We calculate the scalar semileptonic kaon decay in finite volume at the momentum transfer $t_{m} = (m_{K} - m_π)^2$, using chiral perturbation theory. At first we obtain the hadronic matrix element to be calculated in finite volume. We then evaluate the finite size effects for two volumes with $L = 1.83 fm$ and $L= 2.73 fm$ and find that the difference between the finite volume corrections of the two volumes are larger than the difference as quoted in \cite{Boyle2007a}. It appears then that the pion masses used for the scalar form factor in ChPT are large which result in large finite volume corrections. If appropriate values for pion mass are used, we believe that the finite size effects estimated in this paper can be useful for Lattice data to extrapolate at large lattice size.
We present new lattice results of the K -> pion semileptonic form factors obtained from simulations with two flavors of dynamical twisted-mass fermions, using pion masses as light as 260 MeV. Our main result is f+(0) = 0.9560 (84), which, combined with the latest experimental data for Kl3 decays, leads to |V_{us}| = 0.2267 (5)_exp (20)_f+(0). Using the PDG(2008) determinations of |Vud| and |Vub| our result implies for the unitarity relation |Vud|**2 + |Vus|**2 + |Vub|**2 = 1.0004 (15). For the O(p**6) term of the chiral expansion of f+(0) we get Df = f+(0) - 1 - f2 = -0.0214 (84).
We discuss the recent progress in the study of semileptonic kaon and pion decays, including new experimental results, improved electroweak radiative corrections, form factor calculations and isospin-breaking effects. As a result, we obtain $|V_{us}|=0.22309(40)(39)(3)$ from kaon semileptonic decays and $|V_{us}/V_{ud}|=0.22908(66)(41)(40)(2)(1)$ from the ratio between the kaon and pion semileptonic decay rates. We report an apparent violation of the top-row Cabibbo-Kobayashi-Maskawa matrix unitarity at a $3.2\sim 5.6σ$ level, and a discrepancy at a $2.2σ$ level between the value of $|V_{us}/V_…
The quark flavor sector of the Standard Model is a fertile ground to look for new physics effects through a unitarity test of the Cabbibo-Kobayashi-Maskawa (CKM) matrix. We present a lattice QCD calculation of the scalar and the vector form factors (over a large $q^2$ region including $q^2 = 0$) associated with the $D \rightarrow Klν$ semi-leptonic decay. This calculation will then allow us to determine the central CKM matrix element, $V_{cs}$ in the Standard Model, by comparing the lattice QCD results for the form factors and the experimental decay rate. This form factor calculation has been …
We review current issues in exclusive semileptonic tau decays. We present the formalism of structure functions, and then discuss predictions for final states with kaons, for decays into four pions and for radiative corrections to the decay into a single pion. Implications for testing the standard model at low energies are discussed.
An analysis of semileptonic decays of $B$ mesons with the emission of a single soft pion is presented in the framework of the heavy-quark limit using an effective Lagrangian which implements chiral and heavy-quark symmetries. The analysis is performed at leading order of the chiral and inverse heavy mass expansions. In addition to the ground state heavy mesons some of their resonances are included. The estimates of the various effective coupling constants and form factors needed in the analysis are obtained using a chiral quark model. As the main result, a clear indication is found that the $0…
We evaluate the form factors describing the exclusive decay B-> pi l nu by using a Constituent Quark-Meson model based on an effective quark-meson Lagrangian (CQM). The model allows for an expansion in the pion momenta and we consider terms up to the first order in the pion field derivatives. We compute the leading terms in the soft pion limit and consider corrections to this limit.
We derive a new factorization relation for the semileptonic radiative decay B -> π\ell νγin the kinematical region of a slow pion p_π~ Λand an energetic photon E_γ>> Λ, working at leading order in Λ/m_b. In the limit of a soft pion, the nonperturbative matrix element appearing in this relation can be computed using chiral perturbation theory. We present a phenomenological study of this decay, which may be important for a precise determination of the exclusive nonradiative decay.
A brief survey of applications of chiral perturbation theory to both semileptonic and nonleptonic kaon decays is presented. Special emphasis is given to recent theoretical advances related to semileptonic decays, in particular pion pion scattering and Ke4 decays. The systematic approach for including isospin violation and electromagnetic corrections in semileptonic decays is discussed for Kl3 decays.
In this talk I give a short introduction to hard pion Chiral Perturbation Theory and an overview of the available applications $K\toππ$, $B,D\to D,π,K,η$ semileptonic decays and $χ_{c0,2}\toππ,KK$. It is pointed out that the reults for the semileptonic decays obey the LEET relation between $f_+$ and $f_-$.
We calculate the kaon semileptonic form factors in lattice QCD with three flavors of dynamical overlap quarks. Gauge ensembles are generated at pion masses as low as 290 MeV and at a strange quark mass near its physical value. We precisely calculate relevant meson correlators using the all-to-all quark propagator. Twisted boundary conditions and the reweighting technique are employed to vary the momentum transfer and the strange quark mass. We discuss the chiral behavior of the form factors by comparing with chiral perturbation theory and experiments.
We present a lattice determination of the vector and scalar form factors of the $D\toπ(K)\ellν$ semileptonic decays, which are relevant for the extraction of the CKM matrix elements $|V_{cd}|$ and $|V_{cs}|$ from experimental data. Our analysis is based on the gauge configurations produced by the European Twisted Mass Collaboration with $N_f=2+1+1$ flavors of dynamical quarks. We simulated at three different values of the lattice spacing and with pion masses as small as 210 MeV. The matrix elements of both vector and scalar currents are determined for a plenty of kinematical conditions in whic…
We report on our calculation of the kaon semileptonic form factors in Nf=2+1 lattice QCD. Chiral symmetry is exactly preserved by using the overlap quark action for a straightforward comparison with chiral perturbation theory (ChPT). We simulate three pion masses down to 290 MeV at a single lattice spacing of 0.11 fm and at a strange quark mass very close to its physical value. The form factors near zero momentum transfer are precisely calculated by using the all-to-all propagator and twisted boundary conditions. We compare the normalizations and slopes of the form factors with ChPT and experi…
The anomalous value of the tensor form factor in the kaon decay K -> pi e nu and the destructive interference in pion decay pi -> e nu gamma can be simultaneously described by admixture of a tensor interaction in the standard V-A one. It is shown that the same tensor interaction can describe recent HERA data at very large Q^2.
We present a calculation of the B -> pi l nu form factors with domain-wall light quarks and relativistic b-quarks on the lattice. We work with the 2+1 flavor domain-wall fermion and Iwasaki gauge-field ensembles generated by the RBC and UKQCD Collaborations. The chiral-continuum extrapolation is performed using SU(2) hard-pion chiral perturbation theory. To extrapolate the lattice form factors to the full kinematic range, we use the model independent z-parameterization and impose the kinematic constraint f_+(0) = f_0(0) at zero momentum transfer.
We review a novel model-independent approach to the analysis of new-physics effects in the $s\to u\ell^-\barν$ transitions. We apply it to (semi)leptonic kaon decays and study their complementarity with pion and hyperon $β$ decays or with collider searches of new physics.
We present a new calculation of the K->pi semileptonic form factor at zero momentum transfer in domain wall lattice QCD with Nf=2+1 dynamical quark flavours. By using partially twisted boundary conditions we simulate directly at the phenomenologically relevant point of zero momentum transfer. We perform a joint analysis for all available ensembles which include three different lattice spacings (a=0.09-0.14fm), large physical volumes (m_pi*L>3.9) and pion masses as low as 171 MeV. The comprehensive set of simulation points allows for a detailed study of systematic effects leading to the predict…
We solve the coupled system of Dyson-Schwinger and Bethe-Salpeter equations for the quark propagator and the pion Bethe-Salpeter amplitude on a finite volume. To this end we use a truncation scheme that includes pion cloud effects in the quark propagator and light mesons. We study volume effects in the quark condensate, the pion mass and the pion decay constant and compare to corresponding results in other approaches. In general we find large effects for volumes below V=(1.8 fm)^4.
We construct several classes of hadronic matrix elements and relate them to the low-energy constants in Chiral Perturbation Theory that describe the electromagnetic effects in the semileptonic beta decay of the pion and the kaon. We propose to calculate them using lattice QCD, and argue that such a calculation will make an immediate impact to a number of interesting topics at the precision frontier, including the outstanding anomalies in $|V_{us}|$ and the top-row Cabibbo-Kobayashi-Maskawa matrix unitarity.
A calculation of the semi--leptonic decays of the kaon ($K_{l3}$) is presented. The results are direct predictions of a covariant model of the pion and kaon introduced earlier by Ito, Buck, Gross. The weak form factors for $K_{l3}$ are predicted with absolutely no parameter adjustments of the model. We obtained for the form factor parameters: $f_-(q^2=m_l^2)/f_+(q^2=m_l^2)=-0.28$ and $λ_+$= 0.028, both within experimental error bars. Connections of this approach to heavy quark symmetry will also be discussed.
We derive an asymptotic formula a la Luscher for the finite volume correction to the pion decay constant: this is expressed as an integral over the < 3 π| A_μ|0 > amplitude after proper subtraction of the pion pole contribution. We analyze the formula numerically at leading and next-to-leading order in the chiral expansion.
A sample of more than one million Ke4 decay candidates with less than one percent background contamination has been collected by the NA48/2 experiment at the CERN SPS in 2003-2004, allowing a detailed study of the decay properties. The branching ratio, inclusive of radiative Ke4gamma decays, is measured to be BR(Ke4) = (4.257 +/- 0.016_exp +/- 0.031_ext) x 10-5 with a total relative error of 0.8%. This measurement complements the study of S- and P-wave hadronic form factors by assigning absolute values to the relative hadronic form factors obtained earlier in a simultaneous analysis of the pio…
We evaluate the pion mass in finite volume to two loops within Chiral Perturbation Theory. The results are compared with a recently proposed extension of the asymptotic formula of Luscher. We find that contributions, which were neglected in the latter, are numerically very small at the two-loop level and conclude that for Mpi*L>2, L>2fm the finite volume effects in the meson sector are analytically well under control.
We calculate the pion and kaon electromagnetic form factors in finite volume for a generic momentum transfer in space-like region, using SU(3) chiral perturbation theory. To this end, we first find the hadronic matrix element in a new form which is suitable for our calculation in finite volume. We present our numerical findings for the chiral behavior and finite volume corrections and compare these for pion and kaon form factors. As a result, for the pion electromagnetic form factor we find the finite volume correction $ΔF^π_{em} = 0.0041$ at momentum transfer $q^2 = -0.1358$ and for the kaon …
We review some recent progress in the theory of electroweak radiative corrections in semileptonic decay processes. The resurrection of the so-called Sirlin's representation based on current algebra relations permits a clear separation between the perturbatively-calculable and incalculable pieces in the $\mathcal{O}(G_Fα)$ radiative corrections. The latter are expressed as compact hadronic matrix elements that allow systematic non-perturbative analysis such as dispersion relation and lattice QCD. This brings substantial improvements to the precision of the electroweak radiative corrections in s…
We update the lattice calculation of the $B\toπ$ semileptonic form factors, which have important applications to the CKM matrix element $|V_{ub}|$ and the $B\toπ\ell^+\ell^-$ rare decay. We use MILC asqtad ensembles with $N_f=2+1$ sea quarks and over a range of lattice spacings $a \approx 0.045$--$0.12$ fm. We perform a combined chiral and continuum extrapolation of our lattice data using SU(2) staggered chiral perturbation theory in the hard pion limit. To extend the results for the form factors to the full kinematic range, we take a functional approach to parameterize the form factors using …
We investigate how Lüscher's relation between the finite-volume energy of two pions at rest and pion scattering lengths has to be modified in quenched QCD. We find that this relation changes drastically, and in particular, that ``enhanced finite-volume corrections" of order $L^0=1$ and $L^{-2}$ occur at one loop ($L$ is the linear size of the box), due to the special properties of the $η'$ in the quenched approximation. We define quenched pion scattering lengths, and show that they are linearly divergent in the chiral limit. We estimate the size of these various effects in some numerical examp…
I present higher loop order results for several calculations in Chiral perturbation Theory. 1) Two-loop results at finite volume for hadronic vacuum polarization. 2) A three-loop calculation of the pion mass and decay constant in two-flavour ChPT. For the pion mass all needed auxiliary parameters can be determined from lattice calculations of $ππ$-scattering. 3) Chiral corrections to neutron-anti-neutron oscillations.
We determine the relative pion mass shift $M_π(L)/M_π-1$ due to the finite spatial extent $L$ of the box by means of two-flavor chiral perturbation theory and the one-particle Lüscher formula. We use as input the expression for the infinite volume $ππ$ forward scattering amplitude up to next-to-next-to-leading order and can therefore control the convergence of the chiral series. A comparison to the full leading order chiral expression for the pion mass in finite volume allows us to check the size of subleading terms in the large-$L$ expansion.
Semileptonic $B$ meson decays into the four lightest excited charmed meson states ($D_0^*$, $D_1^*$, $D_1$, and $D_2^*$) and their counterparts with $s$ quarks are investigated, including the full lepton mass dependence. We derive the standard model predictions for the differential branching fractions, as well as predictions for the ratios of the semi-tauonic and light lepton semileptonic branching fractions. These can be systematically improved using future measurements of the total or differential semileptonic rates to $e$ and $μ$, as well as the two-body hadronic branching fractions with a …
We evaluate the form factors describing the semileptonic decays $\bar{B^0}\to J/ψ(η_c) D^+ \ell^- \bar ν_\ell$, within the framework of a QCD relativistic potential model. This decay is complementary to $\bar{B^0}\to J/ψ(η_c) D^+ π^-$ in a phase space region where a pion factors out.We estimate the branching ratio for these semileptonic and nonleptonic channels, finding $\mathcal{BR}(\bar{B^0} \to J/ψ(η_c) D^+ \ell ν_\ell) \simeq 10^{-13}$, $\mathcal{BR}(\bar{B^0} \to J/ψD^+ π^-) = 3.1 \times 10^{-8}$ and $\mathcal{BR}(\bar{B^0} \to η_c D^+ π^-) = 3.5 \times 10^{-8}$.
We present the first realistic lattice QCD calculation of the $γW$-box diagrams relevant for beta decays. The nonperturbative low-momentum integral of the $γW$ loop is calculated using a lattice QCD simulation, complemented by the perturbative QCD result at high momenta. Using the pion semileptonic decay as an example, we demonstrate the feasibility of the method. By using domain wall fermions at the physical pion mass with multiple lattice spacings and volumes, we obtain the axial $γW$-box correction to the semileptonic pion decay, $\Box_{γW}^{VA}\big|_π=2.830(11)_{\mathrm{stat}}(26)_{\mathrm…
Using the framework of non-relativistic effective field theory, the finite-volume ground-state energy shift is calculated up-to-and-including $O(L^{-6})$ for the system of three pions in the channel with the total isospin $I=1$. The relativistic corrections are included perturbatively, up to the same order in the inverse of the box size $L$. The obtained explicit expression, together with the known result for the system with maximal isospin $I=3$, can be used for the extraction of two independent effective three-body couplings from the measured ground-state spectrum of three pions.
The quark-level linear sigma model is employed to compute a variety of electromagnetic and weak observables of light mesons, including pion and kaon form factors and charge radii, charged-pion polarizabilities, semileptonic weak $K_{\ell3}$ decay, semileptonic weak radiative pion and kaon form factors, radiative decays of vector mesons, and nonleptonic weak $K_{2π}$ decay. The agreement of all these predicted observables with experiment is striking. In passing, the tight link between the linear sigma model and vector-meson dominance is shown. Some conclusions are drawn on the linear sigma mode…
In this presentation, we briefly review recent investigations on pion and kaon structures from the instanton vacuum. Starting from the low-energy QCD partition function, we have computed the Gasser-Leutwyler low-energy constants, electromagnetic form factors of the pion and kaon, semileptonic decay form factors of the kaon, and light-cone distribution amplitues of the pion and kaon. The results are in good agreement with the experimental and empirical data.
We present a lattice QCD determination of the vector and scalar form factors of the kaon semileptonic decay $K \to π\ell ν$, which is relevant for the determination of the CKM matrix element $|V_{us}|$ from experimental data. Our results are based on the gauge configurations produced by the European Twisted Mass Collaboration with Nf = 2+1+1 dynamical fermions. We simulated at three different values of the lattice spacing and with pion masses as small as 210 MeV. Our estimate for the vector form factor at zero 4-momentum transfer is $f_+(0) = 0.9709 (46)$, where the uncertainty is both statist…
We present a lattice determination of the vector and scalar form factors of the semileptonic $D \to π\ell ν$ decays, which are relevant for the extraction of the CKM matrix element $\lvert V_{cd} \rvert$ from experimental data. Our analysis is based on the gauge configurations produced by the European Twisted Mass Collaboration with $N_f = 2 + 1 + 1$ flavors of dynamical quarks. We simulated at three different values of the lattice spacing and with pion masses as small as 210 MeV. Quark momenta are injected on the lattice using non-periodic boundary conditions. The matrix elements of both vect…
The aim of this paper is to investigate an efficient strategy that allows to obtain pi-pi phase shifts and rho meson properties from QCD lattice data with high precision. For this purpose we evaluate the levels of the pi-pi system in the rho channel in finite volume using chiral unitary theory. We investigate the dependence on the pi mass and compare with other approaches which use QCD lattice calculations and effective theories. We also illustrate the errors induced by using the conventional Luscher approach instead of a more accurate one recently developed that takes into account exactly the…
We present the first calculation of the kaon semileptonic form factor with sea and valence quark masses tuned to their physical values in the continuum limit of 2+1 flavour domain wall lattice QCD. We analyse a comprehensive set of simulations at the phenomenologically convenient point of zero momentum transfer in large physical volumes and for two different values of the lattice spacing. Our prediction for the form factor is f+(0)=0.9685(34)(14) where the first error is statistical and the second error systematic. This result can be combined with experimental measurements of K->pi decays for …
We present a new method aiming at a non-perturbative, model-independent determination of the momentum dependence of the form factors entering semileptonic decays using unitarity and analyticity constraints. We extend the original proposal and, using suitable two-point functions computed non-perturbatively, we determine the form factors at low-momentum transfer $q^2$ from those computed explicitly on the lattice at large $q^2$, without making any assumption about their $q^2$ dependence. As a training ground we apply the new method to the analysis of the lattice data of the semileptonic $D \righ…
We perform a calculation in one-loop chiral perturbation theory of the two-pion matrix elements and correlation functions of an I=0 scalar operator, in finite and infinite volumes for both full and quenched QCD. We show that major difficulties arise in the quenched theory due to the lack of unitarity. Similar problems are expected for quenched lattice calculations of $K \to ππ$ amplitudes with $ΔI=1/2$. Our results raise the important question of whether it is consistent to study $K\toππ$ amplitudes beyond leading order in chiral perturbation theory in quenched or partially quenched QCD.
In this work we present the first non-perturbative determination of the hadronic susceptibilities that constrain the form factors entering the semileptonic $B \to D^{(*)} \ell ν_\ell $ transitions due to unitarity and analyticity. The susceptibilities are obtained by evaluating moments of suitable two-point correlation functions obtained on the lattice. Making use of the gauge ensembles produced by the Extended Twisted Mass Collaboration with $N_f = 2+1+1$ dynamical quarks at three values of the lattice spacing ($a \simeq 0.062, 0.082, 0.089$ fm) and with pion masses in the range $\simeq 210 -…
We use chiral perturbation theory to study the extraction of pion electromagnetic polarizabilities from lattice QCD. Chiral extrapolation formulae are derived for partially quenched QCD, and quenched QCD simulations. On a torus, volume dependence of electromagnetic observables is complicated by SO(4) breaking, as well as photon zero-mode interactions. We determine finite volume corrections to the Compton scattering tensor of pions. We argue, however, that such results cannot be used to ascertain volume corrections to polarizabilities determined in lattice QCD with background field methods. Con…
We compute finite volume effects relevant for lattice QCD simulations using background fields. Focusing on constant electric and magnetic fields on a periodic lattice, we determine volume corrections to pion two-point functions using chiral perturbation theory. Among such corrections are the finite volume shifts to the electric and magnetic polarizabilities, which are numerically shown to be non-negligible. We additionally find that all terms in the single-particle effective action can be renormalized by infrared effects. This includes Born couplings to the pion current and total charge-square…
The two most precise determinations of the CKM matrix element V_us are based on the analyses of leptonic and semileptonic kaon decays. These studies also rely on the lattice QCD calculations of two hadronic parameters, namely the ratio of the kaon and pion decay constants, f_K+/f_pi+, and the kaon semileptonic vector form factor at zero momentum transfer, f_+(0). In this talk, I review the recent lattice results for these quantities, by showing that the sub-percent accuracy required by the phenomenological analyses has been reached by lattice QCD. As best estimates of the lattice calculations …
We calculate the mass shift for the pion in a finite volume with renormalization group (RG) methods in the framework of the quark-mesons model. In particular, we investigate the importance of the quark effects on the pion mass. As in lattice gauge theory, the choice of quark boundary conditions has a noticeable effect on the pion mass shift in small volumes, in addition to the shift due to pion interactions. We compare our results to chiral perturbation theory calculations and find differences due to the fact that chiral perturbation theory only considers pion effects in the finite volume.
We report on the MILC Collaboration calculation of electromagnetic effects on light pseudoscalar mesons. The simulations employ asqtad staggered dynamical quarks in QCD plus quenched photons, with lattice spacings varying from 0.12 to 0.06 fm. Finite volume corrections for the MILC realization of lattice electrodynamics have been calculated in chiral perturbation theory and applied to the lattice data. These corrections differ from those calculated by Hayakawa and Uno because our treatment of zero modes differs from theirs. Updated results for the corrections to "Dashen's theorem" are presente…
Pionless effective field theory in a finite volume (FVEFT$_{π\!/}$) is investigated as a framework for the analysis of multi-nucleon spectra and matrix elements calculated in lattice QCD (LQCD). By combining FVEFT$_{π\!/}$ with the stochastic variational method, the spectra of nuclei with atomic number $A\in\{2,3\}$ are matched to existing finite-volume LQCD calculations at heavier-than-physical quark masses corresponding to a pion mass $m_π=806$ MeV, thereby enabling infinite-volume binding energies to be determined using infinite-volume variational calculations. Based on the variational wave…
We consider the quark-meson-model in a finite three-dimensional volume using the Schwinger proper-time renormalization group. We derive and solve the flow equations for finite volume in local potential approximation. In order to break chiral symmetry in the finite volume, we introduce a small current quark mass. The corresponding effective meson potential breaks chiral O(4) symmetry explicitly, depending on sigma and pion fields separately. We calculate the volume dependence of the pion mass and of the pion decay constant with the renormalization group flow equations and compare with recent re…
We investigate finite volume effects on the pion mass and the pion decay constant with renormalization group (RG) methods in the framework of a phenomenological model for QCD. An understanding of such effects is important in order to interpret results from lattice QCD and extrapolate reliably from finite lattice volumes to infinite volume. We consider the quark-meson-model in a finite Euclidean 3+1 dimensional volume. In order to break chiral symmetry in the finite volume, we introduce a small current quark mass. In the corresponding effective potential for the meson fields, the chiral O(4)-sy…
We present a study of the finite-volume two-pion matrix elements and correlation functions of the I=0 scalar operator, in full and partially quenched QCD, at one-loop order in chiral perturbation theory. In partially quenched QCD, when the sea and valence light quark masses are not equal, the lack of unitarity leads to the same inconsistencies as in quenched QCD and the matrix elements cannot be determined. It is possible, however, to overcome this problem by requiring the masses of the valence and sea quarks to be equal for the u and d quarks while keeping the strange quark (s) quenched (or p…
The elastic $I=1$ $p$-wave $ππ$ scattering amplitude is calculated together with the isovector timelike pion form factor using lattice QCD with $N_{\rm f}=2+1$ dynamical quark flavors. Wilson clover ensembles generated by the Coordinated Lattice Simulations (CLS) initiative are employed at four lattice spacings down to $a = 0.05\,\mathrm{fm}$, several pion masses down to $m_π = 200\,\mathrm{MeV}$, and spatial volumes of extent $L = 3.1-5.5\,\mathrm{fm}$. The set of measurements on these ensembles, which is publicly available, enables an investigation of systematic errors due to the finite latt…
The quantum mechanical formula for Mayer's second cluster integral for the gas of relativistic particles with hard-core interaction is derived. The proper pion volume calculated with quantum mechanical formula is found to be an order of magnitude larger than its classical evaluation. The second cluster integral for the pion gas is calculated in quantum mechanical approach with account for both attractive and hard-core repulsive interactions. It is shown that, in the second cluster approximation, the repulsive pion-pion-interactions as well as the finite width of resonances give important but a…
We discuss preliminary results for the vector form factors $f_+^{\{π,K\}}$ at zero-momentum transfer for the decays $D\toπ\ellν$ and $D\to K \ellν$ using MILC's $N_f = 2+1+1$ HISQ ensembles at four lattice spacings, $a \approx 0.042, 0.06, 0.09$, and 0.12 fm, and various HISQ quark masses down to the (degenerate) physical light quark mass. We use the kinematic constraint $f_+(q^2)= f_0(q^2)$ at $q^2 = 0$ to determine the vector form factor from our study of the scalar current, which yields $f_0(0)$. Results are extrapolated to the continuum physical point in the framework of hard pion/kaon SU(…
