Lattice QCD determination of neutron-antineutron matrix elements with physical quark masses

Matrix elements of six-quark operators are needed to extract new physics constraints from experimental searches for neutron-antineutron oscillations. This work presents, in detail, the first lattice quantum chromodynamics calculations of the necessary neutron-antineutron transition matrix elements including calculation methods and discussions of systematic uncertainties. Implications of isospin and chiral symmetry on the matrix elements, power counting in the isospin limit, and renormalization of a chiral basis of six-quark operators are discussed. Calculations are performed with a chiral-symmetric discretization of the quark action and physical light quark masses in order to avoid the need for chiral extrapolation. Nonperturbative renormalization is performed, including a study of lattice cutoff effects. Excited-state effects are studied using two nucleon operators and multiple values of source-sink separation. Results for the dominant matrix elements are found to be significantly larger compared to previous results from the MIT bag model. Future calculations are needed to fully account for systematic uncertainties associated with discretization and finite-volume effects but are not expected to significantly affect this conclusion.

Figure 1

Contractions for the three-point correlation function of the (anti)neutrons with a $\overline{n}\leftarrow n$ transition operator. The indices of the neutron interpolating operators refer to the standard Dirac-Pauli representation (see Appendix pp1).

Figure 2

Combined correlated ${\chi }^2$ fits of $PP$, $PS$ two-point to Eq. (4.2) in the time range shown in the first row of Table 2. The covariance matrix is estimated with optimal shrinkage ${\lambda }^{*}$ as described in the main text. Corresponding data points show the effective masses $M_n(t)=\ln G_{nn}(t)-\ln G_{nn}(t+1)$ with their statistical uncertainties. Note that $t^{\max }$ in Table 2 indicates the largest separation for $G_{nn}$ considered and that the effective mass is consequently shown for $0\le t\le t^{\max }-1$.

Figure 3

Sensitivity of resulting (bare) matrix elements ${\mathcal{M}}_I$ obtained from three-point function fits [Eq. (4.1)] to the shrinkage parameter $\lambda$ that is used to estimate the covariance matrix as described in the main text (black points). The dark blue points indicate the values obtained with optimal shrinkage parameters ${\lambda }^{*}$ (see Eq. (4.3) and Appendix pp2-s1). The fit ranges are shown in the fourth row of Table 3.

Figure 4

Dependence of correlated ${\chi }^2/N_{\mathrm{dof}}$ values of three-point function fits [Eq. (4.1)] on the shrinkage parameter $\lambda$ that is used to estimate the covariance matrix as described in the main text (black points). The dark blue points indicate the values obtained with optimal shrinkage parameters ${\lambda }^{*}$ (see Eq. (4.3) and Appendix pp2-s1). The fit ranges are shown in the fourth row of Table 3.

Figure 5

Combined correlated ${\chi }^2$ fits of $PS$, $SS$ three-point functions to Eq. (4.1) for operators $Q_1$, $Q_2$, $Q_3$, and $Q_5$ for the time range shown in the first row of Table 3 (shaded bands). The state energies are determined from fits to two-point functions as shown in Fig. 2. Covariance matrix is estimated with optimal shrinkage ${\lambda }^{*}$ as described in the main text. Corresponding data points for ratios of three- to two-point correlation functions for all used source and sink separations $t_{\mathrm{sep}}$ are shown with intermittent square and circle data points. The central values of matrix elements ${\mathcal{M}}_I$ and their statistical uncertainties are shown with gray shaded bands. All the displayed uncertainties are statistical and estimated using bootstrap.

Figure 6

Comparison of ground state matrix elements (lattice units) extracted in fits with different fit ranges that are listed in Table 3. The black point at zero indicates the result of the weighted averaging procedure described in Appendix pp2-s3, and the small and large error bars indicate statistical and statistical-plus-systematic uncertainties, respectively.

Figure 7

Momentum configuration for nonperturbative renormalization of six-quark operators using RI-MOM scheme (only one permutation).

Figure 8

Magnitude of the off-diagonal components of the lattice mixing matrix (5.12) for (approximately) 4d-diagonal momentum $p^2=(5\mathrm{GeV}{)}^2$: (left) with quark external momenta shown in Fig. 7 and (right) averaged over their permutation (5.6). Only the values $|X_{IJ}|\ge {10}^{-4}$ are shown. The operator labels show their chiral isospin structure (see Table 1). The solid lines delineate operators that contain $RRR$, $RRL$, $LLL$, and $LLR$ diquarks.

Figure 9

Scale-independent renormalization factors for vector, scalar, and tensor currents. The axial current renormalization $Z_A$ is trivially constant because its vertex is used to eliminate the quark field renormalization $Z_q$. The close values of the vector and axial-vector renormalization constants indicate that chiral symmetry-breaking effects are negligible.

Figure 10

Perturbative running of the operators $Q_I$ at the two-loop level [17] in the RI-MOM scheme with $N_f=3$ (solid) and $N_f=4$ (dotted) and in the $\overline{\mathrm{MS}}$ scheme with $N_f=3$ (dashed) and $N_f=4$ (dash-dotted). The one-loop results are shown with thin solid lines. The reference point is ${\mu }_1=1.5\mathrm{GeV}\approx m_c$.

Figure 11

Normalized H(4) invariant $p^{[4]}/(p^2{)}^2$ [see Eq. (5.19)] for the lattice momenta included in the analysis.

Figure 12

Fits of lattice renormalization constants $Z^{\mathrm{lat}}(p)$ to the form (5.17), for $1.6\le p\le 4.5\mathrm{GeV}$ (left) and $2.0\le p\le 3.5\mathrm{GeV}$ (right). For each operator, the figures show the $Z^{SI}$ contribution together with the $\propto (ap{)}^2$ discretization correction (dashed lines), plus the nonperturbative correction (5.20) (solid lines), plus the discretization corrections (5.18) (crosses) vs lattice values $Z^{\mathrm{lat}}$ (open symbols). The gray bands indicate the fit regions. The star symbols on the left of each panel show the final $Z^{SI}$ values and their statistical uncertainties.