Lattice Nonrelativistic QCD Research Articles

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.

Bethe-Salpeter amplitudes of Upsilons

Based on lattice non-relativistic QCD (NRQCD) studies we present results for Bethe-Salpeter amplitudes for $\Upsilon(1S)$, $\Upsilon(2S)$ and $\Upsilon(3S)$ in vacuum as well as in quark-gluon plasma. Our study is based on 2+1 flavor $48^3 \times 12$ lattices generated using the Highly Improved Staggered Quark (HISQ) action and with a pion mass of $161$ MeV. At zero temperature the Bethe-Salpeter amplitudes follow the expectations based on non-relativistic potential models. At non-zero temperatures, the interpretation of Bethe-Salpeter amplitudes turns out to be more nuanced, but consistent with our previous lattice QCD study of excited Upsilons in quark-gluon plasma.

Excited bottomonia in quark-gluon plasma from lattice QCD

We present the first lattice QCD study of up to 3S and 2P bottomonia at non-zero temperatures. Correlation functions of bottomonia were computed using novel bottomonium operators and a variational technique, within the lattice non-relativistic QCD framework. We analyzed the bottomonium correlation functions based on simple physically-motivated spectral functions. We found evidence of sequential in-medium modifications, in accordance with the sizes of the bottomonium states.

Thermal broadening of bottomonia: Lattice nonrelativistic QCD with extended operators

We present lattice non-relativistic QCD calculations of bottomonium correlation functions at temperatures $T \simeq 150-350$ MeV. The correlation functions were computed using extended bottomonium operators, and on background gauge-field configurations for 2+1-flavor QCD having physical kaon and nearly-physical pion masses. We analyzed these correlation functions based on simple theoretically-motivated parameterizations of the corresponding spectral functions. The results of our analyses are compatible with significant in-medium thermal broadening of the ground state S- and P-wave bottomonia.

Improving the kinetic couplings in lattice nonrelativistic QCD

We improve the non-relativistic QCD (NRQCD) action by comparing the dispersion relation to that of the continuum through $\mathcal{O}(p^6)$ in perturbation theory. The one-loop matching coefficients of the $\mathcal{O}(p^4)$ kinetic operators are determined, as well as the scale at which to evaluate $\alpha_s$ in the $V$-scheme for each quantity. We utilise automated lattice perturbation theory using twisted boundary conditions as an infrared regulator. The one-loop radiative corrections to the mass renormalisation, zero-point energy and overall energy-shift of an NRQCD $b$-quark are also found. We also explore how a Fat$3$-smeared NRQCD action and changes of the stability parameter $n$ affect the coefficients. Finally, we use gluon field ensembles at multiple lattice spacing values, all of which include $u$, $d$, $s$ and $c$ quark vacuum polarisation, to test how the improvements affect the non-perturbatively determined $\Upsilon(1S)$ and $\eta_b(1S)$ kinetic masses, and the tuning of the $b$ quark mass.

New methods for B decay constants and form factors from Lattice NRQCD

We determine the normalisation of scalar and pseudo scalar current operators made from NonRelativistic QCD (NRQCD) b quarks and Highly Improved Staggered (HISQ) light quarks through O(αs∧QCD/mb). We use matrix elements of these operators to extract B meson decay constants and form factors and compare to those obtained using the standard vector and axial vector operators. We work on MILC second-generation 2+1+1 gluon field configurations, including those with physical light quarks in the sea. This provides a test of systematic uncertainties in these calculations and we find agreement between the results to the 2% level of uncertainty previously quoted.

High statistics study of in-medium S- and P-wave quarkonium states in lattice Non-relativistic QCD

Many measurements of quarkonium suppression at the LHC, e.g. the nuclear modification factor RAA of J/Ψ, are well described by a multitude of different models. Thus pinpointing the underlying physics aspects is difficult and guidance based on first principles is needed. Here we present the current status of our ongoing high precision study of in-medium spectral properties of both bottomonium and charmonium based on NRQCD on the lattice. This effective field theory allows us to capture the physics of quarkonium without modeling assumptions in a thermal QCD medium. In our study a first principles and realistic description of the QCD medium is provided by state-of-the-art lattices of the HotQCD collaboration at almost physical pion mass. Our updated results corroborate a picture of sequential modification of states with respect to their vacuum binding energy. Using a novel low-gain variant of the Bayesian BR method for reconstructing spectral functions we find that remnant features of the Upsilon may survive up to T ∼ 400MeV, while the χb signal disappears around T ∼ 270MeV. The cc‾ analysis hints at melting of χc below T ∼ 190MeV while some J/Ψ remnant feature might survive up to T ∼ 245MeV. An improved understanding of the numerical artifacts in the Bayesian approach and the availability of increased statistics have made possible a first quantitative study of the in-medium ground state masses, which tend to lower values as T increases, consistent with lattice potential based studies.

Erratum: Radiative improvement of the lattice nonrelativistic QCD action using the background field method with applications to quarkonium spectroscopy [Phys. Rev. D88, 014505 (2013)

Erratum: Radiative improvement of the lattice nonrelativistic QCD action using the background field method with applications to quarkonium spectroscopy [Phys. Rev. D<b>88</b>, 014505 (2013)

Hindered M1 radiative decay ofϒ(2S)from lattice NRQCD

We present a calculation of the hindered M$1$ $\Upsilon(2S) \to \eta_b(1S) \gamma$ decay rate using lattice non-relativistic QCD. The calculation includes spin-dependent relativistic corrections to the NRQCD action through $\mathcal{O}(v^6)$ in the quark's relative velocity, relativistic corrections to the leading order current which mediates the transition through the quark's magnetic moment, radiative corrections to the leading spin-magnetic coupling and for the first time a full error budget. We also use gluon field ensembles at multiple lattice spacing values, all of which include $u$, $d$, $s$ and $c$ quark vacuum polarisation. Our result for the branching fraction is $\mathcal{B}(\Upsilon(2S)\to\eta_b(1S)\gamma) = 5.4(1.8)\times 10^{-4} $, which agrees with the current experimental value.

Bottomonium hyperfine splitting on the lattice and in the continuum

We revise the analysis of the bottomonium hyperfine splitting within the lattice nonrelativistic QCD. The Wilson coefficients of the radiatively improved lattice action are evaluated by a semianalytic approach based on the asymptotic expansion about the continuum limit. The nonrelativistic renormalization group is used to estimate the high-order radiative corrections. Our result for the $1S$ hyperfine splitting is $M_{\Upsilon(1S)}-M_{\eta_b(1S)}=52.9\pm 5.5~{\rm MeV}$. It reconciles the predictions of the continuum and lattice QCD and is in very good agreement with the most accurate experimental measurement by Belle collaboration.

Erratum: Radiative Improvement of the Lattice Nonrelativistic QCD Action Using the Background Field Method and Application to the Hyperfine Splitting of Quarkonium States [Phys. Rev. Lett.107, 112002 (2011)

Received 19 June 2015DOI:https://doi.org/10.1103/PhysRevLett.115.039901© 2015 American Physical Society

Bottomonium hyperfine splittings from lattice nonrelativistic QCD including radiative and relativistic corrections

We present a calculation of the hyperfine splittings in bottomonium using lattice Nonrelativistic QCD. The calculation includes spin-dependent relativistic corrections through O(v^6), radiative corrections to the leading spin-magnetic coupling and, for the first time, non-perturbative 4-quark interactions which enter at alpha_s^2 v^3. We also include the effect of u,d,s and c quark vacuum polarisation. Our result for the 1S hyperfine splitting is M(Upsilon,1S) - M(eta_b,1S)= 60.0(6.4) MeV. We find the ratio of 2S to 1S hyperfine splittings (M(Upsilon,2S) - M(eta_b,2S))/ (M(Upsilon,1S) - M(eta_b,1S)) = 0.445(28).

Radiative improvement of the lattice nonrelativistic QCD action using the background field method with applications to quarkonium spectroscopy

We apply the background field (BF) method to Non-Relativistic QCD (NRQCD) on the lattice in order to determine the one-loop radiative corrections to the coefficients of the NRQCD action in a manifestly gauge-covariant manner by matching the NRQCD prediction for particular on-shell processes with those of relativistic continuum QCD. We explain how the BF method is implemented in automated perturbation theory and discuss the technique for matching the relativistic and non-relativistic theories. We compute the one-loop radiative corrections to the sigma.B and Darwin terms for the NRQCD action currently used in simulations, as well as the one-loop coefficients of the spin-dependent O(alpha^2) four-fermion contact terms. The effect of the corrections on the hyperfine splitting of bottomonium is estimated using earlier simulation results; the corrected lattice prediction is found to be in agreement with experiment. Agreement of the hyperfine splitting of bottomonium and the B-meson system is confirmed by recent simulation studies (Dowdall et al.) which include our NRQCD radiative corrections for the first time.

Radiative Improvement of the Lattice Nonrelativistic QCD Action Using the Background Field Method and Application to the Hyperfine Splitting of Quarkonium States

We present the first application of the background field method to nonrelativistic QCD (NRQCD) on the lattice in order to determine the one-loop radiative corrections to the coefficients of the NRQCD action in a manifestly gauge-covariant manner. The coefficients of the σ·B term in the NRQCD action and the four-fermion spin-spin interaction are computed at the one-loop level; the resulting shift of the hyperfine splitting of bottomonium is found to bring the lattice predictions in line with experiment.

Bottomonium spectrum from lattice QCD with2+1flavors of domain wall fermions

Recently, realistic lattice QCD calculations with 2+1 flavors of domain wall fermions and the Iwasaki gauge action have been performed by the RBC and UKQCD Collaborations. Here, results for the bottomonium spectrum computed on their gauge configurations of size 24{sup 3}x64 with a lattice spacing of approximately 0.11 fm and four different values for the light quark mass are presented. Improved lattice nonrelativistic QCD is used to treat the b quarks inside the bottomonium. The results for the radial and orbital energy splittings are found to be in good agreement with experimental measurements, indicating that systematic errors are small. The calculation of the {upsilon}(2S)-{upsilon}(1S) energy splitting provides an independent determination of the lattice spacing. For the most physical ensemble it is found to be a{sup -1}=1.740(25)(19) GeV, where the first error is statistical/fitting and the second error is an estimate of the systematic errors due to the lattice nonrelativistic QCD action.

Υspectrum andmbfrom full lattice QCD

We show results for the $\ensuremath{\Upsilon}$ spectrum calculated in lattice QCD including for the first time vacuum polarization effects for light $u$ and $d$ quarks as well as $s$ quarks. We use gluon field configurations generated by the MILC collaboration. The calculations compare the results for a variety of $u$ and $d$ quark masses, as well as making a comparison to quenched results (in which quark vacuum polarization is ignored) and results with only $u$ and $d$ quarks. The $b$ quarks in the $\ensuremath{\Upsilon}$ are treated in lattice Nonrelativistic QCD through NLO in an expansion in the velocity of the $b$ quark. We concentrate on accurate results for orbital and radial splittings where we see clear agreement with experiment once $u$, $d$ and $s$ quark vacuum polarization effects are included. This now allows a consistent determination of the parameters of QCD. We demonstrate this consistency through the agreement of the $\ensuremath{\Upsilon}$ and $B$ spectrum using the same lattice bare $b$ quark mass. A one-loop matching to continuum QCD gives a value for the $b$ quark mass in full lattice QCD for the first time. We obtain ${m}_{b}^{\overline{\mathrm{MS}}}({m}_{b}^{\overline{\mathrm{MS}}})=4.4(3)\text{ }\text{ }\mathrm{GeV}$. We are able to give physical results for the heavy quark potential parameters, ${r}_{0}=0.469(7)\text{ }\text{ }\mathrm{fm}$ and ${r}_{1}=0.321(5)\text{ }\text{ }\mathrm{fm}$. Results for the fine structure in the spectrum and the $\ensuremath{\Upsilon}$ leptonic width are also presented. We predict the $\ensuremath{\Upsilon}\ensuremath{-}{\ensuremath{\eta}}_{b}$ splitting to be 61(14) MeV, the ${\ensuremath{\Upsilon}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\eta}}_{b}^{\ensuremath{'}}$ splitting as 30(19) MeV and the splitting between the ${h}_{b}$ and the spin-average of the ${\ensuremath{\chi}}_{b}$ states to be less than 6 MeV. Improvements to these calculations that will be made in the near future are discussed.

Differential decay rate ofB→πlνsemileptonic decay with lattice nonrelativistic QCD

We present a lattice QCD calculation of $B\to \pi l \nu$ semileptonic decay form factors in the small pion recoil momentum region. The calculation is performed on a quenched $16^3 \times 48$ lattice at $\beta=5.9$ with the NRQCD action including the full 1/M terms. The form factors $f_1(v\cdot k_{\pi})$ and $f_2(v\cdot k_{\pi})$ defined in the heavy quark effective theory for which the heavy quark scaling is manifest are adpoted, and we find that the 1/M correction to the scaling is small for the $B$ meson. The dependence of form factors on the light quark mass and on the recoil energy is found to be mild, and we use a global fit of the form factors at various quark masses and recoil energies to obtain model independent results for the physical differential decay rate. We find that the $B^*$ pole contribution dominates the form factor $f^+(q^2)$ for small pion recoil energy, and obtain the differential decay rate integrated over the kinematic region $q^2 >$ 18 GeV$^2$ to be $|V_{ub}|^2 \times (1.18 \pm 0.37 \pm 0.08 \pm 0.31)$ psec$^{-1}$, where the first error is statistical, the second is that from perturbative calculation, and the third is the systematic error from finite lattice spacing and the chiral extrapolation. We also discuss the systematic errors in the soft pion limit for $f^0(q^2_{max})$ in the present simulation.

Measurement of the hybrid content of heavy quarkonia using lattice nonrelativistic QCD

Using lowest-order lattice nonrelativistic QCD to create heavy meson propagators and applying the spin-dependent interaction, ${c}_{B}(\ensuremath{-}{g/2m}_{q})\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}\ensuremath{\cdot}\stackrel{\ensuremath{\rightarrow}}{B},$ at varying intermediate time slices, we compute the off-diagonal matrix element of the Hamiltonian for the quarkonium-hybrid two-state system. Thus far, we have results for one set of quenched lattices with an interpolation in quark mass to match the bottomonium spectrum. After diagonalization of the two-state Hamiltonian, we find the ground state of the $\ensuremath{\Upsilon}$ to show a ${0.0035(1)c}_{B}^{2}$ (with ${c}_{B}^{2}\ensuremath{\sim}1.5--3.1)$ probability admixture of hybrid, $|b\overline{b}g〉.$

Spectrum and decay matrix elements of B and D-mesons in lattice NRQCD

We discuss recent results on the excitation spectra of B and D-mesons obtained in the framework of non-relativistic lattice QCD in the quenched approximation. The results allow for the determination of the MS -mass of m b, MS ( m b, MS ) = 4.34(7) GeV in O (α 3 s ) in the perturbative matching. The determination of the decay constants f B s and f D s , is discussed in detail. Results for the matrix elements of semi-leptonic B to D decays are shown.

Renormalization of theΔB=2four-quark operators in lattice nonrelativistic QCD

We calculate perturbative renormalization constants for the \Delta B=2 four-quark operators in lattice NRQCD. Continuum operators \bar{b}\gamma_{\mu}(1-\gamma_5)q~ \bar{b}\gamma_{\mu}(1-\gamma_5)q and \bar{b}(1-\gamma_5)q~\bar{b}(1-\gamma_5)q, which are necessary in evaluating the mass and width differences in $B^0_{d(s)}-\bar{B}^0_{d(s)}$ systems, are matched at one-loop with corresponding lattice operators constructed from the NRQCD heavy quarks and the ${\cal O}(a)$-improved light quarks. Using these perturbative coefficients, we also reanalyse our previous simulation results for the matrix elements of the above operators. Our new results are free from the systematic error of ${\cal O}(\alpha_s/(aM_b))$ in contrast to the previous ones with matching coefficients evaluated in the static limit.