I. HIGHLIGHTS OF THE YEARRIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

CP -odd gluonic operators in QCD spin physics†
Y. Hatta∗1,∗2
The explanation of matter-antimatter asymmetry of
the universe requires new origins of CP -violation beyond the Standard Model (BSM). One of the interesting CP -violating operators that can be induced in
the QCD Lagrangian due to BSM physics is the Weinberg operator.1) In this report I point out a novel relation between the hadronic matrix element of the Weinberg operator and a certain twist-four correction inpolarized Deep Inelastic Scattering (DIS). Such a relation suggests an exciting possibility that polarized
DIS experiments can provide useful information to thephysics of the nucleon electric dipole moment (EDM),
or more generally, BSM-origins of hadronic CP violations.
The Weinberg operator is a dimension-six purely
gluonic operator
νaFbµα Fcα.
OW = gfabc F̃µνν

O4 ≈ ∂ (ψ̄g F̃µν γ ψ),†∗1∗2(8)OW +γ129Nc2O4 .
O4 = OW −γW − γ43Nc2 + 4(9)Since this operator has a rather large anomalous dimension γW ∼ 10, in particular larger than γ4 by a factor of about 2, at high enough renormalization scalesµ2 one has⟨OW ⟩ ≈

9Nc2
⟨O4 ⟩ ≈ 2.61⟨O4 ⟩,3Nc2 + 4(10)(5)

Condensed from the article in Phys. Rev. D 102, 094004(2020)RIKEN Nishina Center
Physics Department, Brookhaven National Laboratory⟨P |ψ̄g F̃ µν γν ψ|P ⟩ = −2f0 M 2 S µ(11)1aν
Fbµσ Fcσ|P ⟩ = 4E ū′ iγ5 u.
⟨P ′ |gf abc F̃µνM3(12)I getE≈(3)

where72
N c + nf33

γ12 = −3Nc .

It immediately follows that the following linear combination is the eigenstate of the RG evolution(2)

to linear order in partial derivative ∂ µ . Such mixing is
usually neglected in the literature because O4 is a total
derivative and hence does
∫ not contribute to the CP violating eﬀective action d4 xO4 = 0. However, whenit comes to hadronic matrix elements, mixing becomes
crucial because only the nonforward matrix element isnonvanishing. Speciﬁcally, their RG equation takes theform(()))(
αs γW γ12dOWOW=−(4)O40
γ4d ln µ2 O44π
γW =(7)In terms of the nucleon matrix elements

Eq. (2) shows that one can choose OW and O4 as theindependent basis of operators and study their mixing.
Due to the equation of motion, one can writeµ⟨0|T{ψ(−k)Aρa (q)ψ̄(p)OW }|0⟩with oﬀ-shell momenta and nonzero momentum transfer ∆ = k − p − q ̸= 0. The result is(1)This operator violates CP and can be induced inthe QCD Lagrangian by physics beyond the StandardModel. It is considered as one of the candidate operators to generate a large EDM of the nucleons andnuclei.
The key observation is the following exact operator
identity←→←→1
OW = −∂ µ (F̃µν D α F να ) − F̃µν D 2 F µν
2
≡ O4 + OD ,is the anomalous dimension of the Weinberg operator.2) The anomalous dimension of O4 is the sameas that of the undiﬀerentiated, twist-four operator
ψ̄g F̃ µν γν ψ and is known to be3)82(6)γ 4 = C F + nf .
33To determine the oﬀ-diagonal component γ12 , I evaluate the following three-point Green’s function

9Nc2f0 ≈ 1.3f0 .
2(3Nc2 + 4)

(13)

Therefore, one can evaluate the matrix element E of
the Weinberg operator through the measurement of thef0 parameter relevant to the twist-four corrections inpolarized DIS.4,5) f0 can be extracted from the g1 (x)structure function measured at the future ElectronIon Collider (EIC) in the U.S. This is a new connection between the EIC and physics beyond the StandardModel. It will demonstrate the EIC’s unique capability to address low-energy nucleon observables in a highenergy collider.
References

1) S. Weinberg, Phys. Rev. Lett. 63, 2333 (1989).
2) A. Y. Morozov, Sov. J. Nucl. Phys. 40, 505 (1984).
3) E. V. Shuryak, A. I. Vainshtein, Nucl. Phys. B 199, 451(1982).
4) E. V. Shuryak, A. I. Vainshtein, Nucl. Phys. B 201, 141(1982).
5) X. D. Ji, P. Unrau, Phys. Lett. B 333, 228 (1994).

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