Matching factors for Delta S=1 four-quark operators in RI/SMOM schemes

Christian Sturm; Christoph Lehner

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Matching factors for Delta S=1 four-quark operators in RI/SMOM schemes

Christoph Lehner, Christian Sturm

High Energy Physics - Phenomenology · Submitted 26 Apr 2011

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim For this purpose, we define regularization-independent symmetric momentum-subtraction (RI/SMOM) schemes for Delta S=1 flavor-changing four-quark operators and provide one-loop matching factors to the MS-bar scheme in naive dimensional regularization.

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The non-perturbative renormalization of four-quark operators plays a significant role in lattice studies of flavor physics. For this purpose, we define regularization-independent symmetric momentum-subtraction (RI/SMOM) schemes for Delta S=1 flavor-changing four-quark operators and provide one-loop matching factors to the MS-bar scheme in naive dimensional regularization. The mixing of two-quark operators is discussed in terms of two different classes of schemes. We provide a compact expression for the finite one-loop amplitudes which allows for a straightforward definition of further RI/SMOM schemes.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The non-perturbative renormalization of four-quark operators plays a significant role in lattice studies of flavor physics.	background 1.00	0.06	0.04	asserted 1.00
s1	For this purpose, we define regularization-independent symmetric momentum-subtraction (RI/SMOM) schemes for Delta S=1 flavor-changing four-quark operators and provide one-loop matching factors to the MS-bar scheme in naive dimensional regularization.	contribution 0.97	0.96	0.43	asserted 1.00
s2	The mixing of two-quark operators is discussed in terms of two different classes of schemes.	contribution 0.89	0.54	0.05	asserted 0.95
s3	We provide a compact expression for the finite one-loop amplitudes which allows for a straightforward definition of further RI/SMOM schemes.	contribution 0.60	0.95	0.21	asserted 1.00

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Original abstract

The non-perturbative renormalization of four-quark operators plays a significant role in lattice studies of flavor physics. For this purpose, we define regularization-independent symmetric momentum-subtraction (RI/SMOM) schemes for Delta S=1 flavor-changing four-quark operators and provide one-loop matching factors to the MS-bar scheme in naive dimensional regularization. The mixing of two-quark operators is discussed in terms of two different classes of schemes. We provide a compact expression for the finite one-loop amplitudes which allows for a straightforward definition of further RI/SMOM schemes.

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References

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Cited by 10 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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First exploratory calculation of the long-distance contributions to the rare kaon decays $K\to\pi\ell^+\ell^-$
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A direct lattice QCD calculation that includes all contractions, including disconnected loops, resolves the Delta I = 1/2 kaon decay amplitude Re(A0) from zero at 25% statistical error, at unphysical kinematics with 4...

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Two-loop RI/SMOM-$\overline{\text{MS}}$ matching factors for the renormalization of bi-local operators contributing to $\epsilon_K$
hep-ph 2026-10 conditional novelty 6.0 of 10

Two-loop O(alpha_s) RI/SMOM-to-MS matching factors for the bi-local operators contributing to epsilon_K are computed for the first time, shifting the Wilson-coefficient-weighted matching by -13% and +17% at the lattic...
Direct CP violation and the $\Delta I=1/2$ rule in $K\to\pi\pi$ decay from the Standard Model
hep-lat 2020-04 conditional novelty 6.0 of 10

Lattice QCD with physical quark masses reproduces the Delta I = 1/2 rule and yields Re(epsilon-prime/epsilon) = 21.7(2.6)(6.2)(5.0) x 10^-4, consistent with the measured 16.6(2.3) x 10^-4.
Lattice determination of the $K \to (\pi\pi)_{I=2}$ Decay Amplitude $A_2$
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A first-principles lattice QCD calculation fixes both the real and imaginary parts of the K to two-pion (I=2) decay amplitude, giving the first Standard Model prediction for the CP-violating imaginary part.
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Adding a coarse-lattice domain wall ensemble with 171 MeV pions to two existing data sets reduces the chiral extrapolation systematic and yields f_pi = 127.1(3.8) MeV, f_K = 152.4(3.4) MeV, and B_K(MSbar, 3 GeV) = 0.535(16).
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How the literature uses it

18 arXiv papers cite this in 35 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 18 sentences without a confident label

Builds on it 2 papers

We extend the RI/SMOM scheme defined in [ref] in two small ways.

Nonperturbative renormalization of the Delta-S=1 weak Hamiltonian including the G_1 operator · Regularization invariant schemes · 1.00
We use formulae provided by Lehner and Sturm which extend their earlier, 2+1 flavor results [ref] for the matching matrix $Δr^RI →MS$ to the four-flavor case being studied here.

Long distance contribution to the $K_L-K_S$ mass difference · The $K_L-K_S$ mass difference · 0.89

Uses its method or tool 6 papers

We will work in the RI/SMOM scheme for the $ΔS = 1$ operators, which is defined in detail in [ref] .

Nonperturbative renormalization of the Delta-S=1 weak Hamiltonian including the G_1 operator · Regularization invariant schemes · 0.81
Specifically we apply the Rome-Southampton method [ref] at $μ= 1.53$ GeV, to introduce RI/SMOM normalization [ref] and then use continuum QCD perturbation theory [ref] to relate this to the Minimal Subtraction ($MS$) normalization used for the Wilson coefficients [ref] .

Standard-model prediction for direct CP violation in $K\to\pi\pi$ decay · Computational Method · 1.00
As described in Appendix [ref] this is done by introducing an intermediate non-perturbative RI/SMOM scheme at the energy scale $μ_=2.15$ GeV and applying the QCD perturbation theory results of Ref. [ref] to express $ℋ_W^ΔS=1$ in terms of operators renormalized in the RI-SMOM scheme: [equation] Specifically we use the ($γ_μ$, $q$) RI-SMOM scheme as described in Section V.A. of Ref. [ref] .

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Uses its data or values 2 papers

The relevant $Δr_ij$ entries are given in Table IX and XI of Ref. [ref] for the $(q -0.15cm ,q -0.15cm )$ and $(γ,γ)$ schemes, respectively.

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(Here we use the conventional operators $Q_1$ and $Q_2$ rather than the combinations $Q_±= Q_1±Q_2$ which belong to different representations of $SU_L(4)$ and do not mix under renormalization.) The conversion matrix for these two operators, $Z^→(μ/μ_0)=I+Δr^→$, has been given by Ref. [ref] at the scale $μ=μ_0$.

Exploratory lattice QCD study of the rare kaon decay $K^+\to\pi^+\nu\bar{\nu}$ · Non-perturbative renormalization using RI/SMOM scheme · 0.81

Cites it as background 2 papers

We briefly review the operator basis for the $ΔS = 1$ three-flavor effective Hamiltonian, following [ref] .

Nonperturbative renormalization of the Delta-S=1 weak Hamiltonian including the G_1 operator · Operator basis · 0.89
The renormalization and mixing of four-quark operators with different flavor content have been investigated before in various perturbative and nonperturbative studies using $$ [ref] , RI$'$/MOM [ref] , RI$'$/SMOM [ref] and Schrödinger Functional [ref] schemes.

Gauge-invariant renormalization of four-quark operators · Introduction · 0.97

What it cites · 47 sentences without a confident label

Uses the method or tool of 4 papers

We adopt the $(q,q)$ and $(q,γ_μ)$ schemes defined in Ref. [ref] so that for the $(27,1)$ operator we use [equation] The $(8,8)$ operators $Q'_7$ and $Q'_8$ also only mix with themselves under renormalization, and therefore we have to provide two projectors.

Continuum Limit of $B_K$ from 2+1 Flavor Domain Wall QCD · Results for the $(q,y)$ schemes · 0.98
We simplify the $$-renormalized amplitudes using the decomposition [ref] [equation] and the Fierz identities [equation] The resulting one-loop expressions are written as [equation] where $Γ_n^(O_j)$ denotes the insertion of the operator $O_j$ of basis I, II or the basis at tree level and the sum over repeated indices is implied.

Two-Loop QCD Anomalous Dimensions of Flavour-Changing Four-Quark Operators Within and Beyond the Standard Model · A compact expression for the amplitudes · 0.96
The diagrams are generated with the program QGRAF [ref] , and the symbolic manipulations are carried out using FORM [ref] .

New features of FORM · A compact expression for the amplitudes · 1.00

Compares with 5 papers

[ref] and [ref] agrees with the result of Ref. [ref] for the conversion from the $(γ_μ,q)$ and the $(γ_μ,γ_μ)$ scheme to the $$ scheme for the (VV+AA)–$(ΔS=2)$ operator.

Continuum Limit of $B_K$ from 2+1 Flavor Domain Wall QCD · The $$ and $(γ_μ,y)$ schemes · 1.00
We would like to point out that the result of Tab. [ref] for $N_c=3$ and $ξ=0$ or $ξ=1$ agrees with the result of Ref. [ref] for the conversion from the $$ scheme to the $$ scheme.

An Upgraded Analysis of $\epsilon\prime/\epsilon$ at the Next-to-Leading Order · The $$ and $(γ_μ,y)$ schemes · 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] .

The Two-Loop Anomalous Dimension Matrix for $\Delta S=1$ Weak Non-Leptonic Decays I: ${\cal O}(\alpha_s^2)$ · The $$ scheme · 0.93

Cites as background 22 papers

Its matching factor and anomalous dimension is known to two-loop order [ref] .

Two-loop matching factors for light quark masses and three-loop mass anomalous dimensions in the RI/SMOM schemes · Introduction · 0.80
Its matching factor and anomalous dimension is known to two-loop order [ref] .

RI'/SMOM scheme amplitudes for quark currents at two loops · Introduction · 0.80
Its matching factor and anomalous dimension is known to two-loop order [ref] .

Renormalization of quark bilinear operators in a momentum-subtraction scheme with a nonexceptional subtraction point · Introduction · 0.80

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