Sum rules for CP-violating operators of Weinberg type

Abstract

We estimate the size of the hadronic matrix elements of CP-violating three-gluon and four-gluon Weinberg operators using sum-rule techniques. In the three-gluon case, we are able to reproduce the expressions given in earlier works, while the four-gluon results obtained in this article are new. Our paper therefore represents the first systematic study of contributions to the electric dipole moment of the neutron due to CP-violating dimension-six and dimension-eight operators. We provide many details on both the derivation of the sum rules as well as the analysis of the uncertainties that plague our final predictions.Here G A µν is the QCD field strength tensor,G A µν = 1/2 µνρλ G A ρλ with 0123 = +1 denotes its dual, f ABC are the fully anti-symmetric structure constants of SU(3) and c ABCD denote the colour structures defined in (5.20). Lacking the expertise in LQCD as well as the needed computer resources, we will present estimates of the hadronic matrix elements of the operators in (1.1) using QCD sum-rule techniques. In the case of the dimension-six contribution O 6 such a calculation has already been performed in [40], but the latter publication does not provide details on the actual computation making an independent reevaluation worthwhile. Our determination of the hadronic matrix elements of the dimension-eight term O 8 is instead new. Both results will be used in a companion paper [59], where we derive model-independent bounds on CP-violating Higgs-gluon interactions in BSM scenarios with vanishing or highly suppressed light-quark Yukawa couplings.Our work is organised as follows. After briefly reviewing the basic idea behind the sum-rule determinations of the hadronic matrix elements of O 6 and O 8 , we discuss in Section 3 the phenomenological side of the sum rules. The operator product expansion (OPE) computation of the dimension-six and dimension-eight contributions is described in Section 4 and Section 5, respectively. The matching and the numerical analysis of the sum rules are performed in Section 6. We conclude in Section 7. Technical details are relegated to several appendices. General idea behind the sum rulesThe central object for the derivation of the sum-rule estimates for the hadronic matrix elements of operators of the type (1.1) is the following correlation function Π(q 2 ) = i d 4 x e iqx Ω T η(x)η(0) Ω EM,O k , (2.1)

Bounds on CP-violating Higgs-gluon interactions: the case of vanishing light-quark Yukawa couplings

“…Estimates of the hadronic matrix elements of the dimension-eight CP-violating fourgluon operators have very recently been obtained in the context of QCD sum rules [93]. Employing these results we find which has a theoretical uncertainty of 80%.…”

“…Motivated by the above observation, we compute in section 2 of this article the leading contributions to the nEDM that involve an insertion of (1.1) and that survive in the limit of vanishing light-quark Yukawa couplings. An extension of our calculation to the case of the full set of dimension-six operators that encode the CP-violating interactions between the Higgs and EW gauge bosons will be presented elsewhere [64]. Based on our results, we then derive in section 3 the bounds on the Wilson coefficient of the CP-violating dimensionsix operator (1.1) from current and future nEDM searches, comparing our findings to the existing LHC limits and their projections.…”

“…The EDM contributions from down and up quarks have been calculated with an accuracy of 5% using lattice QCD (LQCD) [86][87][88], while QCD sum-rule calculations [9,[89][90][91] allow to determine the contributions from the down-quark and up-quark CEDMs with uncertainties of 50%. At present only estimates of the hadronic matrix element of the dimension-six Weinberg operator exist that rely on either QCD sum rules [92,93], the vacuum insertion approximation [94] or naive dimensional analysis [68]. Since only the QCD sum-rules calculations allow for a systematic analysis of theoretical uncertainties, we will in the following rely on them.…”

“…Since only the QCD sum-rules calculations allow for a systematic analysis of theoretical uncertainties, we will in the following rely on them. Adopting our very recent QCD sum-rule estimate [93], which is plagued by an uncertainty of 50%, and employing m where the first, second and third term corresponds to the d q ,d q and C 3G contribution in (2.5), respectively. We emphasise that in the case that new physics enters only through the matching correction C 3G (m h ) the relative signs in (2.6) are all fixed, meaning that the contribution from the dimension-six Weinberg operator necessarily interferes destructively with both the EDM and CEDM contribution.…”

Non-perturbative renormalization scheme for the C P -odd three-gluon operator

“…First-principle calculations, with controlled theoretical uncertainties, of the matrix elements of the gCEDM on the nucleon are necessary to derive the constraints of EDM experiments on this operator, and the implications for BSM physics. At the moment, the best estimates of the nucleon EDM from the gCEDM have been obtained with QCD sum rule calculations [87,88], which are however affected by large theoretical uncertainties, at the level of 50%-100%. While for this operator lattice QCD calculations are still in their infancy [53,54,89], this method can in principle provide fully nonperturbative results, in which all sources of systematic uncertainty can be quantified, controlled, and improved.…”