Electric Dipole Moment of the Nucleon

**Principal Investigator:**

Ulf-G. Meißner(1) and Timo Lähde(2)

**Affiliation:**

(1)Universität Bonn und Forschungszentrum Jülich, (2) Institute for Advanced Simulation, Forschungszentrum Jülich

**Local Project ID:**

jikp05

**HPC Platform used:**

JUQUEEN of JSC

**Date published:**

**Abstract**

*The electric dipole moment of the neutron, measuring the distance of positive and negative charge density in the neutron as shown in Figure 1, provides a unique and sensitive probe to physics beyond the Standard Model. It has played an important part over many decades in shaping and constraining numerous models of CP violation. QCD allows for CP-violating effects that propagate into the hadronic sector via the so-called θ term S _{θ} in the action, S = S + S_{θ }, with S_{θ }= i θ Q, where Q is the topological charge. In this project the electric dipole moment d_{n} of the neutron has been computed from a fully dynamical simulation of lattice QCD with nonvanishing θ term. We find d_{n} = −3.9(2)(9) × 10^{−16} θ e cm, which, when combined with the experimental limit on d_{n}, leads to the upper bound |θ| < 7.4 × 10^{−11}. This anomalously small number is referred to as the strong CP problem, which, together with quark confinement, is the biggest unsolved problem in QCD.*

**Main Text**

While the CP violation observed in K and B meson decays can be accounted for by the phase of the CKM matrix, the baryon asymmetry of the universe cannot be described by this phase alone, suggesting that there are additional sources of CP violation awaiting discovery. Grand unified theories (GUTs) provide a possible solution of the problem. In a wide class of GUTs the diagrams that generate a high baryon-to-photon asymmetry contribute to the renormalization of the CP-violating vacuum angle θ, thus relating the baryon asymmetry to the nucleon EDM. With the increasingly precise experimental efforts to observe the EDM of the nucleon, it is thus important to have a rigorous calculation directly from QCD.

In this project we present a calculation of d_{n} in units of θ with three flavors of dynamical quarks, i.e. up (u), down (d) and strange (s) quarks, using the lattice regularization. The novel feature of our work is that the simulations are performed directly at nonvanishing vacuum angle θ, in contrast to previous lattice studies, which rely on reweighting techniques. The calculation becomes feasible if θ is rotated to purely imaginary values.

The calculation divides into two parts, the lattice calculation and the extrapolation of the lattice results to the physical point using chiral effective theories. The lattice calculations are bound to unphysical quark masses, as the dipole moment vanishes in the chiral limit, which makes an educated extrapolation necessary. For the electric dipole moment this is achieved by chiral effective field theory.

The numerical calculations are done on the JUWELS system at JSC. The JUWELS system is based on Intel Xeon Platinum 8168 processors. This processor belongs to the first generation of Intel Xeon processors supporting the new AVX512 instructions. After having parallelized the compute intensive fermion matrix multiplication with SIMD intrinsics [1], we can fully exploit the capability of these new instructions. The Intel Xeon Platinum 8168 processor provides a relatively low memory bandwidth compared to its compute capabilities. To overcome this issue, we have optimized the use of the caches available on the processor, partially by appropriate scaling of the application, to benefit from super-linear speed-ups.

The cost for generating the gauge field configurations at nonvanishing vacuum angle turns out to be much more computer-time consuming than generating configurations with θ=0. The reason is that we are confronted with a large number of zero modes, which slows down the simulation considerably. The lattice calculations have been performed on lattices up to 48^{ 3 }x 96 and for pion masses as low as 250 MeV.

To extrapolate the numerical result to the physical quark masses, we have calculated the electric dipole moment of the nucleon to next-to-leading order in baryon chiral perturbation theory, in both infinite and finite volumes [2,3], by treating θ as an external field. One can then construct the most general Lagrangian, which is invariant under U(3)_{L} x U(3)_{R }transformations. The final result is given in [3].

Initially [4] we have computed the electric dipole moment of the neutron on modest lattice sizes at medium to large pion masses. What was missing is a meaningful fit to chiral perturbation theory, hand in hand with a reliable extrapolation to the physical point. This has been possible now by including larger lattices and smaller up and down quark masses. In Figure 2 we show our results, which will be published shortly [5].

The calculation follows [5]. We start from the SU(3) flavor symmetric point, m_{u} = m_{d }= m_{s }by keeping the singlet quark mass __m__ = (m_{u }+ m_{d} + m_{s})/3 fixed at its physical value, while δm_{q} = m_{q} - __m__ is varied.

We assume u and d quarks to be mass degenerate. At the physical u, d and s quark masses our calculation gives d_{n} = -0.0039(11) [e fm θ]. Combining this result with the experimental bound |d_{n}| ≤ 2.9 10^{−13 }[e fm], we arrive at |θ| < 7.4 10^{−11}. In Figure 3 we compare the result with other estimates.

**Project Contributors**

Feng-Kun Guo (ITP, Beijing), Roger Horsley (University of Edinburgh), Yoshifumi Nakamura (RIKEN), Dirk Pleiter (JSC), Paul Rakow (University of Liverpool), Akaki Rusetsky (University of Bonn), Gerrit Schierholz (DESY), James Zanotti (University of Adelaide)

**Publications**

[1] T. R. Haar, Y. Nakamura and H. Stüben, EPJ Web Conf. 175 (2018) 14011 [arXiv:1711.03836 [hep-lat]].

[2] T. Akan, F.-K. Guo and U.-G. Meißner, Phys. Lett. B **736** (2014) 163 [arXiv:1406.2882 [hep-ph]].

[3] F.-K. Guo and U.-G. Meißner, JHEP **1212** (2012) 097 [arXiv:1210.5887 [hep-ph]].

[4] F.-K. Guo, R. Horsley, U.-G. Meißner, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, A. Schiller and J. M. Zanotti, Phys. Rev. Lett. **115** (2015) 062001 [arXiv:1502.02295 [hep-lat]].

[4] F.-K. Guo, R. Horsley, U.-G. Meißner, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz and J. M. Zanotti,* in preparation.*

[5] W. Bietenholz, V. Bornyakov, M. Göckeler, R. Horsley, W. G. Lockhart, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller, T. Streuer, H. Stüben, F. Winter and J. Zanotti, Phys. Rev. D **84** (2011) 054509 [arXiv:1102.5300 [hep-lat]].

**Scientific Contact**

Prof. Dr. Ulf-G. Meißner

Universität Bonn und Forschungszentrum Jülich

D-53115 Bonn (Germany)

e-mail: meissner [at] hiskp.uni-bonn.de

*JSC Project ID: jikp05*

*October 2019*