Review of lattice results concerning low-energy particle physics

We review lattice results related to pion, kaon, D-meson, B-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor f_+(0) arising in the semileptonic K rightarrow pi transition at zero momentum transfer, as well as the decay constant ratio f_K/f_pi and its consequences for the CKM matrix elements V_{us} and V_{ud}. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)_Ltimes SU(2)_R and SU(3)_Ltimes SU(3)_R Chiral Perturbation Theory. We review the determination of the B_K parameter of neutral kaon mixing as well as the additional four B parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for m_c and m_b as well as those for D- and B-meson decay constants, form factors, and mixing parameters. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant alpha _s. Finally, in this review we have added a new section reviewing results for nucleon matrix elements of the axial, scalar and tensor bilinears, both isovector and flavor diagonal.

We review lattice results related to pion, kaon, D-meson, B-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor f_+(0) arising in the semileptonic K rightarrow pi transition at zero momentum transfer, as well as the decay constant ratio f_K/f_pi and its consequences for the CKM matrix elements V_{us} and V_{ud}. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)_Ltimes SU(2)_R and SU(3)_Ltimes SU(3)_R Chiral Perturbation Theory. We review the determination of the B_K parameter of neutral kaon mixing as well as the additional four B parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for m_c and m_b as well as those for the decay constants, form factors, and mixing parameters of charmed and bottom mesons and baryons. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant alpha _s. We consider nucleon matrix elements, and review the determinations of the axial, scalar and tensor bilinears, both isovector and flavor diagonal. Finally, in this review we have added a new section reviewing determinations of scale-setting quantities.

We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle-physics community. More specifically, we report on the determination of the light-quark masses, the form factor f_+(0), arising in semileptonic K rightarrow pi transition at zero momentum transfer, as well as the decay-constant ratio f_K/f_pi of decay constants and its consequences for the CKM matrix elements V_{us} and V_{ud}. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of hbox {SU}(2)_Ltimes hbox {SU}(2)_R and hbox {SU}(3)_Ltimes hbox {SU}(3)_R Chiral Perturbation Theory and review the determination of the B_K parameter of neutral kaon mixing. The inclusion of heavy-quark quantities significantly expands the FLAG scope with respect to the previous review. Therefore, we focus here on D- and B-meson decay constants, form factors, and mixing parameters, since these are most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. In addition we review the status of lattice determinations of the strong coupling constant alpha _mathrm{s}.

We review lattice results related to pion, kaon, D -meson, B -meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor f + ( 0 ) arising in the semileptonic K → π transition at zero momentum transfer, as well as the decay-constant ratio f K / f π and its consequences for the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements V u s and V u d . We review the determination of the B K parameter of neutral kaon mixing as well as the additional four B parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for m c and m b as well as those for the decay constants, form factors, and mixing parameters of charmed and bottom mesons and baryons. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant α s . We review the determinations of nucleon charges from the matrix elements of both isovector and flavor-diagonal axial, scalar and tensor local quark bilinears, and momentum fraction, helicity moment and the transversity moment from one-link quark bilinears. We also review determinations of scale-setting quantities. Finally, in this review we have added a new section on the general definition of the low-energy limit of the Standard Model.

We review lattice results relevant for pion and kaon physics with the aim of making them easily accessible to the particle physics community. Specifically, we review the determination of the light-quark masses, the form factor f_+(0), relevant for the semileptonic K -> pi transition at zero momentum transfer as well as the ratio f_K/f_pi of decay constants and discuss the consequences for the elements V_{us} and V_{ud} of the CKM matrix. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)_LxSU(2)_R and SU(3)_LxSU(3)_R Chiral Perturbation Theory and review the determination of the B_K parameter of neutral kaon mixing. We introduce quality criteria and use these when forming averages. Although subjective and imperfect, these criteria may help the reader to judge different aspects of current lattice computations. Our main results are summarized in section 1.2, but we stress the importance of the detailed discussion that underlies these results and constitutes the bulk of the present review.

Using the entire ϒ(4S) dataset, the first two-dimensional unbinned angular analysis of the semileptonic decay B¯→Dℓ−ν¯ℓ is performed, employing hadronic reconstruction of the tag-side B meson from ϒ(4S)→BB¯. Here, ℓ denotes the light charged leptons e and μ. A novel data-driven signal-background separation procedure with minimal dependence on simulation is developed. This procedure preserves all multidimensional correlations present in the data. The expected sin2θℓ dependence of the differential decay rate in the Standard Model is demonstrated, where θℓ is the lepton helicity angle. Including input from the latest lattice QCD calculations and previously available experimental data, the underlying form factors are extracted using both model-independent (BGL) and dependent (CLN) methods. Comparisons with lattice calculations show flavor SU(3) symmetry to be a good approximation in the B(s)→D(s) sector. Using the BGL results, the CKM matrix element |Vcb|=(41.09±1.16)×10−3 and the Standard Model prediction of the lepton-flavor universality violation variable R(D)=0.300±0.004, are extracted. The value of |Vcb| from B¯→Dℓ−ν¯ℓ tends to be higher than that extracted using B¯→D*ℓ−ν¯ℓ. The Standard Model R(D) calculation is at a 1.97σ tension with the latest HFLAV experimental average. Published by the American Physical Society 2024

We present a lattice QCD calculation of the pion scalar form factor and associated radii with fully controlled systematics. Lattice results are computed on a large set of 17 gauge ensembles with Nf=2+1 Wilson Clover-improved sea quarks. These ensembles cover four values of the lattice spacing between a=0.049 fm and a=0.086 fm, a pion mass range of 130–350 MeV, and various physical volumes. A precise determination of the notorious quark-disconnected contributions facilitates an unprecedented momentum resolution for the form factor, particularly on large and fine ensembles in the vicinity of physical quark mass. A large range of source-sink separations 1.0 fm≲tsep≲3.25 fm is used to reliably extract the relevant ground state matrix elements at vanishing and nonvanishing momentum transfer. This allows us for the first time to obtain the scalar radii from a z-expansion parametrization of the Q2 dependence of the resulting form factors rather than a simple, linear approximation at small momentum transfer. The physical extrapolation for the radii is carried out using three-flavor NLO chiral perturbation theory to parametrize the quark mass dependence in terms of three low-energy constants, including the first lattice determination of L4r. Systematic uncertainties on the final results related to the ground state extraction, form factor parametrization as well as the physical extrapolation are accounted for via model averages based on the Akaike information criterion. Our physical results for the pion singlet and octet scalar radii are ⟨rS2⟩π0=0.550(32)stat(39)sys[50]total fm2 and ⟨rS2⟩π8=0.443(16)stat(45)sys[47]total fm2, respectively. The values of the relevant low energy constants are given by f0=116.5(5.5)stat(13.7)sys[14.8]total MeV, L4r(μ=770 MeV)=+0.38(09)stat(15)sys[18]total×10−3 and L5r(μ=770 MeV)=+0.58(0.38)stat(1.08)sys[1.14]total×10−3. For the light quark contribution to the pion scalar radius, we obtain ⟨rS2⟩πl=0.515(28)stat(32)sys[43]total fm2, resulting in a value of ℓ¯4=3.99(15)stat(17)sys[23]total for the pertinent low-energy constant in two-flavor chiral perturbation theory.

We present a comprehensive analysis of form factors for two light pseudoscalar mesons induced by scalar, vector, and tensor quark operators. The theoretical framework is based on a combination of unitarized chiral perturbation theory and dispersion relations. The low-energy constants in chiral perturbation theory are fixed by a global fit to the available data of the two-meson scattering phase shifts. Each form factor derived from unitarized chiral perturbation theory is improved by iteratively applying a dispersion relation. This study updates the existing results in the literature and explores those that have not been systematically studied previously, in particular the two-meson tensor form factors within unitarized chiral perturbation theory. We also discuss the applications of these form factors as mandatory inputs for low-energy phenomena, such as the semi-leptonic decays Bs→ π+π−ℓ+ℓ− and the τ lepton decay τ → π−π0ντ, in searches for physics beyond the Standard Model.

The leptonic decay-constant ratio $f_K/f_\pi$ is calculated from lattice-QCD simulations using $N_f=2+1$ dynamical fermion flavors in the clover-improved formulation and 2-HEX smearing. The simulations were performed for a range of mass-degenerate light quarks including the physical point and at various lattice couplings and volumes, allowing to quantify all relevant sources of systematic uncertainties for our final number of the decay-constant ratio. Utilizing input from chiral perturbation theory, we also quote the charged decay-constant ratio $f_{K^\pm}/f_{\pi^\pm}$. With further input from super-allowed nuclear $\beta$-decays, eventually we obtain an estimate for the CKM-matrix element $V_{\rm us}$.

We calculate the $B \to\pi\ell\nu$ and $B_s \to K \ell\nu$ form factors in dynamical lattice QCD. We use the (2+1)-flavor RBC-UKQCD gauge-field ensembles generated with the domain-wall fermion and Iwasaki gauge actions. For the $b$ quarks we use the anisotropic clover action with a relativistic heavy-quark interpretation. We analyze two lattice spacings $a \approx 0.11, 0.086$ fm and unitary pion masses as light as $M_\pi \approx 290$ MeV. We simultaneously extrapolate our numerical results to the physical light-quark masses and to the continuum and interpolate in the pion/kaon energy using SU(2) "hard-pion" chiral perturbation theory. We provide complete error budgets for the form factors $f_+(q^2)$ and $f_0(q^2)$ at three momenta that span the $q^2$ range accessible in our numerical simulations. We extrapolate these results to $q^2 = 0$ using a model-independent $z$-parametrization and present our final form factors as the $z$-coefficients and the matrix of correlations between them. Our results agree with other lattice determinations using staggered light quarks and provide important independent cross-checks. Both $B \to\pi\ell\nu$ and $B_s \to K \ell\nu$ decays enable a determination of the CKM matrix element $|V_{ub}|$. To illustrate this, we perform a combined $z$-fit of our numerical $B\to\pi\ell\nu$ form-factor data with the experimental branching-fraction measurements leaving the relative normalization as a free parameter; we obtain $|V_{ub}| = 3.61(32) \times 10^{-3}$, where the error includes statistical and systematic uncertainties. This approach can be applied to $B_s\to K \ell\nu$ decay to determine $|V_{ub}|$ once the process has been measured experimentally. Finally, in anticipation of future measurements, we make predictions for $B \to \pi\ell\nu$ and $B_s\to K \ell\nu$ Standard-Model differential branching fractions and forward-backward asymmetries.

Standard Model and beyond with CKM matrix element V_ud

After a short presentation of the FLAG collaboration, we review lattice results related to pion, $K$-, $D$- and $B$-meson physics with the aim of making them easily accessible to the particle-physics community. Only a selection of FLAG averages or estimates is presented. For light flavours, we present results on the form factor $f_+(0)$, arising in semileptonic $K \rightarrow \pi$ transition at zero momentum transfer, as well as the decay-constants $f_K,f_\pi$ and their ratio. The consequences of these results for the CKM matrix elements $|V_{us}|$ and $|V_{ud}|$ are discussed. For heavy flavours we focus on $D$- and $B$-meson decay constants and form factors, as well as the CKM matrix elements $|V_{cs}|$, $|V_{cd}|$ and $|V_{ub}|$. In addition we briefly cover the recent advances stemming from the calculation the $B_K$-parameters and touch upon related current results relevant to the Physics beyond the Standard Model, which will be the subject of the next FLAG edition.

The ${K}_{l3}$ form factors are investigated in the one-loop approximation within the context of a linear renormalizable SU(3) $\ensuremath{\sigma}$ model. The model incorporates SU(3) nonets of pseudoscalar ($\ensuremath{\pi}$,$K$,$\ensuremath{\eta}$,${\ensuremath{\eta}}^{\ensuremath{'}}$) and scalar ($\ensuremath{\epsilon}$,$\ensuremath{\kappa}$,$\ensuremath{\sigma}$,${\ensuremath{\sigma}}^{\ensuremath{'}}$) mesons. The ${\ensuremath{\eta}}^{\ensuremath{'}}$, $\ensuremath{\epsilon}$, $\ensuremath{\kappa}$, $\ensuremath{\sigma}$, and ${\ensuremath{\sigma}}^{\ensuremath{'}}$ mesons are associated with the ${X}^{0}(957)$, $\ensuremath{\delta}(980)$, $\ensuremath{\kappa}(1400)$, ${S}^{*}(980)$, and $\ensuremath{\epsilon}(1300)$, respectively. The Lagrangian contains the most general renormalizable chiral-SU(3) \ifmmode\times\else\texttimes\fi{} SU(3)-invariant couplings as well as explicit linear symmetry-breaking terms belonging to the (3,3*) \ensuremath{\bigoplus} (3*,3) representation of SU(3) \ifmmode\times\else\texttimes\fi{} SU(3). All calculations are carried out in the one-loop approximation. With this model we first obtain a reasonable approximation to the scalar and pseudoscalar mass spectrum and the known leptonic decay constants. Most of the masses and decay constants are reproduced within 10%. In addition, the second-order corrections were usually in the neighborhood of 15-20% or less, supporting the conjecture that higher-order strong-interaction effects may, in most cases, be rather small. Employing these solutions, we calculate the ${K}_{l3}$ form factors and compare these with recent experimental studies. The predictions for ${\ensuremath{\lambda}}_{+}$ are too small probably due to the fact that, at this level of approximation, the model contains no spin-one poles (vector mesons). However, the model predictions for ${\ensuremath{\lambda}}_{0}$ and $\ensuremath{\xi}(0)$ are fairly good. A number of theoretical predictions for the ${K}_{l3}$ form factors based on current algebra and chiral perturbation theory are then investigated. In particular, we were interested in the magnitude of the corrections to various predictions derived using specific symmetry assumptions. As the model reproduces quite closely the model-independent calculation of ${f}_{+}(0)$ from chiral perturbation theory, we feel that our conclusions in this area may have a more general significance. For example, in the tree approximation the Callan-Treiman relation is an identity. In the one-loop approximation, the magnitude of the strong-interaction effects is much larger than the symmetry-breaking effects and the relation is still obeyed quite well (within \ensuremath{\sim} 2%), reflecting the influence of the underlying SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) symmetry. Overall, our results support calculations based on chiral perturbation theory.

We study the effects of a finite cubic volume with twisted boundary conditions on pseudoscalar mesons. We apply Chiral Perturbation Theory in the p-regime and introduce the twist by means of a constant vector field. The corrections of masses, decay constants, pseudoscalar coupling constants and form factors are calculated at next-to-leading order. We detail the derivations and compare with results available in the literature. In some case there is disagreement due to a different treatment of new extra terms generated from the breaking of the cubic invariance. We advocate to treat such terms as renormalization terms of the twisting angles and reabsorb them in the on-shell conditions. We confirm that the corrections of masses, decay constants, pseudoscalar coupling constants are related by means of chiral Ward identities. Furthermore, we show that the matrix elements of the scalar (resp. vector) form factor satisfies the Feynman–Hellman Theorem (resp. the Ward–Takahashi identity). To show the Ward–Takahashi identity we construct an effective field theory for charged pions which is invariant under electromagnetic gauge transformations and which reproduces the results obtained with Chiral Perturbation Theory at a vanishing momentum transfer. This generalizes considerations previously published for periodic boundary conditions to twisted boundary conditions. Another method to estimate the corrections in finite volume are asymptotic formulae. Asymptotic formulae were introduced by Luscher and relate the corrections of a given physical quantity to an integral of a specific amplitude, evaluated in infinite volume. Here, we revise the original derivation of Luscher and generalize it to finite volume with twisted boundary conditions. In some cases, the derivation involves complications due to extra terms generated from the breaking of the cubic invariance. We isolate such terms and treat them as renormalization terms just as done before. In that way, we derive asymptotic formulae for masses, decay constants, pseudoscalar coupling constants and scalar form factors. At the same time, we derive also asymptotic formulae for renormalization terms. We apply all these formulae in combination with Chiral Perturbation Theory and estimate the corrections beyond next-to-leading order. We show that asymptotic formulae for masses, decay constants, pseudoscalar coupling constants are related by means of chiral Ward identities. A similar relation connects in an independent way asymptotic formulae for renormalization terms. We check these relations for charged pions through a direct calculation. To conclude, a numerical analysis quantifies the importance of finite volume corrections at next-to-leading order and beyond. We perform a generic Analysis and illustrate two possible applications to real simulations.

We determine the low-energy constants f0, L4r and L5r of SU(3) chiral perturbation theory from a lattice QCD calculation of the scalar form factors of the pion with fully controlled systematics. Lattice results are computed on a large set of Nf=2+1 gauge ensembles covering four lattice spacings a∈[0.049,0.086] fm, pion masses Mπ∈[130,350] MeV, and various large physical volumes. By determining the notorious quark-disconnected contributions with unprecedented precision and using a large range of source-sink separations tsep∈[1.0,3.25] fm, we are able for the first time to obtain the scalar radii from a z-expansion parametrization of the form factors rather than a simple linear approximation at small momentum transfer. The low-energy constants are obtained from the physical extrapolation of the radii using next-to-leading-order SU(3) next-to-leading-order chiral perturbation theory to parametrize the quark mass dependence. Systematic uncertainties are estimated via model averages based on the Akaike information criterion. Our determination of L4r is the first lattice determination to obtain a result not compatible with zero.