Calculation Of Form Factors Research Articles

Update on |Vus| and |Vus/Vud| from semileptonic kaon and pion decays

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 $3.2\sim 5.6\sigma$ level, and a discrepancy at a $2.2\sigma$ level between the value of $|V_{us}/V_{ud}|$ determined from the vector and axial charged weak interactions. Prospects for future improvements in those comparative precision tests involving $|V_{ud}|$, $|V_{us}|$ and their implications for physics beyond the Standard Model are described.

Geometrical Analysis of Tooth Form Factor and Stress Correction Factor In Calculating Tooth Root Bending Stress on Internal Gear

One of the most critical disadvantages of plastics must be much lower strength than steel, which prevents plastic gears from having high load capacity. Internal gears have larger tooth root thickness and smaller tooth flank sliding than external ones. In addition, they have a concave tooth profile. These facts indicate that even plastic gears could realize relatively high load capacity. Although tooth root bending stresses greatly depend on fillet profile, failure of internal gears is generally not crucial in steel gear pairs, which has provoked few discussions on fillet profile. The present paper describes fillet profiles of internal gears, which the top land of the pinion cutter generates. Geometrical analyses defined the change in the profiles associated with the number of teeth and profile shift coefficient of pinion cutter with the possible maximum radius of top land. According to ISO 6336-3, which is the basic standard on the load capacity of gear pair, calculations of tooth form factors and stress correction factors determined the positions of the critical section for evaluating tooth root bending stress.

Lattice calculation of pion form factors with overlap fermions

We present a precise calculation of the pion form factor using overlap fermions on seven ensembles of 2+1-flavor domain-wall configurations with pion masses varying from 139 to 340 MeV. Taking advantage of the fast Fourier transform and other techniques to access many combinations of source and sink momenta, we find the pion mean square charge radius to be $\langle {r_\pi^2} \rangle= 0.430(5)(13)\ {\rm{fm^2}}$, which agrees well with the experimental result, and includes the systematic uncertainties from chiral extrapolation, lattice spacing and finite-volume dependence. We also find that $\langle {r_\pi^2} \rangle$ depends on both the valence and sea quark masses strongly and predict the pion form factor up to $Q^2 = 1.0 \ {\rm{GeV^2}}$ which agrees with experiments very well.

Improved Vcs determination using precise lattice QCD form factors for D→Kℓν

We provide a 0.8%-accurate determination of ${V}_{cs}$ from combining experimental results for the differential rate of $D\ensuremath{\rightarrow}K$ semileptonic decays with precise form factors that we determine from lattice QCD. This is the first time that ${V}_{cs}$ has been determined with an accuracy that allows its difference from 1 to be seen. Our lattice QCD calculation uses the highly improved staggered quark (HISQ) action for all valence quarks on gluon field configurations generated by the MILC Collaboration that include the effect of $u$, $d$, $s$, and $c$ HISQ quarks in the sea. We use eight gluon field ensembles with five values of the lattice spacing ranging from 0.15 fm to 0.045 fm and include results with physical $u/d$ quarks for the first time. Our calculated form factors cover the full ${q}^{2}$ range of the physical decay process and enable a Standard Model test of the shape of the differential decay rate as well as the determination of ${V}_{cs}$ from a correlated weighted average over ${q}^{2}$ bins. We obtain $|{V}_{cs}|=0.9663(53{)}_{\mathrm{latt}}(39{)}_{\mathrm{exp}}(19{)}_{{\ensuremath{\eta}}_{\mathrm{EW}}}(40{)}_{\mathrm{EM}}$, where the uncertainties come from lattice QCD, experiment, short-distance electroweak, and electromagnetic corrections, respectively. This last uncertainty, neglected for $D\ensuremath{\rightarrow}K\ensuremath{\ell}\ensuremath{\nu}$ hitherto, now needs attention if the uncertainty on ${V}_{cs}$ is to be reduced further. We also determine ${V}_{cs}$ values in good agreement using the measured total branching fraction and the rates extrapolated to ${q}^{2}=0$. Our form factors enable tests of lepton flavor universality violation. We find the ratio of branching fractions for ${D}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}$ with $\ensuremath{\mu}$ and $e$ in the final state to be ${R}_{\ensuremath{\mu}/e}=0.9779(2{)}_{\mathrm{latt}}(50{)}_{\mathrm{EM}}$ in the Standard Model, with the uncertainty dominated by that from electromagnetic corrections.

Semileptonic decay of Bc to ηc and J/ψ on the light front

We study the semileptonic decay of the $B_c (0^-)$ to charmonia through the bottom-to-charm-quark electroweak current in the framework of basis light-front quantization. Explicitly, we calculate the weak transition form factors for processes of $B_c$ decaying into $\eta_c$ or $J/\psi$ based on the corresponding initial and final valence light-front wave functions both obtained from the basis light-front quantization. We also present the corresponding differential decay width and branching ratios, as well as the branching ratios for decays into the $\eta_c'$ and the $\psi'$. We observe unphysical frame dependence of the calculated form factors, which is attributed to only including the valence light-front wave functions. Based on the analysis of current component, we propose a preferred set of frames to calculate these form factors.

Nπ -state contamination in lattice calculations of the nucleon electromagnetic form factors

The nucleon-pion-state contribution to QCD two-point and three-point functions relevant for lattice calculations of the nucleon electromagnetic form factors are studied in chiral perturbation theory. To leading order the results depend on a few experimentally known low-energy constants only, and the nucleon-pion-state contribution to the form factors can be estimated. The nucleon-pion-state contribution to the electric form factor $G_{\rm E}(Q^2)$ is at the +5 percent level for a source-sink separation of 2 fm, and it increases with increasing momentum transfer $Q^2$. For the magnetic form factor the nucleon-pion-state contribution leads to an underestimation of $G_{\rm M}(Q^2)$ by about 5 percent that decreases with increasing $Q^2$. For smaller source-sink separations that are accessible in present-day lattice simulations the impact is larger, although the ChPT results may not be applicable for such small time separations. Still, a comparison with lattice data at $t\approx 1.6$ fm works reasonably well.

Branch point twist field form factors in the sine-Gordon model I: Breather fusion and entanglement dynamics

The quantum sine-Gordon model is the simplest massive interacting integrable quantum field theory whose two-particle scattering matrix is generally non-diagonal. As such, it is a model that has been extensively studied, especially in the context of the bootstrap program. In this paper we compute low particle-number form factors of a special local field known as the branch point twist field, whose correlation functions are building blocks for measures of entanglement. We consider the attractive regime where the theory possesses a particle spectrum consisting of a soliton, an antisoliton (of opposite U(1) charges) and several (neutral) breathers. In the breather sector we exploit the fusion procedure to compute form factors of heavier breathers from those of lighter ones. We apply our results to the study of the entanglement dynamics after a small mass quench and for short times. We show that in the presence of two or more breathers the von Neumann and Rényi entropies display undamped oscillations in time, whose frequencies are proportional to the even breather masses and whose amplitudes are proportional to the breather's one-particle form factor.

The quantum sine-Gordon model with quantum circuits

Analog quantum simulation has the potential to be an indispensable technique in the investigation of complex quantum systems. In this work, we numerically investigate a one-dimensional, faithful, analog, quantum electronic circuit simulator built out of Josephson junctions for one of the paradigmatic models of an integrable quantum field theory: the quantum sine-Gordon (qSG) model in 1+1 space-time dimensions. We analyze the lattice model using the density matrix renormalization group technique and benchmark our numerical results with existing Bethe ansatz computations. Furthermore, we perform analytical form-factor calculations for the two-point correlation function of vertex operators, which closely agree with our numerical computations. Finally, we compute the entanglement spectrum of the qSG model. We compare our results with those obtained using the integrable lattice-regularization based on the quantum XYZ chain and show that the quantum circuit model is less susceptible to corrections to scaling compared to the XYZ chain. We provide numerical evidence that the parameters required to realize the qSG model are accessible with modern-day superconducting circuit technology, thus providing additional credence towards the viability of the latter platform for simulating strongly interacting quantum field theories.

Toward accurate form factors for B -to-light meson decay from lattice QCD

We present the results of a lattice QCD calculation of the scalar and vector form factors for the unphysical $B_s\to\eta_s$ decay, over the full physical range of $q^2$. This is a useful testing ground both for lattice QCD and for our wider understanding of the behaviour of form factors. Calculations were performed using the highly improved staggered quark (HISQ) action on $N_f = 2 + 1 + 1$ gluon ensembles generated by the MILC Collaboration with an improved gluon action and HISQ sea quarks. We use three lattice spacings and a range of heavy quark masses from that of charm to bottom, all in the HISQ formalism. This permits an extrapolation in the heavy quark mass and lattice spacing to the physical point and nonperturbative renormalisation of the vector matrix element on the lattice. We find results in good agreement with previous work using nonrelativistic QCD $b$ quarks and with reduced errors at low $q^2$, supporting the effectiveness of our heavy HISQ technique as a method for calculating form factors involving heavy quarks. A comparison with results for other decays related by SU(3) flavour symmetry shows that the impact of changing the light daughter quark is substantial but changing the spectator quark has very little effect. We also map out form factor shape parameters as a function of heavy quark mass and compare to heavy quark effective theory expectations for mass scaling at low and high recoil. This work represents an important step in the progression from previous work on heavy-to-heavy decays ($b\to c$) to the numerically more challenging heavy-to-light decays.

Revisiting fits to B0→D*−ℓ+νℓ to measure |Vcb| with novel methods and preliminary LQCD data at nonzero recoil

We present a study of fits to exclusive ${B}^{0}\ensuremath{\rightarrow}{D}^{*\ensuremath{-}}{\ensuremath{\ell}}^{+}{\ensuremath{\nu}}_{\ensuremath{\ell}}$ measurements for the determination of the Cabibbo-Kobayashi-Maskawa matrix element magnitude $|{V}_{cb}|$, based on the most recent Belle untagged measurement. Results are obtained with the Caprini-Lellouch-Neubert (CLN) and Boyd-Grinstein-Lebed (BGL) form factor parametrizations, with and without the inclusion of preliminary lattice QCD (LQCD) measurements of form factors at nonzero hadronic recoil from the JLQCD Collaboration. The CLN and BGL fits are also studied in different scenarios with reduced theoretical assumptions and at higher-order expansions, respectively. To avoid bias from high systematic error correlations, we employ a novel technique in the field of $B$-physics phenomenology with a toy Monte Carlo method using a Cholesky decomposition of the covariance matrix. Using additional input from lattice QCD calculations of form factors at nonzero recoil, in collaboration with JLQCD, allows for well-defined fit results with reduced model dependence in CLN and BGL. The results obtained are consistent between different configurations, ultimately providing a method for a more model-independent exclusive measurement of $|{V}_{cb}|$. Using preliminary LQCD inputs and assuming a value of $\mathcal{F}(1)=0.904\ifmmode\pm\else\textpm\fi{}0.012$, we find $|{V}_{cb}|=(38.48\ifmmode\pm\else\textpm\fi{}0.32\ifmmode\pm\else\textpm\fi{}0.97\ifmmode\pm\else\textpm\fi{}0.46)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ in BGL(2,2,2) and $(38.42\ifmmode\pm\else\textpm\fi{}0.35\ifmmode\pm\else\textpm\fi{}1.05\ifmmode\pm\else\textpm\fi{}0.46)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ in CLNnoHQS, where the errors are statistical, systematic, and related to $\mathcal{F}(1)$, respectively.

Electromagnetic form factors of the Ω− baryon in the spacelike and timelike regions

We present complete calculations of the electromagnetic form factors of the $\Omega^-$ in the spacelike region and in the timelike region. The four elastic form factors: electric charge ($G_{E0}$), magnetic dipole ($G_{M1}$), electric quadrupole ($G_{E2}$) and magnetic octupole ($G_{M3}$), are estimated within the covariant spectator quark model, in terms of the square momentum transfer $q^2$. The free parameters of the $\Omega^-$ wave function, including a $S$-wave state and two independent $D$-wave states radial wave functions and the admixture coefficients are fixed by the comparison with the lattice QCD data in the spacelike region ($Q^2=-q^2 \le 0$) and with the recent $e^+ e^- \to \Omega^- \bar \Omega^+$ data from CLEO in the timelike region ($q^2 > 0$). The estimates in the timelike region for square momentum transfer $q^2 \ge 4 M_\Omega^2$ are based on large-$q^2$ asymptotic relations ($M_\Omega$ is the $\Omega^-$ mass). We examine also the impact of the large-$Q^2$ correlations between different form factors and analyze the possible solutions. The electric quadrupole and the magnetic octupole moments of the $\Omega^-$, and the $e^+ e^- \to \Omega^- \bar \Omega^+$ integrated cross sections for very large $q^2$ are estimated based on the model results.

Heavy Tetraquarks in the Relativistic Quark Model

We give a review of the calculations of the masses of tetraquarks with two and four heavy quarks in the framework of the relativistic quark model based on the quasipotential approach and QCD. The diquark-antidiquark picture of heavy tetraquarks is used. The quasipotentials of the quark-quark and diquark-antidiquark interactions are constructed similarly to the previous consideration of mesons and baryons. Diquarks are considered in the colour triplet state. It is assumed that the diquark and antidiquark interact in the tetraquark as a whole and the internal structure of the diquarks is taken into account by the calculated form factor of the diquark-gluon interaction. All parameters of the model are kept fixed from our previous calculations of meson and baryon properties. A detailed comparison of the obtained predictions for heavy tetraquark masses with available experimental data is given. Many candidates for tetraquarks are found. It is argued that the structures in the di-J/ψ mass spectrum observed recently by the LHCb collaboration can be interpreted as ccc¯c¯ tetraquarks.

Confirming the existence of the strong CP problem in lattice QCD with the gradient flow

We calculate the electric dipole moment of the nucleon induced by the QCD theta term. We use the gradient flow to define the topological charge and use $N_f = 2+1$ flavors of dynamical quarks corresponding to pion masses of $700$, $570$, and $410$ MeV, and perform an extrapolation to the physical point based on chiral perturbation theory. We perform calculations at $3$ different lattice spacings in the range of $0.07~{\rm fm} < a < 0.11$ fm at a single value of the pion mass, to enable control on discretization effects. We also investigate finite size effects using $2$ different volumes. A novel technique is applied to improve the signal-to-noise ratio in the form factor calculations. The very mild discretization effects observed suggest a continuum-like behavior of the nucleon EDM towards the chiral limit. Under this assumption our results read $d_{n}=-0.00152(71)\ \bar\theta\ e~\text{fm}$ and $d_{p}=0.0011(10)\ \bar\theta\ e~\text{fm}$. Assuming the theta term is the only source of CP violation, the experimental bound on the neutron electric dipole moment limits $\left|\bar\theta\right| < 1.98\times 10^{-10}$ ($90\%$ CL). A first attempt at calculating the nucleon Schiff moment in the continuum resulted in $S_{p} = 0.50(59)\times 10^{-4}\ \bar\theta\ e~\text{fm}^3$ and $S_{n} = -0.10(43)\times 10^{-4}\ \bar\theta\ e~\text{fm}^3$.

Nuclear shape transitions and elastic magnetic electron scattering

Backward elastic electron scattering from odd-$A$ nuclear targets is characterized by magnetic form factors containing precise information on the nuclear structure. We study the sensitivity of the magnetic form factors to structural effects related to the evolution and shape transitions in both isotopic and isotonic chains. Calculations of magnetic form factors are performed in the plane-wave Born approximation. The nuclear structure is obtained from a deformed self-consistent mean-field calculation based on a Skyrme $\mathrm{HF}+\mathrm{BCS}$ formalism. Collective effects are included in the cranking approximation, whereas nucleon-nucleon correlations are taken into account in the coherent density fluctuation model. The evolution of the magnetic form factors is found to exhibit signatures of shape transitions that show up in selected isotopic and isotonic chains involving both stable and unstable nuclei. Several cases are identified as suitable candidates for showing such fingerprints of shape transitions. A new generation of electron scattering experiments involving electron-radioactive beam colliders will be available in the near future, leading to a renewed interest in this field.

Halo EFT calculation of charge form factor for two-neutron $${}^{6}\text {He}$$ halo nucleus: two-body resonant P-wave interaction

We take a new look at $^{6}\textrm{He}$ halo nucleus and set up a halo effective field theory at low energies to calculate the charge form factor of ${}^{6}\textrm{He}$ system with resonant P-wave interaction. P-wave Lagrangian has been introduced and the charge form factor of ${}^{6}\textrm{He}$ halo nucleus has been obtained at Leading-Order. In this study, the mean-square charge radius of ${}^{6}\textrm{He}$ nucleus relative to ${}^{4}\textrm{He}$ core and the root-mean-square (r.m.s) charge radius of ${}^{6}\textrm{He}$ nucleus have been estimated as $\langle r_{E}^{2}\rangle=1.408\; \textrm{fm}^{2}$ and $\langle r_{E}^{2}\rangle_{{}^{6}He}^{\frac{1}{2}}=2.058 \;\textrm{fm}$, respectively. We have compared our results with the other available theoretical and experimental data.

The Parton Distribution Functions and Calculation of Dirac Neutron Form Factor Considering the EMC and Shadowing Effects

Due to the absence of the free neutron target, the form factors or the structure function of the neutron directly cannot be obtained from the elastic scattering data. Therefore, in this paper, we obtained the neutron structure function using the deep inelastic structure function of the proton and by applying the EMC and the shadowing effects on the structure function of the deuteron. Then, using the quantum chromo dynamics theorem and re-analysis of data from the SLAC experiments E49, E61, E87, and E89 in electron–proton deep inelastic scattering at Q2 = 1–20 GeV2, we have extracted the up and down quark distribution functions, $${{u}_{{v}}}$$ (x), $${{d}_{{v}}}$$ (x), and also, using an appropriate model for the general parton distribution function, we have plotted the Dirac form factor of the neutron. At the end, the curves obtained in this work were compared with the curves obtained in other models.

CP violation in neutral lepton transition dipole moment

The CP violation in the neutrino transition electromagnetic dipole moment is discussed in the context of the Standard Model with an arbitrary number of right-handed singlet neutrinos. A full one-loop calculation of the neutrino electromagnetic form factors is performed in the Feynman gauge. A non-zero CP asymmetry is generated by a required threshold condition for the neutrino masses along with non-vanishing CP violating phases in the lepton flavour mixing matrix. We follow the paradiagm of CP violation in neutrino oscillations to parametrise the flavour mixing contribution into a series of Jarlskog-like parameters. This formalism is then applied to a minimal seesaw model with two heavy right-handed neutrinos denoted N1 and N2. We observe that the CP asymmetries for decays into light neutrinos N → νγ are extremely suppressed, maximally around 10−17. However the CP asymmetry for N2→ N1γ can reach of order unity. Even if the Dirac CP phase δ is the only source of CP violation, a large CP asymmetry around 10−5–10−3 is comfortably achieved.

Heavy baryon wave functions, Bakamjian-Thomas approach to form factors, and observables in Λb→Λc(12±)ℓν¯ transitions

Motivated by the calculation of observables in the decays $\Lambda_b \to \Lambda_c\left({1 \over 2}^\pm \right) \ell \overline{\nu}$, as tests of Lepton Flavor Universality, we present a calculation of form factors in the quark model. Our scheme combines a spectroscopic model, providing the internal wave functions, and the Bakamjian-Thomas relativistic formalism to deduce the wave functions in motion and current matrix elements, that amount in the heavy quark limit to the Isgur-Wise (IW) function. For baryons we meet difficulties using standard spectroscopic models, leading us to propose a simple phenomenological model : a Q-pointlike-diquark model, non-relativistic with harmonic oscillator forces, with a reasonable low-lying spectrum and good slope of the IW function. We extract this slope from Lattice data and find $\rho_\Lambda^2 \sim 2$. We are not able to reproduce the right $\rho_\Lambda^2$ when using standard linear + Coulomb potential models, both with three quarks $Qqq$ or in a Q-pointlike-diquark picture. These difficulties seem to derive from the high sensitivity of $\rho_\Lambda^2$ to the structure of the light quark subsystem. After fixing the parameters of our interim model to yield correct spectrum and $\rho_\Lambda^2$, we compute observables. Bjorken sum rule shows that the inelastic IW function is large, and therefore $\Lambda_b \to \Lambda_c \left({1 \over 2}^-, {3 \over 2}^- \right) \ell \overline{\nu}$ could be studied at LHCb. Some observables in the $\tau$ case present zeroes for specific values of $q^2$ that could be tests of the Standard Model.

Renormalization of the tensor current in lattice QCD and the J/ψ tensor decay constant

Lattice QCD calculations of form factors for rare Standard Model processes such as $B \to K \ell^+ \ell^-$ use tensor currents that require renormalisation. These renormalisation factors, $Z_T$, have typically been calculated within perturbation theory and the estimated uncertainties from missing higher order terms are significant. Here we study tensor current renormalisation using lattice implementations of momentum-subtraction schemes. Such schemes are potentially more accurate but have systematic errors from nonperturbative artefacts. To determine and remove these condensate contributions we calculate the ground-state charmonium tensor decay constant, $f_{J/\psi}^T$, which is also of interest in beyond the Standard Model studies. We obtain $f_{J/\psi}^T(\bar{\text{MS}}, 2\ \mathrm{GeV})=0.3927(27)$ GeV, with ratio to the vector decay constant of 0.9569(52), significantly below 1. We also give $Z_T$ factors, converted to the $\bar{\mathrm{MS}}$ scheme, corrected for condensate contamination. This contamination reaches 1.5\% at a renormalisation scale of 2 GeV (in the preferred RI-SMOM scheme) and so must be removed for accurate results.

Form factors with two operator insertions and the principle of maximal transcendentality

We present the first calculations of two-point two-loop form factors (FFs) with a two identical operators insertion in maximally supersymmetric Yang-Mills theory. In this article, we consider the supersymmetry protected half-BPS primary and unprotected Konishi operators. Unlike the FFs of a single operator insertion of the half-BPS primary, the FFs involving two half-BPS operators are found to contain lower transcendentality weight terms in addition to the highest ones. Moreover, in contrast to Sudakov FFs, the highest weight terms of the FFs of a double half-BPS no longer match with that of a double Konishi. We also find that the principle of maximal transcendentality, which dictates the presence of identical highest weight terms in the scalar FFs of half-BPS and quark/gluon FFs in QCD, does not hold true anymore for insertions of two identical operators. We discover the absence of any additional ultraviolet counterterm that could arise from the contact interaction between two composite operators.