal C
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The $Q_{1,2}$–$Q_7$ interference contributions to $b \rightarrow s \gamma $ at ${\mathcal O}(\alpha _{\mathrm s}^2)$ for the physical value of $m_c$
by M. Czaja, M. Czakon, T. Huber, M. Misiak, M. Niggetiedt, A. Rehman, K. Schönwald, M. Steinhauser
The $$\bar{B}\rightarrow X_s\gamma $$ $\bar{B} \to X_{s} γ$ branching ratio is currently measured with around $$5\%$$ $5 %$ accuracy. Further improvement is expected from Belle II. To match such a precision on the theoretical side, evaluation of $${\mathcal O}(\alpha _{\mathrm s}^2)$$ $O (α_{s}^{2})$ corrections to the partonic decay $$b \rightarrow X_s^\textrm{part}\gamma $$ $b \to X_{s}^{part} γ$ are necessary, which includes the $$b \rightarrow s \gamma $$ $b \to s γ$ , $$b \rightarrow s g\gamma $$ $b \to s g γ$ , $$b \rightarrow s gg\gamma $$ $b \to s g g γ$ , $$b \rightarrow sq\bar{q}\gamma $$ $b \to s q \bar{q} γ$ decay channels. Here, we evaluate the unrenormalized contribution to $$b \rightarrow s \gamma $$ $b \to s γ$ that stems from the interference of the photonic dipole operator $$Q_7$$ $Q_{7}$ and the current–current operators $$Q_1$$ $Q_{1}$ and $$Q_2$$ $Q_{2}$ . Our results, obtained in the cut propagator approach at the 4-loop level, agree with those found in parallel by Fael et al. who have applied the amplitude approach at the 3-loop level. Partial results for the same quantities recently determined by Greub et al. agree with our findings, too.
Cited by 8
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Cited by 29
- 30 November 2021
- Published by Springer Nature in Journal of High Energy Physics
General non-leptonic ∆F = 1 WET at the NLO in QCD
by Jason Aebischer, Christoph Bobeth, Andrzej J. Buras, Jacky Kumar, Mikołaj Misiak
We reconsider the complete set of four-quark operators in the Weak Effective Theory (WET) for non-leptonic ∆F = 1 decays that govern s → d and b → d, s transitions in the Standard Model (SM) and beyond, at the Next-to-Leading Order (NLO) in QCD. We discuss cases with different numbers N_f of active flavours, intermediate threshold corrections, as well as the issue of transformations between operator bases beyond leading order to facilitate the matching to high-energy completions or the Standard Model Effective Field Theory (SMEFT) at the electroweak scale. As a first step towards a SMEFT NLO analysis of K → ππ and non-leptonic B-meson decays, we calculate the relevant WET Wilson coefficients including two-loop contributions to their renormalization group running, and express them in terms of the Wilson coefficients in a particular operator basis for which the one-loop matching to SMEFT is already known.
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- 30 November 2021
- Published by Springer Nature in Journal of High Energy Physics
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General non-leptonic ∆F = 1 WET at the NLO in QCD
by Jason Aebischer, Christoph Bobeth, Andrzej J. Buras, Jacky Kumar, Mikołaj Misiak
We reconsider the complete set of four-quark operators in the Weak Effective Theory (WET) for non-leptonic ∆F = 1 decays that govern s → d and b → d, s transitions in the Standard Model (SM) and beyond, at the Next-to-Leading Order (NLO) in QCD. We discuss cases with different numbers N_f of active flavours, intermediate threshold corrections, as well as the issue of transformations between operator bases beyond leading order to facilitate the matching to high-energy completions or the Standard Model Effective Field Theory (SMEFT) at the electroweak scale. As a first step towards a SMEFT NLO analysis of K → ππ and non-leptonic B-meson decays, we calculate the relevant WET Wilson coefficients including two-loop contributions to their renormalization group running, and express them in terms of the Wilson coefficients in a particular operator basis for which the one-loop matching to SMEFT is already known.
Cited by 29
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Cited by 211
- 29 June 2020
- Published by Springer Nature in Journal of High Energy Physics
Towards $ \overline{B}\to {X}_s\gamma $ at the NNLO in QCD without interpolation in mc
by Miko-Laj Misiak, Abdur Rehman, Matthias Steinhauser
Strengthening constraints on new physics from the $$ \overline{B}\to {X}_s\gamma $$ $\bar{B} \to X_{s} γ$ branching ratio requires improving accuracy in the measurements and the Standard Model predictions. To match the expected Belle-II accuracy, Next-to-Next-to-Leading Order (NNLO) QCD corrections must be calculated without the so-far employed interpolation in the charm- quark mass m _c. In the process of evaluating such corrections at the physical value of mc, we have finalized the part coming from diagrams with closed fermion loops on the gluon lines that contribute to the interference of the current-current and photonic dipole operators. We confirm several published results for corrections of this type, and supplement them with a previously uncalculated piece. Taking into account the recently improved estimates of non-perturbative contributions, we find ℬ_sγ = (3.40±0.17)×10⁻⁴ and R _γ ≡ $$ {\mathrm{\mathcal{B}}}_{\left(s+d\right)\gamma }/{\mathrm{\mathcal{B}}}_{c\mathrm{\ell}\overline{v}} $$ $ℬ_{(s + d) γ} / ℬ_{c ℓ \bar{v}}$ = (3.35 ± 0.16) × 10⁻³ for E _γ > 1.6 GeV in the decaying meson rest frame.
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- 29 June 2020
- Published by Springer Nature in Journal of High Energy Physics
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Towards $ \overline{B}\to {X}_s\gamma $ at the NNLO in QCD without interpolation in mc
by Miko-Laj Misiak, Abdur Rehman, Matthias Steinhauser
Strengthening constraints on new physics from the $$ \overline{B}\to {X}_s\gamma $$ $\bar{B} \to X_{s} γ$ branching ratio requires improving accuracy in the measurements and the Standard Model predictions. To match the expected Belle-II accuracy, Next-to-Next-to-Leading Order (NNLO) QCD corrections must be calculated without the so-far employed interpolation in the charm- quark mass m _c. In the process of evaluating such corrections at the physical value of mc, we have finalized the part coming from diagrams with closed fermion loops on the gluon lines that contribute to the interference of the current-current and photonic dipole operators. We confirm several published results for corrections of this type, and supplement them with a previously uncalculated piece. Taking into account the recently improved estimates of non-perturbative contributions, we find ℬ_sγ = (3.40±0.17)×10⁻⁴ and R _γ ≡ $$ {\mathrm{\mathcal{B}}}_{\left(s+d\right)\gamma }/{\mathrm{\mathcal{B}}}_{c\mathrm{\ell}\overline{v}} $$ $ℬ_{(s + d) γ} / ℬ_{c ℓ \bar{v}}$ = (3.35 ± 0.16) × 10⁻³ for E _γ > 1.6 GeV in the decaying meson rest frame.
Cited by 211
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Cited by 1
- 27 June 2020
- Published by arXiv in arXiv
Towards B -> X_s gamma at the NNLO in QCD without interpolation in m_c
by M. Misiak, A. Rehman, M. Steinhauser
Strengthening constraints on new physics from the B -> X_s gamma branching ratio requires improving accuracy in the measurements and the Standard Model predictions. To match the expected Belle-II accuracy, Next-to-Next-to-Leading Order (NNLO) QCD corrections must be calculated without the so-far employed interpolation in the charm-quark mass m_c. In the process of evaluating such corrections at the physical value of m_c, we have finalized the part coming from diagrams with closed fermion loops on the gluon lines that contribute to the interference of the current-current and photonic dipole operators. We confirm several published results for corrections of this type, and supplement them with a previously uncalculated piece. Taking into account the recently improved estimates of non-perturbative contributions, we find B_{s gamma} = (3.40 \pm 0.17) * 10^{-4} and R_gamma = B_{(s+d) gamma}/B_{c l nu} = (3.35 \pm 0.16) * 10^{-3} for E_gamma > 1.6 GeV in the decaying meson rest frame.
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- 27 June 2020
- Published by arXiv in arXiv
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Towards B -> X_s gamma at the NNLO in QCD without interpolation in m_c
by M. Misiak, A. Rehman, M. Steinhauser
Strengthening constraints on new physics from the B -> X_s gamma branching ratio requires improving accuracy in the measurements and the Standard Model predictions. To match the expected Belle-II accuracy, Next-to-Next-to-Leading Order (NNLO) QCD corrections must be calculated without the so-far employed interpolation in the charm-quark mass m_c. In the process of evaluating such corrections at the physical value of m_c, we have finalized the part coming from diagrams with closed fermion loops on the gluon lines that contribute to the interference of the current-current and photonic dipole operators. We confirm several published results for corrections of this type, and supplement them with a previously uncalculated piece. Taking into account the recently improved estimates of non-perturbative contributions, we find B_{s gamma} = (3.40 \pm 0.17) * 10^{-4} and R_gamma = B_{(s+d) gamma}/B_{c l nu} = (3.35 \pm 0.16) * 10^{-3} for E_gamma > 1.6 GeV in the decaying meson rest frame.
Cited by 1
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- 7 February 2020
- Published by arXiv in arXiv
On charm-mass dependent NNLO corrections to B -> Xs gamma
by M. Misiak, A. Rehman, M. Steinhauser
The inclusive radiative decay of the B meson is known to provide strong constraints on many popular extensions of the Standard Model. Such constraints crucially depend on precision of the Standard Model predictions. One of the main contributions to the theoretical uncertainty is due to certain Next-to-Next-to-Leading Order QCD corrections whose values at the physical charm quark mass m_c have been estimated using interpolation between the m_c=0 and m_c >> m_b limits. A direct determination of such corrections at the physical value of m_c requires calculating hundreds of two-scale four-loop propagator diagrams with unitarity cuts. Applying the integration-by-parts method, we express the corrections in terms of master integrals. Asymptotic expansions of these integrals at m_c >> m_b serve as boundary conditions for differential equations in z=m_c^2/m_b^2 that are being numerically solved. Here, we present our final results for the diagrams involving massless and massive fermion loops on the gluon lines. For the two-body cuts, we confirm the analytical expressions and/or numerical fits that are already present in the literature. In the four-body case, we make the correction complete by including several diagrams that have previously been only estimated using interpolation in m_c. We also report the status of the ongoing calculation of the remaining diagrams where no closed fermion loops on the gluon lines are present.
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- 7 February 2020
- Published by arXiv in arXiv
Open Access
On charm-mass dependent NNLO corrections to B -> Xs gamma
by M. Misiak, A. Rehman, M. Steinhauser
The inclusive radiative decay of the B meson is known to provide strong constraints on many popular extensions of the Standard Model. Such constraints crucially depend on precision of the Standard Model predictions. One of the main contributions to the theoretical uncertainty is due to certain Next-to-Next-to-Leading Order QCD corrections whose values at the physical charm quark mass m_c have been estimated using interpolation between the m_c=0 and m_c >> m_b limits. A direct determination of such corrections at the physical value of m_c requires calculating hundreds of two-scale four-loop propagator diagrams with unitarity cuts. Applying the integration-by-parts method, we express the corrections in terms of master integrals. Asymptotic expansions of these integrals at m_c >> m_b serve as boundary conditions for differential equations in z=m_c^2/m_b^2 that are being numerically solved. Here, we present our final results for the diagrams involving massless and massive fermion loops on the gluon lines. For the two-body cuts, we confirm the analytical expressions and/or numerical fits that are already present in the literature. In the four-body case, we make the correction complete by including several diagrams that have previously been only estimated using interpolation in m_c. We also report the status of the ongoing calculation of the remaining diagrams where no closed fermion loops on the gluon lines are present.

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