RBC-UKQCD Collaboration Research Articles

Charm physics with overlap fermions on 2+1-flavor domain wall fermion configurations* *Supported in part by the National Key Research and Development Program of China (2020YFA0406400, 2023YFA1606002), the National Natural Science Foundation of China (12075253, 11935017, 12192264, 12293060, 12293065, 12293063, 12070131001), CRC 110 by DFG and NNSFC. Keh-Fei Liu is supported by the U.S. DOE Grant (DE-SC0013065) and DOE Grant (DEAC05-06OR23177),

Decay constants of pseudoscalar mesons D, , and vector mesons , , are determined from the lattice QCD at a lattice spacing fm. For vector mesons, the decay constants defined by tensor currents are given in the scheme at GeV. The calculation is performed on domain wall fermion configurations generated by the RBC-UKQCD collaborations and the overlap fermion action is used for the valence quarks. Comparing the current results with our previous results at a coarser lattice spacing fm provides a better understanding of the discretization error. We obtain with a better precision than our previous result. Combining our MeV with the total width of determined in a recent study gives a branching fraction for leptonic decay.

Varepsilon '/varepsilon in the Standard Model at the Dawn of the 2020s

We reanalyse the ratio varepsilon '/varepsilon in the Standard Model (SM) using most recent hadronic matrix elements from the RBC-UKQCD collaboration in combination with most important NNLO QCD corrections to electroweak penguin contributions and the isospin-breaking corrections. We illustrate the importance of the latter by using their latest estimate from chiral perturbation theory (ChPT) based on the octet approximation for lowest-lying mesons and a very recent estimate in the nonet scheme that takes into account the contribution of eta _0. We find (varepsilon '/varepsilon )^{(8)}_text {SM} = (17.4 pm 6.1) times 10^{-4} and (varepsilon '/varepsilon )^{(9)}_text {SM} = (13.9 pm 5.2) times 10^{-4}, respectively. Despite a very good agreement with the measured value (varepsilon '/varepsilon )_text {exp} = (16.6 pm 2.3) times 10^{-4}, the large error in (varepsilon '/varepsilon )_text {SM} still leaves room for significant new physics (BSM) contributions to this ratio. We update the 2018 master formula for (varepsilon '/varepsilon )_text {BSM} valid in any extension beyond the SM without additional light degrees of freedom. We provide new values of the penguin parameters B_6^{(1/2)}(mu ) and B_8^{(3/2)}(mu ) at the mu -scales used by the RBC-UKQCD collaboration and at lower scales mathcal {O}(1, text {GeV}) used by ChPT and Dual QCD (DQCD). We present semi-analytic formulae for (varepsilon '/varepsilon )_text {SM} in terms of these parameters and hat{Omega }_text {eff} that summarizes isospin-breaking corrections to this ratio. We stress the importance of lattice calculations of the mathcal {O}(alpha _{text {em}}) contributions to the hadronic matrix elements necessary for the removal of renormalization scheme dependence at mathcal {O}(alpha _{text {em}}) in the present analyses of varepsilon '/varepsilon .

\u03f5K and \u03f5\u2032/\u03f5 in a diquark model

Based on the calculations using the lattice QCD by the RBC-UKQCD collaboration and a large Nc dual QCD, the resulting ϵ′/ϵ, which is less than the experimental data by more than a 2σ in the standard model (SM), suggests the necessity of a new physics effect. In order to complement the insufficient ϵ′/ϵ, we study the extension of the SM with a colored scalar in a diquark model. In addition to the pure diquark box diagrams, it is found that the box diagrams with one W-boson and one diquark, ignored in the literature, play an important role in the ΔS = 2 process. The mass difference between KL and KS in the diquark model is well below the current data, whereas the Kaon indirect CP violation ϵK gives a strict constraint on the new parameters. Three mechanisms are classified in the study of ϵ′/ϵ. They include a tree-level diagram, QCD and electroweak (EW) penguins, and chromomagnetic operators (CMOs). Taking the Kobayashi-Maskawa phase as the unique CP source, we analyze each contribution of the three mechanisms in detail and conclude that with the exception of QCD and EW penguins, the tree and CMO effects can singly enhance ϵ′/ϵ to be of mathcal{O} (10−3), depending on the values of the free parameters, when the bound from ϵK is satisfied.

αs from the Hadronic Vacuum Polarisation

We present our result for the strong coupling constant computed from the u-d vector Hadronic Vacuum Polarisation function. We use nf = 2 + 1 flavours of Domain Wall fermions at 3 lattice spacings, generated by the RBC-UKQCD collaboration. We identify several possible pitfalls in this method for determining the coupling and illustrate how to resolve them.

The KL - KS Mass Difference

We review the status of the RBC-UKQCD collaborations’ computations of the KL-KS mass difference. After a brief discussion of the theoretical framework which had been developed previously by the collaboration, we describe our latest computation, performed at physical quark masses, and present our preliminary result mKL - mKS = (5.5 ± 1.70) × 10-12 MeV.

Quark chiral condensate from the overlap quark propagator**Supported by National Natural Science Foundation of China (11575197, 11575196, 11335001, 11405178), joint funds of NSFC (U1632104, U1232109), YC and ZL acknowledge the support of NSFC and DFG (CRC110)

From the overlap lattice quark propagator calculated in the Landau gauge, we determine the quark chiral condensate by fitting operator product expansion formulas to the lattice data. The quark propagators are computed on domain wall fermion configurations generated by the RBC-UKQCD Collaborations with Nf = 2+1 flavors. Three ensembles with different light sea quark masses are used at one lattice spacing 1/a = 1.75(4) GeV. We obtain in the SU(2) chiral limit.

Recent progress on CP violation in K → ππ decays in the SM and a supersymmetric solution

Using the recent first lattice results of the RBC-UKQCD collaboration for K → ππ decays, we perform a phenomenological analysis of and find a discrepancy between SM prediction and experiments by ∼ 3σ. We discuss an explanation by new physics. The well-understood value of εK, which quantifies indirect CP violation, however, typically prevents large new physics contributions to . In this talk, we show a solution of the discrepancy in the Minimal Supersymmetric Standard Model with squark masses above 3 TeV without fine-tuning of CP phases. In this solution, the Trojan penguin diagram gives large isospin-breaking contributions which enhance , while the contribution to εK is suppressed thanks to the Majorana nature of gluinos.

Kaon BSMB-parameters using improved staggered fermions fromNf=2+1unquenched QCD

We present results for the matrix elements of the additional $\Delta S=2$ operators that appear in models of physics beyond the Standard Model (BSM), expressed in terms of four BSM $B$-parameters. Combined with experimental results for $\Delta M_K$ and $\epsilon_K$, these constrain the parameters of BSM models. We use improved staggered fermions, with valence HYP-smeared quarks and $N_f=2+1$ flavors of "asqtad" sea quarks. The configurations have been generated by the MILC collaboration. The matching between lattice and continuum four-fermion operators and bilinears is done perturbatively at one-loop order. We use three lattice spacings for the continuum extrapolation: $a\approx 0.09$, $0.06$ and $0.045\;$fm. Valence light-quark masses range down to $\approx m_s^{\rm phys}/13$ while the light sea-quark masses range down to $\approx m_s^{\rm phys}/20$. Compared to our previous published work, we have added four additional lattice ensembles, leading to better controlled extrapolations in the lattice spacing and sea-quark masses. We report final results for two renormalization scales, $\mu=2\;\text{GeV}$ and $3\;\text{GeV}$, and compare them to those obtained by other collaborations. Agreement is found for two of the four BSM $B$-parameters ($B_2$ and $B_3^\text{SUSY}$). The other two ($B_4$ and $B_5$) differ significantly from those obtained using RI-MOM renormalization as an intermediate scheme, but are in agreement with recent preliminary results obtained by the RBC-UKQCD collaboration using RI-SMOM intermediate schemes.

Upper bounds on ε′/ε parameters B 6 (1/2) and B 8 (3/2) from large N QCD and other news

We demonstrate that in the large N approach developed by the authors in collaboration with Bardeen, the parameters B_6^{(1/2)} and B_8^{(3/2)} parametrizing the K\to\pi\pi matrix elements <Q_6>_0 and <Q_8>_2 of the dominant QCD and electroweak operators receive both negative O(1/N) corrections such that B_6^{(1/2)} < B_8^{(3/2)}<1 in agreement with the recent lattice results of the RBC-UKQCD collaboration. We also point out that the pattern of the size of the hadronic matrix elements of all QCD and electroweak penguin operators Q_i contributing to the K\to \pi \pi amplitudes A_0 and A_2, obtained by this lattice collaboration, provides further support to our large N approach. In particular, a very precise lattice result for the matrix element <Q_8>_0 implies for the corresponding parameter B_8^{(1/2)}=1.0\pm 0.2 to be compared with large N value B_8^{(1/2)}=1.1\pm 0.1. We discuss briefly the implications of these findings for the ratio epsilon'/epsilon. In fact, with the precise value for B_8^{(3/2)} from RBC-UKQCD collaboration, our upper bound on B_6^{(1/2)} implies epsilon'/epsilon in the SM roughly by a factor of two below its experimental value (16.6\pm 2.3)\times 10^{-4}. We also briefly comment on the parameter \hat B_K and the \Delta I=1/2$ rule.

Improved anatomy of ε′/ε in the Standard Model

We present a new analysis of the ratio epsilon'/epsilon within the Standard Model (SM) using a formalism that is manifestly independent of the values of leading (V-A)x(V-A) QCD penguin, and EW penguin hadronic matrix elements of the operators Q_4, Q_9, and Q_10, and applies to the SM as well as extensions with the same operator structure. It is valid under the assumption that the SM exactly describes the data on CP-conserving K -> pi pi amplitudes. As a result of this and the high precision now available for CKM and quark mass parameters, to high accuracy epsilon'/epsilon depends only on two non-perturbative parameters, B_6^(1/2) and B_8^(3/2), and perturbatively calculable Wilson coefficients. Within the SM, we are separately able to determine the hadronic matrix element <Q_4>_0 from CP-conserving data, significantly more precisely than presently possible with lattice QCD. Employing B_6^(1/2) = 0.57+-0.19 and B_8^(3/2) = 0.76+-0.05, extracted from recent results by the RBC-UKQCD collaboration, we obtain epsilon'/epsilon = (1.9+-4.5) 10^-4, substantially more precise than the recent RBC-UKQCD prediction and 2.9 sigma below the experimental value (16.6+-2.3) 10^-4, with the error being fully dominated by that on B_6^(1/2). Even discarding lattice input completely, but employing the recently obtained bound B_6^(1/2) <= B_8^(3/2) <= 1 from the large-N approach, the SM value is found more than 2 sigma below the experimental value. At B_6^(1/2) = B_8^(3/2) = 1, varying all other parameters within one sigma, we find epsilon'/epsilon = (8.6+-3.2) 10^-4. We present a detailed anatomy of the various SM uncertainties, including all sub-leading hadronic matrix elements, briefly commenting on the possibility of underestimated SM contributions as well as on the impact of our results on new physics models.

Vacuum Insertion Approximation and the [formula omitted] rule: A lattice QCD test of the naïve factorization hypothesis for K, D, B and static mesons

Motivated by a recent paper by the RBC–UKQCD Collaboration, which observes large violations of the naïve factorization hypothesis in K→ππ decays, we study in this paper the accuracy of the Vacuum Insertion Approximation (VIA) for the matrix elements of the complete basis of four-fermion ΔF=2 operators. We perform a comparison between the matrix elements in QCD, evaluated on the lattice, and the VIA predictions. We also investigate the dependence on the external meson masses by computing matrix elements for K, Ds, Bs and static mesons. In commonly used renormalization schemes, we find large violations of the VIA in particular for one of the two relevant Wick contractions in the kaon sector. These deviations, however, decrease significantly as the meson mass increases and the VIA predictions turn out to be rather well verified for B-meson matrix elements and, even better, in the infinite mass limit.

Large $$N$$ N approach to Kaon decays and mixing 28 years later: $$\Delta I=1/2$$ Δ I = 1 / 2 rule, $$\hat{B}_K$$ B ^ K , and $$\Delta M_K$$ Δ M K

We review and update our results for $$K\rightarrow \pi \pi $$ decays and $$K^0$$ – $$\bar{K}^0$$ mixing obtained by us in the 1980s within an analytic approximate approach based on the dual representation of QCD as a theory of weakly interacting mesons for large $$N$$ , where $$N$$ is the number of colors. In our analytic approach the Standard Model dynamics behind the enhancement of $$\hbox {Re}A_0$$ and suppression of $$\hbox {Re}A_2$$ , the so-called $$\Delta I=1/2$$ rule for $$K\rightarrow \pi \pi $$ decays, has a simple structure: the usual octet enhancement through the long but slow quark–gluon renormalization group evolution down to the scales $$\mathcal{O}(1\, {\hbox { GeV}})$$ is continued as a short but fast meson evolution down to zero momentum scales at which the factorization of hadronic matrix elements is at work. The inclusion of lowest-lying vector meson contributions in addition to the pseudoscalar ones and of Wilson coefficients in a momentum scheme improves significantly the matching between quark–gluon and meson evolutions. In particular, the anomalous dimension matrix governing the meson evolution exhibits the structure of the known anomalous dimension matrix in the quark–gluon evolution. While this physical picture did not yet emerge from lattice simulations, the recent results on $$\hbox {Re}A_2$$ and $$\hbox {Re}A_0$$ from the RBC-UKQCD collaboration give support for its correctness. In particular, the signs of the two main contractions found numerically by these authors follow uniquely from our analytic approach. Though the current–current operators dominate the $$\Delta I=1/2$$ rule, working with matching scales $$\mathcal{O}(1 \, {\hbox { GeV}})$$ we find that the presence of QCD-penguin operator $$Q_6$$ is required to obtain satisfactory result for $$\hbox {Re}A_0$$ . At NLO in $$1/N$$ we obtain $$R=\hbox {Re}A_0/\hbox {Re}A_2= 16.0\pm 1.5$$ which amounts to an order of magnitude enhancement over the strict large $$N$$ limit value $$\sqrt{2}$$ . We also update our results for the parameter $$\hat{B}_K$$ , finding $$\hat{B}_K=0.73\pm 0.02$$ . The smallness of $$1/N$$ corrections to the large $$N$$ value $$\hat{B}_K=3/4$$ results within our approach from an approximate cancelation between pseudoscalar and vector meson one-loop contributions. We also summarize the status of $$\Delta M_K$$ in this approach.

Opening the Rome-Southampton window for operator mixing matrices

We show that the running of operators which mix under renormalization can be computed fully nonperturbatively as a product of continuum step-scaling matrices. These step-scaling matrices are obtained by taking the ``ratio'' of $Z$ matrices computed at different energies in an RI-MOM type scheme for which twisted boundary conditions are an essential ingredient. Our method allows us to relax the bounds of the Rome-Southampton window. We also explain why such a method is important in view of the light quark physics program of the RBC-UKQCD Collaborations. To illustrate our method, using ${n}_{f}=2+1$ domain-wall fermions, we compute the nonperturbative running matrix of four-quark operators needed in $K\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ decay and neutral kaon mixing. Our results are then compared to perturbation theory.

Continuum limit physics from2+1flavor domain wall QCD

We present physical results obtained from simulations using 2+1 flavors of domain wall quarks and the Iwasaki gauge action at two values of the lattice spacing a, [a−1=1.73(3) GeV and a−1=2.28(3) GeV]. On the coarser lattice, with 243×64×16 points (where the 16 corresponds to Ls, the extent of the 5th dimension inherent in the domain wall fermion formulation of QCD), the analysis of C. Allton et al. (RBC-UKQCD Collaboration), Phys. Rev. D 78 is extended to approximately twice the number of configurations. The ensembles on the finer 323×64×16 lattice are new. We explain in detail how we use lattice data obtained at several values of the lattice spacing and for a range of quark masses in combined continuum-chiral fits in order to obtain results in the continuum limit and at physical quark masses. We implement this procedure for our data at two lattice spacings and with unitary pion masses in the approximate range 290–420 MeV (225–420 MeV for partially quenched pions). We use the masses of the π and K mesons and the Ω baryon to determine the physical quark masses and the values of the lattice spacing. While our data in the mass ranges above are consistent with the predictions of next-to-leading order SU(2) chiral perturbation theory, they are also consistent with a simple analytic ansatz leading to an inherent uncertainty in how best to perform the chiral extrapolation that we are reluctant to reduce with model-dependent assumptions about higher order corrections. In some cases, particularly for fπ, the pion leptonic decay constant, the uncertainty in the chiral extrapolation dominates the systematic error. Our main results include fπ=124(2)stat(5)syst MeV, fK/fπ=1.204(7)(25) where fK is the kaon decay constant, msMS¯(2 GeV)=(96.2±2.7) MeV and mudMS¯(2 GeV)=(3.59±0.21) MeV (ms/mud=26.8±1.4) where ms and mud are the mass of the strange quark and the average of the up and down quark masses, respectively, [ΣMS¯(2 GeV)]1/3=256(6) MeV, where Σ is the chiral condensate, the Sommer scale r0=0.487(9) fm and r1=0.333(9) fm.52 MoreReceived 10 November 2010DOI:https://doi.org/10.1103/PhysRevD.83.074508© 2011 American Physical Society