Physical Pion Research Articles

Glueballs at physical pion mass* *Supported by the Strategic Priority Research Program of Chinese Academy of Sciences (XDB34030300, XDPB15), the National Key Research and Development Program of China (2020YFA0406400), and the National Natural Science Foundation of China (NNSFC) under Grants ( 11935017, 12070131001) (CRC 110 by DFG and NNSFC). The computations were performed on the CAS ORISE computing environment, the

Glueballs are investigated through gluonic operators on two RBC/UKQCD gauge ensembles at the physical pion mass. The statistical errors of glueball correlation functions are considerably reduced through the cluster decomposition error reduction (CDER) method. The Bethe-Salpeter wave functions are obtained for the scalar, tensor, and pseudoscalar glueballs by using spatially extended glueball operators defined through the gauge potential in the Coulomb gauge. These wave functions exhibit similar features of non-relativistic two-gluon systems and are used to optimize the signals of the related correlation functions at the early time regions, where the ground state masses are extracted. These masses are close to those from the quenched approximation and indicate the possible existence of glueballs at the physical point. The resonance feature of glueballs and the mixing with conventional mesons and multi-hadron systems should be considered in a more systematic lattice study.

Overlap quark propagator near the physical pion mass

The Landau-gauge quark propagator is calculated using overlap valence fermions on 2+1-flavour dynamical clover fermion gauge fields from the PACS-CS collaboration with pion mass mπ∼156 MeV and spatial volume ∼(3 fm)3. The observed features of the mass and renormalisation functions are discussed, including a comparison with recent results using O(a)-improved Wilson fermions on 2-flavour dynamical gauge fields.

Muon g-2 with overlap valence fermions

We present a lattice calculation of the leading order (LO) hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment for the connected light and strange quarks, $a^{\rm W}_{{\rm con}, l/s}$ in the widely used window $t_0=0.4~\mathrm{fm}$, $t_1=1.0~\mathrm{fm}$, $\Delta=0.15~\mathrm{fm}$, and also of $a^{\rm S}_{{\rm con}, l/s}$ in the short distance region. We use overlap fermions on 4 physical-point ensembles. Two 2+1 flavor RBC/UKQCD ensembles use domain wall fermions (DWF) and Iwasaki gauge actions at $a = 0.084$ and 0.114 fm, and two 2+1+1 flavor MILC ensembles use the highly improved staggered quark (HISQ) and Symanzik gauge actions at $a = 0.088$ and 0.121 fm. We have incorporated infinite volume corrections from 3 additional DWF ensembles at ${\rm L}$ = 4.8, 6.4 and 9.6 fm and physical pion mass. For $a^{\rm W}_{{\rm con}, l}$, we find that our results on the two smaller lattice spacings are consistent with those using the unitary setup, but those at the two coarser lattice spacings are slightly different. Eventually, we predict $a^{\rm W}_{{\rm con}, l}=206.7(1.5)(1.0)$ and $a^{\rm W}_{{\rm con}, s}=26.8(0.1)(0.3)$, using linear extrapolation in $a^2$, with systematic uncertainties estimated from the difference of the central values from the RBC/UKQCD and MILC ensembles.

Speeding up lattice QCD simulation on Arm architectures with multigrid algorithm

Lattice quantum chromodynamics (lattice QCD) is the non-perturbative definition of the QCD theory from first principle and can be systematically improved. Meanwhile, it is one of the most important high performance computing application in high energy physics. The physics research of lattice QCD benefited enormously from the development of computer hardware and algorithms, and particle physicists can now study strong interaction at physical pion mass, smaller lattice spacing and larger volume. On the other hand, with the emerging of supercomputers and clusters based on Arm architectures, it is necessary to implement the algorithms and conduct the simulations on these machines to maximize the utilization of the limited computing resources. In this paper, we implemented and ported the multigrid (MG) algorithm, the state of the art algorithm to solve the sparse linear Dirac equation Dx = b on the lattice for clover fermion on a cluster with Arm based processors.

Lattice QCD calculation of the light sterile neutrino contribution in 0ν2β decay

We present a lattice QCD study of the neutrinoless double beta decay involving light sterile neutrinos. The calculation is performed at physical pion mass using five gauge ensembles generated with the $2+1$-flavor domain wall fermions. We obtain the low-energy constants $g_{\text{LR}}^{\pi\pi}(m_\nu)$ with the neutrino mass $m_\nu$ from $0$ GeV to $3$ GeV. The lattice results are reasonably consistent with the previous interpolation method with a $\sim20\%$ deviation at small $m_\nu$. We provide an explanation on the discrepancy at vanishing neutrino mass. At large $m_\nu$, a good consistence between our results and the previous lattice determination of $g_{4}^{\pi\pi}(\mu)$ is found at $\mu=m_\nu=3$ GeV.

Pion and Kaon Distribution Amplitudes from Lattice QCD.

We present a state-of-the-art lattice QCD calculation of the pion and kaon light-cone distribution amplitudes (DAs) using large-momentum effective theory. The calculation is done at three lattice spacings a≈{0.06,0.09,0.12} fm and physical pion and kaon masses, with the meson momenta P_{z}={1.29,1.72,2.15} GeV. The result is nonperturbatively renormalized in a recently proposed hybrid scheme with self-renormalization, and extrapolated reliably to the continuum as well as the infinite momentum limit. We find a significant deviation of the pion and kaon DAs from the asymptotic form, and a large SU(3) flavor breaking effect in the kaon DA.

Search for b¯b¯us and b¯c¯ud tetraquark bound states using lattice QCD

We use lattice QCD to investigate the existence of strong-interaction-stable antiheavy-antiheavy-light-light tetraquarks. We study the $\overline{b}\overline{b}us$ system with quantum numbers ${J}^{P}={1}^{+}$ as well as the $\overline{b}\overline{c}ud$ systems with quantum numbers $I({J}^{P})=0({0}^{+})$ and $I({J}^{P})=0({1}^{+})$. We carry out computations on five gauge-link ensembles with $2+1$ flavors of domain-wall fermions, including one at the physical pion mass. The bottom quarks are implemented using lattice nonrelativistic QCD, and the charm quarks using an anisotropic clover action. In addition to local diquark-antidiquark and local meson-meson interpolating operators, we include nonlocal meson-meson operators at the sink, which facilitates the reliable determination of the low-lying energy levels. We find clear evidence for the existence of a strong-interaction-stable $\overline{b}\overline{b}us$ tetraquark with binding energy $(\ensuremath{-}86\ifmmode\pm\else\textpm\fi{}22\ifmmode\pm\else\textpm\fi{}10)\text{ }\text{ }\mathrm{MeV}$ and mass $(10609\ifmmode\pm\else\textpm\fi{}22\ifmmode\pm\else\textpm\fi{}10)\text{ }\text{ }\mathrm{MeV}$. For the $\overline{b}\overline{c}ud$ systems we do not find any indication for the existence of bound states, but cannot rule out their existence either.

Proton momentum and angular momentum decompositions with overlap fermions

We present a calculation of the proton momentum and angular momentum decompositions using overlap fermions on a $2+1$-flavor RBC/UKQCD domain-wall lattice at 0.143 fm with a pion mass of 171 MeV which is close to the physical one. A complete determination of the momentum and angular momentum fractions carried by up, down, strange, and glue inside the proton has been done with valence pion masses varying from 171 to 391 MeV. We have utilized fast Fourier transform on the stochastic-sandwich method for connected-insertion parts and the cluster-decomposition error reduction technique technique for disconnected-insertion parts has been used to reduce statistical errors. The full nonperturbative renormalization and mixing between the quark and glue operators are carried out. The final results are normalized with the momentum and angular momentum sum rules and reported at the physical valence pion mass at $\overline{\mathrm{MS}}(\ensuremath{\mu}=2\text{ }\text{ }\mathrm{GeV})$. The renormalized momentum fractions for the quarks and glue are $⟨x{⟩}^{q}=0.491(20)(23)$ and $⟨x{⟩}^{g}=0.509(20)(23)$, respectively, and the renormalized total angular momentum fractions for quarks and glue are $2{J}^{q}=0.539(22)(44)$ and $2{J}^{g}=0.461(22)(44)$, respectively. The quark spin fraction is $\mathrm{\ensuremath{\Sigma}}=0.405(25)(37)$ from our previous work and the quark orbital angular momentum fraction is deduced from $2{L}^{q}=2{J}^{q}\ensuremath{-}\mathrm{\ensuremath{\Sigma}}$ to be 0.134(22)(44).

Optimized two-baryon operators in lattice QCD

A set of optimized interpolating operators which are dominantly coupled to each eigenstate of two baryons on the lattice is constructed by the HAL QCD method. To test its validity, we consider heavy dibaryons ${\mathrm{\ensuremath{\Omega}}}_{3Q}{\mathrm{\ensuremath{\Omega}}}_{3Q}$ ($Q=s$, $c$) calculated by ($2+1$)-flavor lattice QCD simulations with nearly physical pion mass. The optimized two-baryon operators are shown to provide effective energies of the ground and excited states, separately stable as a function of the Euclidean time. Also they agree with the eigenenergies in a finite lattice box obtained from the leading-order HAL QCD potential $V(\mathbit{r})$ within statistical errors. The overlapping factors between the optimized sink operators and the state created by the wall-type source operator indicate that $V(\mathbit{r})$ can be reliably extracted, no matter whether the spacetime correlation of two baryons is dominated by the ground state or the excited state. It is suggested that the optimized set of operators is useful for variational studies of hadron-hadron interactions.

Proton decay matrix elements on the lattice at physical pion mass

Proton decay is a major prediction of grand-unified theories (GUT) and its observation would indicate baryon number violation that is required for baryogenesis. Many decades of searching for proton decay have constrained its rate and ruled out some of the simplest GUT models. Apart from the baryon number-violating interactions, this rate also depends on transition amplitudes between the protons and mesons or leptons produced in the decay, which are matrix elements of three-quark operators. We report the nonperturbative calculation of these matrix elements for the most studied two-body decay channels into a meson and antilepton done on a lattice with physical light and strange quark masses and lattice spacings $a\ensuremath{\approx}0.14\text{ }\text{ }\mathrm{fm}$ and 0.20 fm. We perform nonperturbative renormalization and excited state analysis to control associated systematic effects. Our results largely agree with previous lattice calculations done with heavier quark masses and thus remove ambiguity in ruling out some simple GUT theories due to quark mass dependence of hadron structure.

Lattice QCD Calculation of the Pion Mass Splitting.

We use the infinite volume reconstruction method to calculate the charged and neutral pion mass difference. The hadronic tensor is calculated using lattice QCD and then combined with an analytic photon propagator, and the mass splitting is calculated with exponentially suppressed finite-volume errors. The calculation is performed using six gauge ensembles generated with 2+1-flavor domain wall fermions, and five ensembles are at the physical pion mass. Both Feynman and Coulomb gauges are adopted in the calculation and agree well when the lattice spacing approaches zero. After performing the continuum extrapolation and examining the residual finite-volume effects, we obtain the pion mass splitting Δm_{π}=4.534(42)_{stat}(43)_{sys} MeV, which agrees well with experimental measurements.

Pion form factor and charge radius from lattice QCD at the physical point

We present our results on the electromagnetic form factor of pion over a wide range of $Q^2$ using lattice QCD simulations with Wilson-clover valence quarks and HISQ sea quarks. We study the form factor at the physical point with a lattice spacing $a=0.076$ fm. To study the lattice spacing and quark mass effects, we also present results for 300 MeV pion at two different lattice spacings $a=0.04$ and 0.06 fm. The lattice calculations at the physical quark mass appear to agree with the experimental results. Through fits to the form factor, we estimate the charge radius of pion for physical pion mass to be $\langle r_{\pi}^2 \rangle=0.42(2)~{\rm fm}^2$.

Lattice determination of I=0 and 2 ππ scattering phase shifts with a physical pion mass

Phase shifts for $s$-wave $\pi\pi$ scattering in both the $I=0$ and $I=2$ channels are determined from a lattice QCD calculation performed on 741 gauge configurations obeying G-parity boundary conditions with a physical pion mass and lattice size of $32^3\times 64$. These results support our recent study of direct CP violation in $K\to\pi\pi$ decay \cite{Abbott:2020hxn}, improving our earlier 2015 calculation \cite{Bai:2015nea}. The phase shifts are determined for both stationary and moving $\pi\pi$ systems, at three ($I=0$) and four ($I=2$) different total momenta. We implement several $\pi\pi$ interpolating operators including a scalar bilinear "$\sigma$" operator and paired single-pion bilinear operators with the constituent pions carrying various relative momenta. Several techniques, including correlated fitting and a bootstrap determination of p-values have been used to refine the results and a comparison with the generalized eigenvalue problem (GEVP) method is given. A detailed systematic error analysis is performed which allows phase shift results to be presented at a fixed energy.

Quark and Gluon Momentum Fractions in the Pion from N_{f}=2+1+1 Lattice QCD.

We perform the first full decomposition of the pion momentum into its gluon and quark contributions. We employ an ensemble generated by the Extended Twisted Mass Collaboration with N_{f}=2+1+1 Wilson twisted mass clover fermions at maximal twist tuned to reproduce the physical pion mass. We present our results in the MS[over ¯] scheme at 2GeV. We find ⟨x⟩_{u+d}=0.601(28), ⟨x⟩_{s}=0.059(13), ⟨x⟩_{c}=0.019(05), and ⟨x⟩_{g}=0.52(11) for the separate contributions, respectively, whose sum saturates the momentum sum rule.

Pion-Nucleon Sigma Term from Lattice QCD.

We present an analysis of the pion-nucleon σ-term σ_{πN} using six ensembles with 2+1+1-flavor highly improved staggered quark action generated by the MILC Collaboration. The most serious systematic effect in lattice calculations of nucleon correlation functions is the contribution of excited states. We estimate these using chiral perturbation theory (χPT) and show that the leading contribution to the isoscalar scalar charge comes from Nπ and Nππ states. Therefore, we carry out two analyses of lattice data to remove excited-state contamination, the standard one and a new one including Nπ and Nππ states. We find that the standard analysis gives σ_{πN}=41.9(4.9) MeV, consistent with previous lattice calculations, while our preferred χPT-motivated analysis gives σ_{πN}=59.6(7.4) MeV, which is consistent with phenomenological values obtained using πN scattering data. Our data on one physical pion mass ensemble were crucial for exposing this difference, therefore, calculations on additional physical mass ensembles are needed to confirm our result and resolve the tension between lattice QCD and phenomenology.

Nucleon helicity generalized parton distribution at physical pion mass from lattice QCD

The generalized parton distributions (GPDs) offer a window on three-dimensional imaging of the nucleon, providing understanding of how the fundamental properties of the nucleon, such as its mass and spin, arise from the underlying quark and gluon degrees of freedom. In this work, we present the first lattice calculation of the nucleon isovector helicity GPD at physical pion mass, using an a≈0.09 fm lattice ensemble with 2+1+1 flavors of highly improved staggered quarks generated by MILC Collaboration. We perform the GPD calculation in Breit frame using averaged nucleon boost momentum Pz≈2.2 GeV with nonzero momentum transfers in [0.2,1.0] GeV2. Nonperturbative renormalization in RI/MOM scheme is used to obtain the quasi-distribution before matching to the lightcone GPDs. The three-dimensional distribution H˜(x,Q2) is presented, along with the three-dimensional nucleon tomography and impact-parameter–dependent distribution for selected Bjorken x at μ=3 GeV in MS‾ scheme.

Positivity and geometric function theory constraints on pion scattering

This paper presents the fascinating correspondence between the geometric function theory and the scattering amplitudes with O(N) global symmetry. A crucial ingredient to show such correspondence is a fully crossing symmetric dispersion relation in the z-variable, rather than the fixed channel dispersion relation. We have written down fully crossing symmetric dispersion relation for O(N) model in z-variable for three independent combinations of isospin amplitudes. We have presented three independent sum rules or locality constraints for the O(N) model arising from the fully crossing symmetric dispersion relations. We have derived three sets of positivity conditions. We have obtained two-sided bounds on Taylor coefficients of physical Pion amplitudes around the crossing symmetric point (for example, π+π−→ π0π0) applying the positivity conditions and the Bieberbach-Rogosinski inequalities from geometric function theory.

Constraints for the semileptonic B→D(*) form factors from lattice QCD simulations of two-point correlation functions

In this work we present the first nonperturbative determination of the hadronic susceptibilities that constrain the form factors entering the semileptonic $B\ensuremath{\rightarrow}{D}^{(*)}\ensuremath{\ell}{\ensuremath{\nu}}_{\ensuremath{\ell}}$ 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 (ETMC) with ${N}_{f}=2+1+1$ dynamical quarks at three values of the lattice spacing ($a\ensuremath{\simeq}0.062$, 0.082, 0.089 fm) and with pion masses in the range $\ensuremath{\simeq}210--450\text{ }\text{ }\mathrm{MeV}$, we evaluate the longitudinal and transverse susceptibilities of the vector and axial-vector polarization functions at the physical pion point and in the continuum and infinite volume limits. The ETMC ratio method is adopted to reach the physical $b$-quark mass ${m}_{b}^{\mathrm{phys}}$. At zero momentum transfer for the $b\ensuremath{\rightarrow}c$ transition we get ${\ensuremath{\chi}}_{{0}^{+}}({m}_{b}^{\mathrm{phys}})=7.58(59)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, ${\ensuremath{\chi}}_{{1}^{\ensuremath{-}}}({m}_{b}^{\mathrm{phys}})=6.72(41)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}2}$, ${\ensuremath{\chi}}_{{0}^{\ensuremath{-}}}({m}_{b}^{\mathrm{phys}})=2.58(17)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ and ${\ensuremath{\chi}}_{{1}^{+}}({m}_{b}^{\mathrm{phys}})=4.69(30)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}2}$ for the scalar, vector, pseudoscalar and axial susceptibilities, respectively. In the case of the vector and pseudoscalar channels the one-particle contributions due to ${B}_{c}^{*}$ and ${B}_{c}$ mesons are evaluated and subtracted to improve the bounds, obtaining ${\ensuremath{\chi}}_{{1}^{\ensuremath{-}}}^{\mathrm{sub}}({m}_{b}^{\mathrm{phys}})=5.84(44)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}2}$ and ${\ensuremath{\chi}}_{{0}^{\ensuremath{-}}}^{\mathrm{sub}}({m}_{b}^{\mathrm{phys}})=2.19(19)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, respectively.

Nucleon isovector scalar charge from overlap fermions

We calculate the nucleon isovector scalar charge in lattice QCD using overlap fermions on five ensembles of gauge configurations generated by the RBC/UKQCD collaboration using the domain-wall quark action with $2+1$ dynamical flavors. The five ensembles cover five pion masses, $m_\pi \approx$ 139, 171, 302, 337 and 371 MeV, and four lattice spacings, $a \approx $ 0.06, 0.08, 0.11 and 0.14 fm. Three to six valence quark masses are computed on each ensemble to investigate the pion mass dependence. The extrapolation to the physical pion mass, continuum and infinite volume limits is obtained by a global fit of all data to a formula originated from partially quenched chiral perturbation theory. The excited-states contamination is carefully analyzed with 3--5 sink-source separations and multi-state fits. Our final result, in the $\overline{\text{MS}}$ scheme at 2 GeV, is $g_{S}^{u-d}= 0.94 (10)_{stat}(8)_{sys}$, where the first error is the statistical error and the second is the systematic error.

Nucleon Tomography and Generalized Parton Distribution at Physical Pion Mass from Lattice QCD.

We present the first lattice calculation of the nucleon isovector unpolarized generalized parton distribution at the physical pion mass using a lattice ensemble with 2+1+1 flavors of highly improved staggered quarks generated by the MILC Collaboration, with lattice spacing a≈0.09 fm and volume 64^{3}×96. We use momentum-smeared sources to improve the signal at nucleon boost momentum P_{z}≈2.2 GeV and report results at nonzero momentum transfers in [0.2,1.0] GeV^{2}. Nonperturbative renormalization in a regularization-independent momentum-subtraction scheme is used to obtain the quasidistribution before matching to the light-cone generalized parton distributions. The three-dimensional distributions H(x,Q^{2}) and E(x,Q^{2}) at ξ=0 are presented, along with the three-dimensional nucleon tomography and impact parameter-dependent distribution for selected Bjorken x at μ=3 GeV in a modified minimal subtraction scheme.