Neutron electric dipole moment from lattice QCD

Abstract

We carry out a feasibility study for the lattice QCD calculation of the neutron electric dipole moment (NEDM) in the presence of the $\theta$ term. We develop the strategy to obtain the nucleon EDM from the CP-odd electromagnetic form factor $F_3$ at small $\theta$, in which NEDM is given by $\lim_{q^2\to 0}\theta F_3(q^2)/(2m_N)$ where $q$ is the momentum transfer and $m_N$ is the nucleon mass. We first derive a formula which relates $F_3$, a matrix element of the electromagnetic current between nucleon states, with vacuum expectation values of nucleons and/or the current. In the expansion of $\theta$, the parity-odd part of the nucleon-current-nucleon three-point function contains contributions not only from the parity-odd form factors but also from the parity-even form factors multiplied by the parity-odd part of the nucleon two-point function, and therefore the latter contribution must be subtracted to extract $F_3$. We then perform an explicit lattice calculation employing the domain-wall quark action with the RG improved gauge action in quenched QCD at $a^{-1}\simeq 2$ GeV on a $16^3\times 32\times 16$ lattice. At the quark mass $m_f a =0.03$, corresponding to $m_\pi/m_\rho \simeq 0.63$, we accumulate 730 configurations, which allow us to extract the parity-odd part in both two- and three-point functions. Employing two different Dirac $\gamma$ matrix projections, we show that a consistent value for $F_3$ cannot be obtained without the subtraction described above. We obtain $F_3(q^2\simeq 0.58 \textrm{GeV}^2)/(2m_N) =$ $-$0.024(5) $e\cdot$fm for the neutron and $F_3(q^2\simeq 0.58 \textrm{GeV}^2)/(2m_N) =$ 0.021(6) $e\cdot$fm for the proton.Comment: LaTeX2e, 43 pages, 42 eps figures, uses revtex4 and graphicx, comments added and typos corrected, final version to appear in Phys. Rev.

Recent progress in hadron structure from Lattice QCD

“…[71]. Besides extracting the nEDM from the CP-odd form factor [72,76,77,71], there are alternative methods to compute the nEDM, such are the implementation of an imaginary θ [74], or with an application of an external electric field and measuring the associated energy shifts [75]. A collection of lattice results are displayed in Fig.…”

“…2) [71] (blue square) corresponding to a weighted average using different methods for extracting F 3 (0)), b) N f =0 DWF [72] (magenta upward triangles), N f =2+1 DWF [76] (red circles) and N f =0 clover fermions [77] (green downward triangles) by extracting the CP-odd F 3 (Q 2 ) and fitting its Q 2 -dependence, c) N f =2 Clover fermions [75] (turquoise left triangles) obtained using a background electric field, d) N f =2+1 clover fermions [74] (orange diamonds) by implementing an imaginary θ .…”

Extrapolation formulas for neutron EDM calculations in lattice QCD

“…Lattice calculations of the neutron edm [2,3,4,5,6,7] in terms of the strong CP-violating parameter are continually evolving toward a reliable estimate that can be directly compared with experimental limits and possible future observations. The latest generation of lattice calculations respect chiral symmetry and lattice spacing effects have been relegated to O(a 2 ).…”

New Physics Searches from Nucleon Matrix Elements in Lattice QCD

“…Similar methods have been extended for the calculation of other matrix elements [49]. So far we discussed the extraction of the nEDM from the CP-odd form factor [47,[50][51][52], which is based on the small-θ expansion. However, there are alternative methods to compute the nEDM, such as, simulating the action with an imaginary θ [53], or with an application of an external electric field and measuring the associated energy shifts [54].…”

“…The large statistical error is a demonstration of the difficulty to extract this quantity and the need for development of new techniques. π for: a) N f =2+1+1 TMF [47] (blue square) corresponding to a weighted average using different methods for extracting F 3 (0)), b) N f =0 DWF [50] (magenta upward triangles), N f =2+1 DWF [51] (red circles) and N f =0 Clover fermions [52] (green downward triangles) by extracting the CP-odd F 3 (Q 2 ) and fitting its Q 2 -dependence, c) N f =2 Clover fermions [54] (turquoise left triangles) obtained using a background electric field, d) N f =2+1 Clover fermions [53] (orange diamonds) by implementing an imaginary θ.…”