REVIEW
The Anatomy of $\varepsilon'/\varepsilon$ Beyond Leading Logarithms with Improved Hadronic Matrix Elements
High Energy Physics - Phenomenology · Submitted 19 Mar 1993 · 60 references
Pith's verdict: A landmark NLO ΔS=1 calculation whose headline numbers are one point in a scan, not a prediction with a total error bar.
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Referee report
The paper
The paper claims
The Standard Model's prediction for direct CP violation in kaon decay, computed one order deeper in QCD and anchored to measured kaon-to-pion-pion rates, can reproduce either of the two conflicting experiments depending on a single hadronic
The reading
What carries the argument
The load-bearing device is the choice of renormalization scale $mu = m_c$. At that scale, in the 't Hooft-Veltman and NDR schemes, the penguin coefficients and the charm-sector operators $Q^c_{1,2}$ either vanish or are negligible, so the CP-conserving amplitudes reduce to a two-operator problem: $mathrm{Re},A_2$ fixes $langle Q_1rangle_2 = langle Q_2rangle_2$ directly as a function of $Lambda_{overline{MS}}$ and the scheme, and $mathrm{Re},A_0$ fixes $langle Q_1rangle_0$ as a function of one free parameter $B_2^{(1/2)}(m_c)$. Operator relations then hand over the matrix elements of the $(V-A)otimes(V-A)$ penguins $Q_3, Q_4, Q_9, Q_{10}$, and renormalization-group evolution in $mu$ and conversion between schemes propagate everything elsewhere. That leaves four inputs, $Lambda_{overline{MS}}$, $B_2^{(1/2)}(m_c)$, $B_6^{(1/2)}(m_c)$ and $B_8^{(3/2)}(m_c)$, in terms of which $varepsilon'/varepsilon$ becomes a linear expansion, $P^{(1/2)} = a_0^{(1/2)} + a_2^{(1/2)} B_2 + a_6^{(1/2)} B_6$ and $P^{(3/2)} = a_0^{(3/2)} + a_8^{(3/2)} B_8$, with coefficients tabulated against $Lambda_{overline{MS}}$, $m_t$ and scheme. The second piece of machinery is the demonstration that the scheme dependence of the Wilson coefficients can only be cancelled by a matching scheme dependence in the matrix elements, which requires $O(alpha)$ corrections to $langle Q_i rangle$ that no available method supplies.
Extended reading notes
Core claim
The central claim is that a complete next-to-leading-order treatment of the $Delta S = 1$ Hamiltonian, combined with matrix elements constrained by CP-conserving data, yields $varepsilon'/varepsilon = (6.7 pm 2.6)times 10^{-4}$ in the NDR scheme for $B_6^{(1/2)} = B_8^{(3/2)} = 1$, matching E731, and rises to $(20.0 pm 6.5)times 10^{-4}$ when the matrix element of $Q_6$ is merely doubled, matching NA31. Two contributions are fixed almost independently of the QCD scale and of the renormalization scheme because they are pinned by the measured $A_0$ and $A_2$: a negative, top-mass-independent piece dominated by $Q_4$, and a smaller positive piece from $Q_9 + Q_{10}$. The remaining competition is between the QCD penguin $Q_6$, which pushes $varepsilon'/varepsilon$ up, and the electroweak penguin $Q_8$, which pushes it down and grows quickly with the top mass, so that above roughly $m_t approx 200$ GeV the ratio turns negative. The authors also report a systematic surprise: the next-to-leading corrections suppress the $Delta I = 1/2$ coefficient $z_-$ and enhance $z_+$, the opposite of what had been inferred from earlier two-loop work, because those earlier conclusions were drawn from scheme-rotated coefficients rather than the ones multiplying the operators directly.
Load-bearing premise
The matrix elements of the two operators that dominate the answer, $Q_6$ and $Q_8$, are assumed to keep the vacuum-insertion shape of Eqs. (5.10) and (5.18) up to a single multiplicative number each, and since no method determines those numbers, they are set to one or scanned, which leaves a residual scheme dependence the authors explicitly acknowledge.
Editorial extensions
If this is right
- The $Lambda_{overline{MS}}$ extracted from deep-inelastic scattering and LEP can for the first time be inserted meaningfully into $varepsilon'/varepsilon$, because the prediction is dominated by penguin operators with large anomalous dimensions and is therefore genuinely sensitive to the strength of the coupling.
- The $Delta I = 1/2$ rule and the $Delta I = 3/2$ suppression are built into the analysis by construction, so any simultaneous explanation of the two CP-conserving amplitudes and of $varepsilon'/varepsilon$ must live inside the same four-parameter space.
- The ratio $varepsilon'/varepsilon$ is predicted to fall with increasing top mass and with increasing $B_8^{(3/2)}$, and to rise with increasing $Lambda_{overline{MS}}$ and $B_6^{(1/2)}$, giving a definite directional signature for future measurements to test.
- The matrix elements of $Q_{9}$ and $Q_{10}$ implied by $mathrm{Re},A_2$ come out roughly a factor of two below the lattice estimates then in use, which propagates into a smaller electroweak-penguin contribution than other groups had assumed.
- The operators $Q^c_{1,2}$, sometimes invoked as important for $varepsilon'/varepsilon$ at $mu approx 2$ GeV, are shown to be calculable from matching conditions and to matter little below 2 GeV, contrary to earlier educated guesses.
Reading between the lines
- If a lattice or large-$N$ calculation eventually pins $langle Q_6rangle$ and $langle Q_8rangle$ to within roughly 20 percent, the spread exhibited here collapses and $varepsilon'/varepsilon$ becomes a genuine test of the CKM picture rather than a parameter scan; the tables of coefficients in this paper are already in the form needed to convert such a calculation directly into a prediction.
- The observation that the $B$-parameters of all $(V-A)otimes(V+A)$ operators are nearly scale-independent, while those of $(V-A)otimes(V-A)$ operators are not, suggests that the vacuum-insertion-like structure happens to be a good approximation precisely for the operators that dominate $varepsilon'/varepsilon$ and fails for the ones that do not, which is a lucky accident worth understanding rather
- Because the two contributions fixed by CP-conserving data sum to a small negative number, any independent determination of $mathrm{Im},lambda_t$ that comes out below the values used here would push the whole prediction toward or below zero, making the sign of $varepsilon'/varepsilon$ itself a diagnostic of the CKM phase rather than only of hadronic physics.
Original abstract
We use the recently calculated two--loop anomalous dimensions of current-current operators, QCD and electroweak penguin operators to construct the effective Hamiltonian for $\Delta S=1$ transitions beyond the leading logarithmic approximation. We solve the renormalization group equations and give the numerical values of Wilson coeff. functions. We propose a new semi-phenomenological approach to hadronic matrix elements which incorporates the data for $CP$-conserving $K \rightarrow \pi\pi$ amplitudes and allows to determine the matrix elements of all $(V-A)\otimes (V-A)$ operators in any renormalization scheme and do a renormalization group analysis of all hadronic matrix elements $\langle Q_i(\mu) \rangle$. We compare critically our treatment of these matrix elements with those given in the literature. We find in the NDR scheme $\epe = (6.7 \pm 2.6)\times 10^{-4}$ in agreement with the experimental findings of E731. We point out however that the increase of $\langle Q_6 \rangle$ by only a factor of two gives $\epe = (20.0 \pm 6.5)\times 10^{-4}$ in agreement with the result of NA31. The dependence of $\epe$ on $\Lambda_{\bar{MS}}$, $m_t$ and $\langle Q_{6,8} \rangle$ is presented.
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
We use the recently calculated two--loop anomalous dimensions of current-current operators, QCD and electroweak penguin operators to construct the effective Hamiltonian for $\Delta S=1$ transitions beyond the leading logarithmic approximation. We solve the renormalization group equations and give the numerical values of Wilson coeff. functions. We propose a new semi-phenomenological approach to hadronic matrix elements which incorporates the data for $CP$-conserving $K \rightarrow \pi\pi$ amplitudes and allows to determine the matrix elements of all $(V-A)\otimes (V-A)$ operators in any renormalization scheme and do a renormalization group analysis of all hadronic matrix elements $\langle Q_i(\mu) \rangle$. We compare critically our treatment of these matrix elements with those given in the literature. We find in the NDR scheme $\epe = (6.7 \pm 2.6)\times 10^{-4}$ in agreement with the experimental findings of E731. We point out however that the increase of $\langle Q_6 \rangle$ by only a factor of two gives $\epe = (20.0 \pm 6.5)\times 10^{-4}$ in agreement with the result of NA31. The dependence of $\epe$ on $\Lambda_{\bar{MS}}$, $m_t$ and $\langle Q_{6,8} \rangle$ is presented.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We use the recently calculated two--loop anomalous dimensions of current-current operators, QCD and electroweak penguin operators to construct the effective Hamiltonian for $\Delta S=1$ transitions beyond the leading logarithmic approximation. | method 0.77 | 0.86 | 0.04 | asserted 1.00 |
| s1 | We solve the renormalization group equations and give the numerical values of Wilson coeff. functions. | method 0.82 | 0.95 | 0.03 | asserted 1.00 |
| s2 | We propose a new semi-phenomenological approach to hadronic matrix elements which incorporates the data for $CP$-conserving $K \rightarrow \pi\pi$ amplitudes and allows to determine the matrix elements of all $(V-A)\otimes (V-A)$ operators in any renormalization scheme and do a renormalization group analysis of all hadronic matrix elements $\langle Q_i(\mu) \rangle$. | method 0.63 | 0.90 | 0.08 | asserted 0.95 |
| s3 | We compare critically our treatment of these matrix elements with those given in the literature. | contribution 0.71 | 0.75 | 0.03 | asserted 0.62 |
| s4 | We find in the NDR scheme $\epe = (6.7 \pm 2.6)\times 10^{-4}$ in agreement with the experimental findings of E731. | result 1.00 | 0.95 | 0.48 | asserted 1.00 |
| s5 | We point out however that the increase of $\langle Q_6 \rangle$ by only a factor of two gives $\epe = (20.0 \pm 6.5)\times 10^{-4}$ in agreement with the result of NA31. | result 0.86 | 0.82 | 0.15 | asserted 0.84 |
| s6 | The dependence of $\epe$ on $\Lambda_{\bar{MS}}$, $m_t$ and $\langle Q_{6,8} \rangle$ is presented. | result 0.51 | 0.83 | 0.06 | asserted 0.88 |
What was examined, and what was not
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Figures
Showing 10 of the paper's 20 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/9303284 by the authors.
References
How the literature uses it
37 arXiv papers cite this in 122 sentences
Papers whose main result relates to this one's
States the same result 1 paper
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Taking $\mt=130\,GeV$, $\Lms=300\,MeV$ and calculating $\langle Q_6 \rangle$ and $\langle Q_8 \rangle$ in the $1/N$ approach, we find in the NDR scheme $\epe = (6.7 \pm 2.6)\times 10^{-4}$ in agreement with the experimental findings of E731.
The Anatomy of eps'/eps Beyond Leading Logarithms with Improved Hadronic Matrix Elements · 1.00
Papers citing it · 92 sentences without a confident label
Contests it 3 papers
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Unfortunately, not knowing $1/N$ corrections to $$ and $$ in 1993, both parameters have been set at $μ=m_c$ in [ref] to unity, which is clearly wrong.
$\epsilon^\prime/\epsilon$-2018: A Christmas Story · The DQCD View · contest 0.81
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Unfortunately, not knowing $1/N$ corrections to $$ and $$ in 1993, both parameters have been set at $μ=m_c$ in [ref] to unity, which is clearly wrong.
The $\epsilon'/\epsilon$-Story: 1976-2021 · The Fourth Period: $2015-2019$ · contest 0.76
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The proof of scheme dependence including QED in Ref. [ref] fails in our case as the anomalous dimension is itself scheme independent.)
Two-loop Electroweak Corrections to the Top-Quark Contribution to $\epsilon_K$ · Scheme independence at order $α$ · contest 0.81
Uses its method or tool 8 papers
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The RG evolution at NLO in QCD and QED leads to the effective Hamiltonian at a scale $μ≲μ_c ∼m_c$ ($N_f=3$) [equation] after decoupling of $b$- and $c$-quarks at scales $μ_b,c$ [ref] , where $y_a ≡v_a - z_a$ and all Wilson coefficients are at the scale $μ$.
Leptoquarks meet $\varepsilon'/\varepsilon$ and rare Kaon processes · $d_j→d_i qq̅$ and $$ · 0.96
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We take the basis operators $Q_i$ to be the ones specified by Buras et al. in Section 2 of Ref. [ref] .
Eradication of singularities in the next-to-leading order RG evolution for the \Delta S = 1 effective Hamiltonian with 3 quark flavours · Introduction · 0.83
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Here we summarize the $ΔS = 1$ Wilson coefficients at the various scales used in our analysis in the NDR- scheme using the NLO RG evolution from [ref] .
$\varepsilon'/\varepsilon$ in the Standard Model at the Dawn of the 2020s · Wilson coefficients · 0.98
Uses its data or values 1 paper
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The corresponding results for the QED matrices $M̂_01^(5)$ and $M̂_01^(4)$ are given as [ref] [equation] where [equation]
General non-leptonic $\Delta F=1$ WET at the NLO in QCD · Threshold corrections · 0.91
Compares with it 1 paper
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Consequently, at scales $(1)$ relevant for the phenomenology both $$ and $$ are predicted to be below unity and there is strong indication that $<$. * While the dependence of $$ and $$ on $Λ< 1$ is stronger than their dependence on $μ$ in the perturbative regime, these two properties of $$ and $$ are at the qualitative level consistent with the numerical analysis performed for $$ and $$ by means of the standard renormalization group running in [ref] .
Upper Bounds on epsilon'/epsilon Parameters B_6^{(1/2)} and B_8^{(3/2)} from Large N QCD and other News · Derivations · 0.84
Cites it as support 1 paper
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As a result of this the $μ$ dependence of the parameters $$ and $$ is negligible for $μ≥1$ [ref] .
New Physics Patterns in $\varepsilon^\prime/\varepsilon$ and $\varepsilon_K$ with Implications for Rare Kaon Decays and $\Delta M_K$ · Basic formula for $$ · 0.96
Cites it as background 10 papers
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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
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We will consider the usual basis of operators contributing to $K→ππ$ amplitudes [ref] , namely
Upper Bounds on epsilon'/epsilon Parameters B_6^{(1/2)} and B_8^{(3/2)} from Large N QCD and other News · Preliminaries · 0.93
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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
Discussion
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