Catalog · EDM / neutrino · E009

E009 $w, C_{\tilde G}$

Weinberg three-gluon operator

Status SUBTLETY-ADDED PARTIAL High Code: NO Priority Medium

PDG / equivalent values

Observable	Value	Year	Experiment / source	Provenance
Pospelov-Ritz neutron EDM response to the Weinberg operator	22 MeV	2005	Pospelov and Ritz, Annals Phys. 318, 119-169 (2005); arXiv:hep-ph/0504231	source ↑
Reference scale for the Pospelov-Ritz Weinberg coefficient	1 GeV	2005	Pospelov and Ritz, Annals Phys. 318, 119-169 (2005); arXiv:hep-ph/0504231	source ↑
Single-source benchmark bound on abs(w(1 GeV))	4.1e-11 GeV^-2	2026	Derived from PDG Live 2026 S017EDM neutron EDM datablock plus Pospelov and Ritz 2005	source ↑
Haisch-Hala O6 neutron EDM response	74 MeV	2019	Haisch and Hala, JHEP 11, 154 (2019); arXiv:1909.08955	source ↑
Single-source benchmark bound on $abs(C_6$) in the Haisch-Hala O6 convention	1.2e-11 GeV^-2	2026	Derived from PDG Live 2026 S017EDM neutron EDM datablock plus Haisch and Hala 2019	source ↑

Why this constrains the RS scan

Anarchic warped or partial-compositeness models carry new CP phases in the colored sector. Integrating out KK fermions, heavy quarks, Higgs-sector states, or colored resonances can generate quark chromo-EDMs and finite heavy-threshold contributions to the Weinberg operator. The operator is therefore a compact EDM-adjacent diagnostic of colored CP violation that complements E008.

What's changed since the original paper

Relative to the CFW 2008 RS-flavor baseline (CsakiFalkowskiWeiler:RSFlavor2008), the neutron anchor is now the Abel 2020 result adopted by PDG Live 2026 (Abel:NeutronEDM2020; PDG2026:NeutronEDM). Composite and warped dipole analyses after 2008 emphasized that heavy-quark CEDM running generates a finite threshold correction to the three-gluon Weinberg operator (KoenigNeubertStraub:CompositeDipoles2014). In the Pospelov--Ritz normalization, $|d_n(w)|\simeq e\,22~\mathrm{MeV}\,w(1~\mathrm{GeV})$, which gives the central one-source benchmark $|w(1~\mathrm{GeV})|<4.1\times10^{-11}\ \mathrm{GeV}^{-2}$ from the current neutron limit when other CP-odd sources are set to zero (PospelovRitz:EDMReview2005; PDG2026:NeutronEDM). Haisch and Hala 2019 quote, in their $O_6$ convention, $(d_n/e)_{O_6}=74(1\pm0.5)~\mathrm{MeV}$, corresponding to the central benchmark $|C_6|<1.2\times10^{-11}\ \mathrm{GeV}^{-2}$ (HaischHala:WeinbergSumRules2019; PDG2026:NeutronEDM). Lattice work using gradient flow now studies the dimension-six gluonic CP-violating matrix element and its operator mixing (Bhattacharya:WeinbergLattice2022).

Validity and model dependence

The neutron EDM limit is robust. The Weinberg-operator bound is not: it depends on operator normalization, RG running, heavy-quark thresholds, hadronic matrix elements, and assumptions about cancellations with $\bar\theta$, quark EDMs, qCEDMs, semileptonic terms, and four-quark operators. In global EDM language, the same neutron and diamagnetic data probe more CP-odd sources than there are independent low-energy observables. For anarchic CP phases, EDM constraints should be reviewed jointly with the flavor branching rows that share the same Wilson coefficients (relevant cross-refs: K001, K003, B011, B033, B034), not as an isolated appendix.

Code coverage in this repo

NO. Required greps over quarkConstraints/, qcd/, flavorConstraints/, neutrinos/, yukawa/, warpConfig/, solvers/, scanParams/, and tests/ found no Weinberg, neutron-EDM, or CP-odd gluonic operator constraint. As recorded in the sidecar code-coverage block, the only nearby dipole implementation is the unrelated $\mu\to e\gamma$ code in flavorConstraints/muToEGamma.py:3, :21, and :81.

Implementation difficulty

HIGH. A live E009 constraint needs new CP-odd colored operators, RS loop or threshold matching, QCD running and mixing into other EDM sources, and a chosen hadronic matrix-element convention. The existing $\Delta F=2$ basis and muon LFV dipole path do not cover this observable.

Reason: Requires new CP-odd colored-sector operators, RS loop or heavy-threshold matching, QCD running and operator mixing, and a hadronic matrix-element convention. Existing $\Delta F = 2$ and $\mu \to e \gamma$ code paths do not provide this observable.

Key references

Process-local source keys before bibliography consolidation: PDG2026:NeutronEDM, Abel:NeutronEDM2020, Weinberg:ThreeGluon1989, PospelovRitz:EDMReview2005, ChuppRamseyMusolf:EDMGlobal2014, KoenigNeubertStraub:CompositeDipoles2014, HaischHala:WeinbergSumRules2019, Bhattacharya:WeinbergLattice2022, and CsakiFalkowskiWeiler:RSFlavor2008.

value_summary |d_n| < 1.8 x 10^-26 e cm at 90% CL

RESOLVED

Match snapshot line 20

L17: Canonical direct result:
L18:   <0.18, CL = 90%, ABEL 2020, method MRS UCN.
L19: Converted value:
L20:   |d_n| < 1.8 x 10^-26 e cm at 90% CL.
L21: 
L22: PDG measurement note for ABEL 2020:
L23:   d = (0.0 +/- 1.1 +/- 0.2) x 10^-26 e cm, corresponding to the listed

value_summary d_n = (0.0 +/- 1.1_stat +/- 0.2_sys) x 10^-26 e cm; |d_n| < 1.8 x 10^-26 e cm at 90% CL

RESOLVED

Match snapshot line 15

L12: 
L13: Relevant experimental statement:
L14:   The measured value of the neutron EDM is
L15:   d_n = (0.0 +/- 1.1_stat +/- 0.2_sys) x 10^-26 e cm.
L16: 
L17: Canonical limit as used by PDG:
L18:   |d_n| < 1.8 x 10^-26 e cm at 90% CL.

value_summary |d_n(w)| approximately e * 22 MeV * w(1 GeV)

RESOLVED

Match snapshot line 22

L19: Weinberg-operator neutron EDM estimate:
L20:   |d_n(w)| approximately equals |mu_n| [3 g_s m_0^2 /(32 pi^2)]
L21:   w ln(M^2/mu_IR^2), and for M/mu_IR = 2 with g_s = 2.1 the review quotes
L22:   |d_n(w)| about e * 22 MeV * w(1 GeV).
L23: 
L24: Use in E009:
L25:   Paper-era normalization and benchmark translation from the Weinberg

value_summary The Pospelov-Ritz benchmark writes the coefficient as w(1 GeV).

RESOLVED

Match snapshot line 26

L23: 
L24: Use in E009:
L25:   Paper-era normalization and benchmark translation from the Weinberg
L26:   coefficient w(1 GeV) to the neutron EDM. This is a theory normalization, not
L27:   a measured PDG value.

value_summary (d_n/e)_O6 = 74(1 +/- 0.5) MeV

RESOLVED

Match snapshot line 24

L21:   from f^{ABC}, the dual gluon field strength, and two gluon field strengths.
L22: 
L23: Numerical result in the Haisch-Hala O6 normalization:
L24:   (d_n / e)_O6 = 74 (1 +/- 0.5) MeV.
L25: 
L26: Uncertainty statement:
L27:   The final relative error is 50%; the dominant source is the scale-ratio