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

TL;DR

This paper refines the upper limits on CP violating eta-pi-pi couplings and the neutron EDM using a phenomenological hadronic model, including nonminimal photon couplings, providing constraints relevant for upcoming experiments.

Contribution

It extends previous analyses by incorporating nonminimal photon couplings into the model, leading to more accurate upper limits on CP violating couplings and the theta parameter.

Findings

Upper limits on eta-pi-pi couplings: |f_eta pi pi|<4.4e-11 and |f_eta' pi pi|<3.8e-11.

Derived upper limit on the CP violating theta parameter: <4.7e-10.

Results are relevant for planned experiments at JLab Eta Factory.

Abstract

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 $\bar{\mathrm{\theta}}<4.7 \times 10^{-10}$.

Tables

Table 1. Table 1: Numerical results for the c η subscript 𝑐 𝜂 c_{\eta} and c η ′ subscript 𝑐 superscript 𝜂 ′ c_{\eta^{\prime}} couplings.

Diagram	Coupling $c_{\eta }$				Coupling $c_{{\eta }^{\prime }}$
(Figs.1 and 2)	PP	MC	NC	PP+MC+NC	PP	MC	NC	PP+MC+NC
1a + 1b / 2i+2k	-	0.58	-0.92	-0.34	-	0.71	-1.57	-0.86
1c + 1d / 2g + 2h	-	0.56	-0.88	-0.32	-	0.67	-1.5	-0.83
1e + 1k	1.12	-	-	1.12	1.29	-	-	1.29
1g + 1h	1.02	-	-	1.02	1.13	-	-	1.01
1f + 1i	0.1	-	-	0.1	0.13	-	-	0.13
2a + 2b	-	-	0.77	0.77	-	-	1.32	1.32
2c + 2d	-	-	0.47	0.47	-	-	0.8	0.8
2e + 2f	-	-	0.49	0.49	-	-	0.84	0.84
Total	4.48	2.28	-0.14	6.62	5.1	2.76	-0.22	7.64

Equations

$${\mathcal{L}}_{\mathrm{eff}}={\mathcal{L}}_0+{\mathcal{L}}_{\mathrm{int}}.$$

$${\mathcal{L}}_0=\bar{N}(i/\partial -M_N)N+\frac{1}{2}\vec{\pi }(\square -M_{\pi }^2)\vec{\pi }+\frac{1}{2}H(\square -M_H^2)H-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu },$$

$${\mathcal{L}}_{\mathrm{int}}$$

$${\mathcal{L}}_{\pi NN}$$

$${\mathcal{L}}_{HNN}$$

$${\mathcal{L}}_{H\pi \pi }^{\mathrm{CP}}$$

$${\mathcal{L}}_{\gamma NN}$$

$${\mathcal{L}}_{\gamma \pi \pi }$$

$$\mathrm{Br}(H\to \pi \pi )=n_{\Gamma }\frac{\sqrt{M_H^2-4M_{\pi }^2}}{4\pi {\Gamma }_H^{tot}}{f}_{H\pi \pi }^2,$$

$$\mathrm{Br}(\eta ({\eta }^{\prime })\to \pi \pi )$$

$$f_{\eta \pi \pi }^{\bar{\theta }}=-\frac{1}{\sqrt{3}}\frac{\bar{\theta }{M}_{\pi }^{2}R}{F_{\pi }{M}_{\eta }(1+R{)}^2},f_{{\eta }^{\prime }\pi \pi }^{\bar{\theta }}=\sqrt{2}{f}_{\eta \pi \pi }^{\bar{\theta }}\frac{M_{\eta }}{M_{{\eta }^{\prime }}},$$

$$M_{\mathrm{inv}}$$

$$-{\bar{u}}_N(p_2)d_N^i{\sigma }^{\mu \nu }{q}_{\nu }{\gamma }^5{u}_N(p_1)+\dots$$

$$I_{\mathrm{loop}}$$

$$I_d(M_1,M_2,M_3)$$

$$A_{ij}$$

$$d_N$$

$$d_n^{E,k}\simeq (c_{\eta }{f}_{\eta \pi \pi }+c_{{\eta }^{\prime }}{f}_{{\eta }^{\prime }\pi \pi })\times 10^{-16}\text{e}\cdot \text{cm},$$

$$c_{\eta }=-0.14,c_{{\eta }^{\prime }}=-0.22.$$

$$d_n^E\simeq (c_{\eta }{f}_{\eta \pi \pi }+c_{{\eta }^{\prime }}{f}_{{\eta }^{\prime }\pi \pi })\times 10^{-16}\text{e}\cdot \text{cm},$$

$$c_{\eta }=6.62,c_{{\eta }^{\prime }}=7.64.$$

$$\mathrm{Br}(\eta \to {\pi }^{+}{\pi }^{-})<5.54\times 10^{-17},$$

$$\mathrm{Br}(\eta \to {\pi }^0{\pi }^0)<2.27\times 10^{-17},$$

$$|f_{\eta \pi \pi }(m_{\eta }^2)|<4.4\times 10^{-11},|f_{{\eta }^{\prime }\pi \pi }(m_{{\eta }^{\prime }}^2)|<3.8\times 10^{-11}.$$

$$|f_{\eta \pi \pi }(m_{\eta }^2)|<4.3\times 10^{-11},|f_{{\eta }^{\prime }\pi \pi }(m_{{\eta }^{\prime }}^2)|<3.7\times 10^{-11}.$$

$${\bar{\theta }}^{\eta }<9.1\times 10^{-10},{\bar{\theta }}^{{\eta }^{\prime }}<9.7\times 10^{-10},$$

$${\bar{\theta }}^{\eta }<8.6\times 10^{-10},{\bar{\theta }}^{{\eta }^{\prime }}<9.1\times 10^{-10},$$

$$d_n^E\simeq (c_n^E\times 10^{-16})\cdot \bar{\theta }\text{e}\cdot \text{cm},$$

$$c_n^E=0.65(\mathrm{ChPT}),c_n^E=0.62(\mathrm{LQCD}).$$

$$\bar{\theta }=4.4\times 10^{-10}(\mathrm{ChPT}),\bar{\theta }=4.7\times 10^{-10}(\mathrm{LQCD}).$$

Full text

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 $\eta$ and $\eta^{\prime}$ 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. Pospelov and Ritz (2005); Chupp et al. (2019)). Our interest in hadron EDMs is motivated by the possibility to extract limits on the CP-violating strong coupling between hadrons and the $\bar{\theta}$ parameter (CP-violating gluon-gluon coupling). From study of the nEDM one can estimate the QCD $\bar{\theta}$ parameter and the $\pi NN$, $\eta^{(\prime)}NN$, and $\eta^{(\prime)}\pi\pi$ couplings. In series of papers Kuckei et al. (2007); Dib et al. (2006); Faessler et al. (2006a, b); Castillo-Felisola et al. (2015); Gutsche et al. (2017); Zhevlakov et al. (2019) we gave several analyses of CP violating physics focused on the nEDM and strong CP violating phenomena with the relation to the QCD $\bar{\theta}$ 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. Gutsche et al. (2017); Zhevlakov et al. (2019) we focused on the determination of the CP violating couplings $\eta\pi\pi$ and $\eta^{\prime}\pi\pi$ from the nEDM. Our formalism was based on a phenomenological Lagrangian describing the interaction of nucleons with pseudoscalar mesons (pions and $\eta(\eta^{\prime})$). 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 ${\cal L}_{\rm eff}$ formulated in terms of hadronic degrees of freedom (nucleons $N=(p,n)$, pions $\pi=(\pi^{\pm},\pi^{0})$, $H=(\eta,\eta^{\prime})$ mesons) and photons $A_{\mu}$ which separates into a free ${\cal L}_{0}$ and interaction part ${\cal L}_{\rm int}$ with

[TABLE]

${\cal L}_{0}$ includes the usual free terms of nucleons, mesons, and photons

[TABLE]

where $\Box=-\partial_{\mu}\partial^{\mu}$, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the stress tensor of the electromagnetic field, $M_{N}$, $M_{\pi}$, and $M_{H}$ are the masses of nucleons, pions, and $\eta(\eta^{\prime})$ mesons, respectively. The interaction Lagrangian ${\cal L}_{\rm int}$ 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 ${\cal L}_{\pi NN}$ and $\eta(\eta^{\prime})$ mesons ${\cal L}_{HNN}$ and the CP-violating $\eta(\eta^{\prime})\pi\pi$ coupling ${\cal L}^{\rm CP}_{H\pi\pi}$. The second part includes the electromagnetic interaction terms, describing the coupling of charged pions and nucleons with the photon (${\cal L}_{\gamma NN}$ and ${\cal L}_{\gamma\pi\pi}$, respectively):

[TABLE]

where $g_{\pi NN}=\frac{g_{A}}{F_{\pi}}M_{N}$, $g_{A}=1.275$ is the nucleon axial charge, $F_{\pi}=92.4$ MeV is the pion decay constant, $g_{HNN}$ and $f_{H\pi\pi}$ are corresponding CP-even and CP-odd couplings between pions and $\eta(\eta^{\prime})$, $\gamma^{\mu}$ and $\gamma^{5}$ are the Dirac matrices, and $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$. The values of $g_{\eta NN}$ and $g_{\eta^{\prime}NN}$ are taken from Ref. Tiator et al. (2018): $g_{\eta NN}=g_{\eta^{\prime}NN}=0.9$. Note that in the case of nucleons we include both minimal and nonminimal electromagnetic couplings. Here $Q_{N}={\rm diag}(1,0)$ and $k_{N}={\rm diag}(k_{p},k_{n})$ 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 Weinberg (1968), which is equivalent to the pseudovector (PV) coupling as demonstrated in Refs. Weinberg (1968); Faessler et al. (2006c); Lensky and Pascalutsa (2010). As we have shown in Ref. Zhevlakov et al. (2019), 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.

The CP-violating term ${\cal L}^{\rm CP}_{H\pi\pi}$ induces a contribution to the nEDM. The $\eta(\eta^{\prime})\pi\pi$ couplings define the corresponding two-body decay branching ratios as

[TABLE]

where $\Gamma_{H}^{tot}$ is the total width of $H$; $n_{\Gamma}$ is a final-state factor, which equals 1/2 for the $\pi^{0}\pi^{0}$ and 1 for the $\pi^{+}\pi^{-}$ final states. Upper limits for these decays are set by the LHCb results Aaij et al. (2017)

[TABLE]

As discussed in Ref. Zhevlakov et al. (2019), there are two possible mechanisms for the generation of the $\eta(\eta^{\prime})\pi\pi$ effective couplings. In the first mechanism this coupling is generated by the QCD $\bar{\theta}$-term Crewther et al. (1979); Shifman et al. (1980)

[TABLE]

where $\bar{\theta}$ 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 $\eta(\eta^{\prime})\pi\pi$ couplings are proportional to $\bar{\theta}$, which in turn is originally constrained by the experimental bounds on the neutron EDM Harris et al. (1999); Baker et al. (2006). In the the second scenario the nEDM and the CP violating $\eta\to\pi\pi$ 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 $\eta(\eta^{\prime})\pi\pi$ couplings, which would induce a contribution to the nEDM at the two-loop level (see details in Ref. Zhevlakov et al. (2019)).

III Neutron EDM induced by the CP violating $\eta(\eta^{\prime})\pi\pi$ 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. Zhevlakov et al. (2019) 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 nonminimal couplings of the nucleon to the electromagnetic field due to the anomalous magnetic moments $k_{N}$ [see Fig. 2]. contained in the interaction Lagrangian ${\cal L}_{\gamma NN}$.

The matrix element corresponding to the diagrams of Figs. 1 and 2 or the electromagnetic vertex function of the neutron is expanded in terms of four relativistic form factors $F_{E}$ (electric), $F_{M}$ (magnetic), $F_{D}$ (electric dipole) and $F_{A}$ (anapole) as

[TABLE]

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 $d_{n}^{E}=-F_{D}(0)/(2M_{N})$.

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. Zhevlakov et al. (2019). Also in Ref. Zhevlakov et al. (2019) 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:

[TABLE]

$I_{\rm loop}$ is sum of two topologically equivalent Feynman diagrams

[TABLE]

where $S_{N}(k)=(\not\!k-M_{N})^{-1}$ and $S_{M_{i}}(k)=(k^{2}-M_{i}^{2})^{-1}$ are the nucleon and meson (with mass $M_{i}$) propagators, respectively. Here $A_{ij}$ is the $2\times 2$ matrix

[TABLE]

$A^{-1}$ and ${\rm det}A$ are its inverse and determinant, respectively, and $B_{1}=p_{1}\alpha_{1}\,,\ B_{2}\,=\,p_{1}\alpha_{3}+p_{2}\alpha_{2}\,,\ \Delta=M_{1}^{2}\alpha_{4}+M_{2}^{2}\alpha_{5}+M_{3}^{2}\alpha_{6}$, $\beta_{1}=\alpha_{1}A_{11}^{-1}+\alpha_{23}A_{12}^{-1}$ and $\beta_{2}=\alpha_{1}A_{12}^{-1}+\alpha_{23}A_{22}^{-1}$. $I_{d}(M_{1},M_{2},M_{3})$ 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

[TABLE]

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 $\eta(\eta^{\prime})\pi\pi$ couplings and the anomalous magnetic moments of nucleons is

[TABLE]

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. Zhevlakov et al. (2019) restricted to the case of minimal coupling:

[TABLE]

In Table 1 we present the detailed numerical results for the contribution of each diagram and their total contribution to the couplings $c_{\eta}$ and $c_{\eta^{\prime}}$. 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.

The bounds for the branching ratios of the rare decays $\Gamma_{\eta\pi\pi}$ and $\Gamma_{\eta^{\prime}\pi\pi}$ are strongly suppressed when compared to existing data Aaij et al. (2017)

[TABLE]

When we deduce these new bounds we suppose that the CP-violating $\eta\pi\pi$ and $\eta^{\prime}\pi\pi$ couplings are independent and use the current experimental bound on the nEDM: $|d_{n}^{E}|<2.9\times 10^{-26}\,\text{e}\cdot\text{cm}$. These limits are about $\sim$ 12-14 orders of magnitude more stringent than given by the recent data from the LHCb Collaboration Aaij et al. (2017). The planned study of the $\eta(\eta^{\prime})\pi\pi$ decays at the JLab Eta Factory (JEF) Gan (2017) could shed light on the possible impact of New Physics on these CP-violating processes.

The CP-violating $\eta\pi\pi$ and $\eta^{\prime}\pi\pi$ couplings are estimated using Eq. (IV) and limits on the nEDM Tanabashi et al. (2018):

[TABLE]

Note that these results are very close to the ones obtained in Ref. Zhevlakov et al. (2019) restricted to the minimal coupling of charged pions and proton with the photon:

[TABLE]

Using Eq. (8) we also derive upper limits for the $\bar{\theta}$ parameter:

[TABLE]

for a current quark mass ratio $R=0.556$ taken from ChPT at 1 GeV scale Gasser and Leutwyler (1982) and

[TABLE]

for $R=0.468$ taken from QCD lattice (LQCD) data at scale of 2 GeV Tanabashi et al. (2018).

One can also pursue another way to obtain an upper estimate for the $\bar{\theta}$ parameter. In particular, one can extract $\bar{\theta}$ substituting the QCD relations between $f^{\bar{\theta}}_{\eta(\eta^{\prime})\pi\pi}$ and $\bar{\theta}$ (8) into the expression for the nEDM (IV). For the quark mass ratios taken from ChPT and LQCD we get:

[TABLE]

Using data on the nEDM Graner et al. (2016) we extract the following upper limits for $\bar{\theta}$

[TABLE]

The second predictions of upper limits for $\bar{\theta}$ are by a factor 2 smaller than the first ones and are closer to the prediction done in Ref. Sahoo (2017). In particular, in Ref. Sahoo (2017) the value $\bar{\theta}\sim 1.1\times 10^{-10}$ was extracted using data for the nEDM with $|d_{n}^{E}|<1.6\times 10^{-26}\,\text{e}\cdot\text{cm}$. For this limit on the nEDM we deduce $\bar{\theta}=2.4\times 10^{-10}\ ({\rm ChPT})$ and $\bar{\theta}=2.6\times 10^{-10}\ ({\rm LQCD})$.

In conclusion, we studied limits on the QCD CP- violating parameter $\bar{\theta}$ and branchings of the CP- violating rare decays $\eta\to\pi\pi$ and $\eta^{\prime}\to\pi\pi$ 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 $\eta(\eta^{\prime})\to\pi\pi$ couplings. Obtained results will be important for the planned experiments on rare $\eta$ and $\eta^{\prime}$ meson decays at JEF Gan (2017).

Acknowledgements.

The work of A.S.Zh. was funded by Russian Science Foundation grant (RSCF 18-72-00046), in particular part of numerical calculation. The work of V.E.L. was funded by the Carl Zeiss Foundation under Project “Kepler Center für Astro- und Teilchenphysik: Hochsensitive Nachweistechnik zur Erforschung des unsichtbaren Universums (Gz: 0653-2.8/581/2)”, and by CONICYT (Chile) under Grant No. PIA/Basal FB0821.