Robert V. Harlander

We demonstrate the use of the gradient flow combined with the short flow-time expansion ( GF + SFTX ) as a renormalization procedure for four-quark operator matrix elements and associated bag parameters relevant to neutral heavy-meson mixing ( Δ Q = 2 ) and heavy-meson lifetimes ( Δ Q = 0 ). Using six RBC/UKQCD 2 + 1 -flavor domain-wall fermion ensembles, we calculate for a charm-strange system with physical quark masses flowed bag parameters and match them to the MS ¯ scheme using perturbative SFTX coefficients up to next-to-next-to-leading order in QCD. We employ a multiscale matching strategy and a renormalization-group improved flow-time evolution which allows for a reliable estimate of systematic uncertainties. For a fictitious neutral D s meson, we obtain the Δ Q = 2 MS ¯ bag parameter B 1 MS ¯ ( 3 GeV ) = 0.7673 ( 123 ) , consistent with existing short-distance D 0 mixing determinations. For the Δ Q = 0 lifetime-ratio operator basis, we find the MS ¯ results B 1 MS ¯ ( 3 GeV ) = 1.0524 ( 97 ) , B 2 MS ¯ ( 3 GeV ) = 0.9621 ( 71 ) , ε 1 MS ¯ ( 3 GeV ) = − 0.2275 ( 76 ) , and ε 2 MS ¯ ( 3 GeV ) = − 0.0005 ( 8 ) . We provide conversion formulae to reexpress these results for an arbitrary choice of evanescent operators. These results demonstrate that GF + SFTX can deliver precise determinations of dimension-six four-quark operators and establish a framework for future lattice computations including more complex operator bases, where the challenge of power-divergent mixing is shifted to the continuum and handled in the SFTX.

We calculate the dimension-six Δ Q = 0 four-quark matrix elements describing heavy-meson lifetime ratios using the gradient flow with its short flow-time expansion (SFTX) as a renormalization procedure. On six RBC/UKQCD 2 + 1 -flavor domain-wall fermion ensembles, we determine flowed bag parameters for physical charm and strange quarks and match to the MS ¯ scheme with perturbative SFTX coefficients through next-to-next-to-leading order (NNLO). A multiscale matching procedure using renormalization-group running improves the extrapolation to zero flow time. For the operators relevant to τ ( D s ) / τ ( D 0 ) at the SU ( 3 ) F symmetric point, we obtain B 1 MS ¯ ( 3 GeV ) = 1.0524 ( 97 ) , B 2 MS ¯ ( 3 GeV ) = 0.9621 ( 70 ) , ε 1 MS ¯ ( 3 GeV ) = − 0.2275 ( 76 ) , and ε 2 MS ¯ ( 3 GeV ) = − 0.0005 ( 8 ) using a specific choice of evanescent operators. This is the first lattice-QCD determination of Δ Q = 0 four-quark operators with a full error budget. It opens the path toward higher-precision predictions of heavy-meson lifetimes and similar quantities exhibiting operator mixing under renormalization.

A bstract: We provide results for the vacuum expectation values of the flowed action density, the quark condensate, and the quark kinetic operator in the gradient-flow formalism. We work in N _F -flavor QCD, keeping the heaviest quark massive and all others massless. The vacuum expectation values of these operators are calculated numerically through next-to-next-to-leading order QCD, providing important input for the extraction of fundamental QCD parameters from lattice calculations. While the focus is on charm- and bottom-quark mass effects, we provide the results in a form that is independent of the specific quark mass.

We propose a new and simple method for determining the renormalized quark masses from lattice simulations. Renormalized quark masses are an important input to many phenomenological applications, including searching and modeling physics beyond the Standard Model. The nonperturbative renormalization is performed using gradient flow combined with the short-flow-time expansion that is improved by renormalization group (RG) running to match to the MS ¯ scheme. Implementing the RG running perturbatively, we demonstrate this method works reliably at least up to the charm-quark mass and exhibits an easily attainable “windowing condition.” Using RBC/UKQCD’s ( 2 + 1 )-flavor Shamir domain-wall fermion ensembles with Iwasaki gauge action, we find m s MS ¯ ( μ = 2 GeV ) = 89 ( 3 ) MeV and m c MS ¯ ( μ = 3 GeV ) = 972 ( 16 ) MeV . These results predict the scale-independent ratio m c / m s = 12.1 ( 4 ) . Generalization to other observables is possible, providing an efficient approach to determine nonperturbatively renormalized fermionic observables like form factors or bag parameters from lattice simulations.

We evaluate the coupling of quantum electrodynamics (QED) in the gradient-flow scheme in three and four space-time dimensions. Our general result applies to any theory with a U(1) gauge field coupled to arbitrary other fields via arbitrary interactions. As an example, we consider QED with n f flavors in three and four space-time dimensions and evaluate the corresponding β functions. In four dimensions, we find that the perturbative expansion of the β function behaves much better than the corresponding expression in quantum chromodynamics. In three dimensions, we recover both the ultraviolet as well as the infrared fixed points of the QED coupling in the large- n f limit.

Parton distribution functions (PDFs) are central to precision QCD phenomenology. Their Mellin moments can be computed on the lattice, but direct determinations using local operators, besides ⟨ x ⟩ , face severe challenges from reduced hypercubic symmetry, limiting results to the lowest moments. A recently proposed method resolves these issues using gradient flow. We demonstrate the efficacy of this method by computing ratios of flavor nonsinglet pion PDF moments up to ⟨ x 5 ⟩ on four lattice spacings at m π ≃ 411 MeV . The moments and reconstructed PDF agree quantitatively with recent phenomenological extractions.

We study the impact of the pointlike q q ¯ ′ W H interactions originating from a subset of dimension-six operators of the Standard Model effective field theory (SMEFT) on the Higgs boson associated production p p → W + H at the LHC. For consistency, we also take into account the corresponding q q ¯ ′ W vertices implied by these operators. We compute the next-to-next-to-leading order quantum chromodynamics (QCD) corrections to this process and compare these corrections in the Standard Model and SMEFT production mechanisms.