Updated limits on the CP violating $\eta\pi\pi$ and $\eta'\pi\pi$ couplings derived from the neutron EDM

We complete our derivation of upper limits on the CP violating $\eta\pi\pi$ and $\eta'\pi\pi$ couplings from an analysis of their two-loop contributions to the neutron electric dipole moment (nEDM). We use a phenomenological Lagrangian approach which is formulated in terms of hadronic degrees of freedom - nucleons and pseudoscalar mesons. The essential part of the Lagrangian contains the CP violating couplings between $\eta(\eta')$ and pions. Previously, we included photons using minimal substitution in case of the proton and charged pions. Now we extend our Lagrangian by adding the nonminimal couplings, i.e. anomalous magnetic couplings of nucleons with the photon. The obtained numerical upper limits for the $\eta\pi\pi$ and $\eta'\pi\pi$ couplings $|f_{\eta\pi\pi}(M_\eta^2)|<4.4 \times10^{-11}$ and $|f_{\eta^\prime\pi\pi}(M_{\eta'}^2)|<3.8 \times10^{-11}$ can be useful for the related, planned experiments at the JLab Eta Factory. Using present experimental limits on the nEDM, we derive upper limits on the CP violating $\bar\theta$ parameter of $\bar{\mathrm{\theta}}<4.7 \times 10^{-10}$.


I. INTRODUCTION
Since the 1950s the study of T-or CP-violation in hadronic processes is a relevant topic in particle physics since it helps to shed light on the entries of the Cabibbo-Kabayashi-Maskawa mixing matrix and the related oscillations of neutral kaons, D and B mesons. Some phenomena, like CP violation in processes involving K and B mesons, have been explained in the framework of the Standard Model (SM). The study of other CP-violating effects, such as strong CP-violation, a neutron electric dipole moment (nEDM), decays of η and η mesons into two pions, etc. clearly call for the search of possible New Physics mechanisms, which are outside the scope of the SM. In particular, SM predictions for the nEDM are up to several orders of magnitude lower than existing experimental limits. Clearly the study of EDMs of hadrons and nuclei could probe New Physics beyond the SM (for a review see, e.g., Refs. [1,2]). Our interest in hadron EDMs is motivated by the possibility to extract limits on the CP-violating strong coupling between hadrons and theθ parameter (CP-violating gluon-gluon coupling). From study of the nEDM one can estimate the QCDθ parameter and the πN N , η ( ) N N , and η ( ) ππ couplings. In series of papers [3][4][5][6][7][8][9] we gave several analyses of CP violating physics focused on the nEDM and strong CP violating phenomena with the relation to the QCDθ term, aspects of the phenomenology of axions, CP-violating hadronic couplings, intrinsic electric and chromoelectric dipole moments of quarks, CP-violating quark-gluon, three-gluon and four-quark couplings, etc.
In Refs. [8,9] we focused on the determination of the CP violating couplings ηππ and η ππ from the nEDM. Our formalism was based on a phenomenological Lagrangian describing the interaction of nucleons with pseudoscalar mesons (pions and η(η )). The interaction of charged particles with photons has been introduced by minimal substitution. In the present paper we extend the previous considerations by also including nonminimal couplings induced by the anomalous magnetic moments of nucleons.

II. FRAMEWORK
In this section we review our formalism to link the nEDM to the CP violating couplings. It is based on a phenomenological Lagrangian L eff formulated in terms of hadronic degrees of freedom (nucleons N = (p, n), pions π = (π ± , π 0 ), H = (η, η ) mesons) and photons A µ which separates into a free L 0 and interaction part L int with L 0 includes the usual free terms of nucleons, mesons, and photons is the stress tensor of the electromagnetic field, M N , M π , and M H are the masses of nucleons, pions, and η(η ) mesons, respectively. The interaction Lagrangian L int is given by a sum of two part. The first part contains the strong interaction terms, which describe the CP-even couplings of nucleons with pions L πN N and η(η ) mesons L HN N and the CP-violating η(η )ππ coupling L CP Hππ . The second part includes the electromagnetic interaction terms, describing the coupling of charged pions and nucleons with the photon (L γN N and L γππ , respectively): where g πN N = g A Fπ M N , g A = 1.275 is the nucleon axial charge, F π = 92.4 MeV is the pion decay constant, g HN N and f Hππ are corresponding CP-even and CP-odd couplings between pions and η(η ), γ µ and γ 5 are the Dirac matrices, and σ µν = i 2 [γ µ , γ ν ]. The values of g ηN N and g η N N are taken from Ref. [10]: g ηN N = g η N N = 0.9. Note that in the case of nucleons we include both minimal and nonminimal electromagnetic couplings. Here Q N = diag(1, 0) and k N = diag(k p , k n ) are the diagonal matrices of nucleon charges and anomalous magnetic moments, respectively, where k p = 1.793 and k n = −1.913. For the CP-even interactions of nucleons with pseudoscalar mesons we use the pseudoscalar (PS) coupling [11], which is equivalent to the pseudovector (PV) coupling as demonstrated in Refs. [11][12][13]. As we have shown in Ref. [9], matrix elements in the two theories can differ by a divergent term, which can always be absorbed by an appropriative choice of a counterterm. In particular, in the case of the matrix element describing the nEDM the PS theory does not contain a logarithmic divergence, while it occurs in the PV theory.
As discussed in Ref. [9], there are two possible mechanisms for the generation of the η(η )ππ effective couplings. In the first mechanism this coupling is generated by the QCDθ-term [15,16] whereθ is the QCD vacuum angle and R = m u /m d is the ratio of u and d current quark masses. In this scenario, the η(η )ππ couplings are proportional toθ, which in turn is originally constrained by the experimental bounds on the neutron EDM [17,18]. In the the second scenario the nEDM and the CP violating η → ππ vertices are generated by two distinct mechanisms, without specifying details of a particular model in which this scenario would be realized. Thereby one can expect that the yet unknown mechanisms due to New Physics could enlarge the η(η )ππ couplings, which would induce a contribution to the nEDM at the two-loop level (see details in Ref. [9]).

III. NEUTRON EDM INDUCED BY THE CP VIOLATING η(η )ππ COUPLINGS AT THE TWO-LOOP LEVEL
In our approach a neutron EDM is described by a set of two-loop diagrams shown in Fig. 1 and 2. In Ref. [9] we already evaluated the diagrams generated by the minimal electromagnetic coupling [see Fig.1], i.e., by the coupling of virtual charged pions and the proton to the electromagnetic field. Here we extend our analysis by inclusion of the where p 1 and p 2 are the momenta of the initial and final neutron states, q 2 = (p 2 − p 1 ) 2 is the transfer momentum squared. The nEDM is defined as d E n = −F D (0)/(2M N ). To proceed we have to evaluate two-loop diagrams in the framework of the PS approach for the coupling between nucleons and pseudoscalar mesons. We want to point out again that the diagrams generated by the minimal coupling of charged pions and the proton with the electromagnetic field have been calculated in Ref. [9]. Also in Ref. [9] one can find details of the calculational technique, which is the same for the diagrams in Fig. 2 involving the nonminimal electromagnetic couplings of nucleons (anomalous magnetic moments). The diagrams in Fig. 2 can be grouped into sets with the same topology: (2a and 2b), (2c and 2d), (2e and 2f), (2g and 2h), and (2i and 2k).
The generic contribution of these diagrams to the nEDM is written as: I loop is sum of two topologically equivalent Feynman diagrams where S N (k) = ( k − M N ) −1 and S Mi (k) = (k 2 − M 2 i ) −1 are the nucleon and meson (with mass M i ) propagators, respectively. Here A ij is the 2 × 2 matrix A −1 and detA are its inverse and determinant, respectively, and is the scalar function deduced after calculation of the generic two-loop diagram from the set in Fig. 2.
Summing all graphs of Fig. 2 we obtain the resulting expression for the nEDM Here the factor 1/2 corresponds to the graphs with neutral pion loops.

IV. RESULTS AND DISCUSSION
Our numerical result for the nEDM induced by the CP violating η(η )ππ couplings and the anomalous magnetic moments of nucleons is The full result including both minimal and nonminimal electromagnetic couplings of nucleons can easily be computed by taking into account our previous results of Ref. [9] restricted to the case of minimal coupling: In Table I we present the detailed numerical results for the contribution of each diagram and their total contribution to the couplings c η and c η . For each diagram we specify (if it occurs) the contribution of charged pion-photon (PP) coupling, nucleon-photon minimal (MC) and nonminimal (NC) couplings and also indicate their total contribution (PP+MC+NC). The contributions coming from the nonminimal coupling of proton and neutron to the electromagnetic field have the same order of magnitude as the one induced by minimal coupling of the proton, but they compensate each other due to their oposite sign. The total numerical contribution of the nonminimal couplings of the nucleon is relatively suppressed (by one order of magnitude) compared to the total contribution of the minimal coupling of the proton.
In conclusion, we studied limits on the QCD CP-violating parameterθ and branchings of the CP-violating rare decays η → ππ and η → ππ using a phenomenological Lagrangian approach. We particularly took into account both minimal and nonminimal couplings of the nucleon to the photon. The nEDM was induced by the CP violating η(η ) → ππ couplings. Obtained results will be important for the planned experiments on rare η and η meson decays at JEF [19].