REVIEW
The $\Delta S=1$ Effective Hamiltonian Including Next-to-Leading Order QCD and QED Corrections
High Energy Physics - Phenomenology · Submitted 15 Apr 1993 · 30 references
Pith's verdict: A solid, well-cross-checked two-loop calculation whose genuinely new HV leg is validated mostly by internal consistency rather than by anything external.
T0 · AI review
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
A two-loop calculation of the short-distance mixing matrix for ΔS=1 four-quark operators, carried out in two regularization schemes and tied together by a scheme-translation identity checked diagram by diagram.
The reading
What carries the argument
The load-bearing object is the two-loop anomalous dimension matrix $γ̂^{(1)}$, extracted from the single poles of the operator renormalization matrix via $γ̂^{(1)}_s = -4Ẑ^{(2)}_1 - 2β_0Ẑ^{(1)}_0 + 2(Ẑ^{(1)}_1Ẑ^{(1)}_0 + Ẑ^{(1)}_0Ẑ^{(1)}_1)$. The subtlety that makes the calculation long is that in dimensional regularization the Dirac algebra does not close on the four-dimensional operator basis, so 'effervescent' operators appear at intermediate stages and must be carried through the mixing before being projected away at the end. The scheme-translation identity above is what turns two independent computations into a single cross-checked result, and the finite one-loop matrix $r̂$ that appears in it is also the object that, combined with the two-loop matrix, defines scheme-independent coefficients.
Extended reading notes
Core claim
The paper's claim is that the full two-loop anomalous dimension matrix for the ten-operator $|ΔS|=1$ basis is now known, including the $α_s^2$ and $α_eα_s$ terms, in both NDR and HV, and that the two results are consistent. Consistency here is not a hand-wave: the authors state and use the relation $Δγ̂^{(1)} = [Δr̂, γ̂^{(0)}] + 2β_0 Δr̂$, where $Δr̂$ is the difference of the one-loop finite parts between the schemes, and they verify it for each individual two-loop diagram after all counter-terms and effervescent-operator insertions are included. The two matrices themselves are collected in tables 3 and 4. Along the way the authors report that the earlier HV results for two penguin diagrams, P8 and F8, fail this relation and that the difference is compensated elsewhere in the strong matrix but not in the electromagnetic one. They also show how to redefine the renormalized operators so that the evolution matrix, not just the product of coefficients and matrix elements, becomes scheme independent.
Load-bearing premise
The NDR side of the agreement assumes that an anticommuting $γ_5$ together with a prescription for closed odd-parity fermion loops taken from a program manual yields correct results, and the paper itself says this is not guaranteed, so the cross-scheme check inherits that unproven step.
Editorial extensions
If this is right
- The entries of tables 3 and 4 become the standard short-distance input for next-to-leading-order Wilson coefficients at scales of a few GeV.
- Predictions for $ε'/ε$ can be quoted with the two-loop QCD and QED mixing included, and the residual scale and scheme dependence can be estimated rather than guessed.
- The redefinition of renormalized operators given in the paper allows Wilson coefficients to be combined with non-perturbative matrix elements, for instance from lattice QCD, without scheme mismatch.
- The earlier HV results for diagrams P8 and F8 are superseded, which changes the electromagnetic part of the two-loop matrix in that scheme.
Reading between the lines
- Because the scheme-translation identity holds diagram by diagram, it can serve as a debugging tool for any future two-loop calculation in this basis: a mismatch localises the error to a single diagram rather than to the whole matrix.
- The same machinery transfers almost unchanged to the $|ΔC|=1$ and $|ΔB|=1$ bases, so the tables effectively seed a family of two-loop mixing calculations for charm and bottom decays.
- The authors postpone the numerical and uncertainty analysis to a separate publication, so the phenomenological size of the two-loop correction to $ε'/ε$ is not settled here; the natural next step is to ask how much of the earlier leading-order prediction moves.
- Since the scheme-independent definition of operators depends on the external states used to fix it, any comparison between a scheme-independent coefficient and a lattice matrix element must state that choice explicitly.
Original abstract
In this paper we present a calculation of the $\Delta S=1$ effective weak Hamiltonian including next-to-leading order QCD and QED corrections. At a scale $\mu$ of the order of few GeV, the Wilson coefficients of the operators are given in terms of the renormalization group evolution matrix and of the coefficients computed at a large scale $\sim M_W$. The expression of the evolution matrix is derived from the two-loop anomalous dimension matrix which governs the mixing of the relevant current-current and penguin operators, renormalized in some given regularization scheme. We have computed the anomalous dimension matrix up to and including order $\alphas^2$ and $\alphae \alphas$ in two different renormalization schemes, NDR and HV, with consistent results. We give many details on the calculation of the anomalous dimension matrix at two loops, on the determination of the Wilson coefficients at the scale $M_W$ and of their evolution from $M_W$ to $\mu$. We also discuss the dependence of the Wilson coefficients/operators on the regularization scheme.
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
In this paper we present a calculation of the $\Delta S=1$ effective weak Hamiltonian including next-to-leading order QCD and QED corrections. At a scale $\mu$ of the order of few GeV, the Wilson coefficients of the operators are given in terms of the renormalization group evolution matrix and of the coefficients computed at a large scale $\sim M_W$. The expression of the evolution matrix is derived from the two-loop anomalous dimension matrix which governs the mixing of the relevant current-current and penguin operators, renormalized in some given regularization scheme. We have computed the anomalous dimension matrix up to and including order $\alphas^2$ and $\alphae \alphas$ in two different renormalization schemes, NDR and HV, with consistent results. We give many details on the calculation of the anomalous dimension matrix at two loops, on the determination of the Wilson coefficients at the scale $M_W$ and of their evolution from $M_W$ to $\mu$. We also discuss the dependence of the Wilson coefficients/operators on the regularization scheme.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper we present a calculation of the $\Delta S=1$ effective weak Hamiltonian including next-to-leading order QCD and QED corrections. | contribution 1.00 | 0.95 | 0.41 | asserted 1.00 |
| s1 | At a scale $\mu$ of the order of few GeV, the Wilson coefficients of the operators are given in terms of the renormalization group evolution matrix and of the coefficients computed at a large scale $\sim M_W$. | method 0.44 | 0.53 | 0.10 | asserted 1.00 |
| s2 | The expression of the evolution matrix is derived from the two-loop anomalous dimension matrix which governs the mixing of the relevant current-current and penguin operators, renormalized in some given regularization scheme. | method 0.92 | 0.79 | 0.05 | asserted 1.00 |
| s3 | We have computed the anomalous dimension matrix up to and including order $\alphas^2$ and $\alphae \alphas$ in two different renormalization schemes, NDR and HV, with consistent results. | result 0.48 | 0.97 | 0.15 | asserted 1.00 |
| s4 | We give many details on the calculation of the anomalous dimension matrix at two loops, on the determination of the Wilson coefficients at the scale $M_W$ and of their evolution from $M_W$ to $\mu$. | contribution 0.75 | 0.94 | 0.06 | asserted 0.99 |
| s5 | We also discuss the dependence of the Wilson coefficients/operators on the regularization scheme. | contribution 0.94 | 0.66 | 0.05 | unclear 0.50 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Showing 3 of the paper's 14 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/9304257 by the authors.
References
How the literature uses it
54 arXiv papers cite this in 80 sentences
Papers citing it · 57 sentences without a confident label
Uses its method or tool 4 papers
-
[ref] was either in the or in the schemes [ref] ; iii) the matching of the renormalised operators in the SMOM schemes to the operators renormalised in the $$ scheme in which the Wilson coefficient functions have been computed at the NLO [ref] ; iv) the combination of the operators in the $$ scheme and the Wilson coefficients computed at the NLO to compute the relevant amplitudes.
New UTfit Analysis of the Unitarity Triangle in the Cabibbo-Kobayashi-Maskawa scheme · New: the lattice determination of $ε^'/ε$ · 0.90
-
We use the effective Hamiltonian for $ΔS=1$ transitions of [ref] [equation]
Improved anatomy of epsilon'/epsilon in the Standard Model · Effective Hamiltonian · 0.96
-
All our calculations are performed with the naive dimensional regularization (NDR) scheme with anti-commuting $γ_5$ in $D=4-2ϵ$ space-time dimensions [ref] , which matches exactly the one used for evaluations of the short-distance Wilson coefficients [ref] .
$\bar{B}_{s,d}^{0} \to J/\psi \mu^{+}\mu^{-}$ Decays in QCD Factorization · Non-factorizable one-loop vertex corrections · 0.83
Uses its data or values 2 papers
-
We collect all tables of pole coefficients present in the literature [ref] , involved in the calculation of the two-loop four-fermion ADMs (in the NDR scheme), in Tables [ref], [ref], and [ref].
Two-Loop Anomalous Dimensions in the LEFT: Dimension-Six Four-Fermion Operators in NDR · Tables of Pole Coefficients · 0.99
-
For $=3$ and arbitrary $$ they read [equation] Since the VLL,$u$ operators are part of the SM operator basis, their QED ADMs are known, too [ref] [equation]
General non-leptonic $\Delta F=1$ WET at the NLO in QCD · Anomalous dimensions · 0.86
Compares with it 3 papers
-
By explicit calculation, we reproduce the known results in the physical sector up to NNLO, [equation] in agreement with the literature [ref] ; see Refs. [ref] for the original calculations, using different bases of evanescent operators.
Four-loop QCD mixing of current-current operators · Operator basis and anomalous dimension · 1.00
-
For on-shell matrix elements we use Eq. [ref] in Eq. [ref] and thus reproduce the results for the anomalous dimensions of Refs. [ref] .
Matching factors for Delta S=1 four-quark operators in RI/SMOM schemes · The $$ scheme · 0.93
-
Nevertheless, some other classic past results in Refs. [ref] overlap with some of the results presented here, as do some more recent works [ref] .
Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology · Previous work · 0.86
Cites it as background 11 papers
-
Weak $ΔS=1$ decays of hadrons can be described in the framework of the Standard Model by an effective Hamiltonian $_̋eff$ obtained by integrating out the degrees of freedom associated with heavy quarks and gauge bosons [ref] .
Eradication of singularities in the next-to-leading order RG evolution for the \Delta S = 1 effective Hamiltonian with 3 quark flavours · Introduction · 0.96
-
As far as short-distance contributions encoded in the Wilson coefficients are concerned, they have been known for the SM operators already for 25 years at the NLO level [ref] and for the BSM operators two-loop anomalous dimensions have been known [ref] for almost two decades.
Master formula for $\varepsilon'/\varepsilon$ beyond the Standard Model · Introduction · 0.92
-
The Wilson coefficients $z_i$ and $y_i$ have been calculated at the NLO level more than twenty years ago [ref] , and some pieces of NNLO corrections are also available [ref] .
Improved anatomy of epsilon'/epsilon in the Standard Model · Effective Hamiltonian · 0.95
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.