NRQCD Action Research Articles

Construction of bbu¯d¯ tetraquark states on lattice with NRQCD bottom and HISQ up and down quarks

We construct $bb\bar{u}\bar{d}$ states on lattice using NRQCD action for bottom and HISQ action for the light up/down quarks. The NRQCD-HISQ tetraquark operators are constructed for "bound" $[bb][\bar{u}\bar{d}]$ and "molecular" $[b\bar{u}] [b\bar{d}]$ states. Corresponding to these different operators, two different appropriately tuned light quark masses are needed to obtain the desired spectra. We explain this requirement of different $m_{u/d}$ in the light of relativised quark model involving Hartree-Fock calculation. The mass spectra of double bottom tetraquark states are obtained on MILC $N_f=2+1$ Asqtad lattices at three different lattice spacings. Variational analysis has been carried out to obtain the relative contribution of "bound" and "molecular" states to the energy eigenstates.

Heavy baryon spectrum on the lattice with NRQCD bottom and HISQ lighter quarks

We determine the mass spectra of heavy baryons containing one or more bottom quarks along with their hyperfine splittings and various mass differences on MILC 2+1 Asqtad lattices at three different lattice spacings. NRQCD action is used for bottom quarks whereas relativistic HISQ action for the lighter up/down, strange and charm quarks. We consider all possible combinations of bottom and lighter quarks to construct the bottom baryon operators for the states $J^P=1/2^+$ and $3/2^+$.

Neutral B -meson mixing from full lattice QCD at the physical point

We calculate the bag parameters for neutral $B$-meson mixing in and beyond the Standard Model, in full four-flavour lattice QCD for the first time. We work on gluon field configurations that include the effect of $u$, $d$, $s$ and $c$ sea quarks with the Highly Improved Staggered Quark (HISQ) action at three values of the lattice spacing and with three $u/d$ quark masses going down to the physical value. The valence $b$ quarks use the improved NRQCD action and the valence light quarks, the HISQ action. Our analysis was blinded. Our results for the bag parameters for all five operators are the most accurate to date. For the Standard Model operator between $B_s$ and $B_d$ mesons we find: $\hat{B}_{B_s}=1.232(53)$, $\hat{B}_{B_d}=1.222(61)$. Combining our results with lattice QCD calculations of the decay constants using HISQ quarks from the Fermilab/MILC collaboration and with experimental values for $B_s$ and $B_d$ oscillation frequencies allows determination of the CKM elements $V_{ts}$ and $V_{td}$. We find $V_{ts} = 0.04189(93)$, $V_{td} = 0.00867(23)$ and $V_{ts}/V_{td} = 0.2071(27)$. Our results agree well (within $2\sigma$) with values determined from CKM unitarity constraints based on tree-level processes (only). Using a ratio to $\Delta M$ in which CKM elements cancel in the Standard Model, we determine the branching fractions ${\text{Br}}(B_s\rightarrow \mu^+\mu^-) = 3.81(18) \times 10^{-9}$ and ${\text{Br}}(B_d\rightarrow \mu^+\mu^-) = 1.031(54) \times 10^{-10}$. We also give results for matrix elements of the operators $R_0$, $R_1$ and $\tilde{R}_1$ that contribute to neutral $B$-meson width differences.

Improving the theoretical prediction for the Bs - B̅s width difference: matrix elements of next-to-leading order ΔB = 2 operators

We present lattice QCD results for the matrix elements of R2 and other dimension-7, ΔB = 2 operators relevant for calculations of Δs, the Bs - B̅s width difference. We have computed correlation functions using 5 ensembles of the MILC Collaboration’s 2+1 + 1-flavour gauge field configurations, spanning 3 lattice spacings and light sea quarks masses down to the physical point. The HISQ action is used for the valence strange quarks, and the NRQCD action is used for the bottom quarks. Once our analysis is complete, the theoretical uncertainty in the Standard Model prediction for ΔΓs will be substantially reduced.

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.

Mass of thebquark from lattice NRQCD and lattice perturbation theory

We present a determination of the b-quark mass accurate through O(\alpha_s^2) in perturbation theory and including partial contributions at O(\alpha_s^3). Nonperturbative input comes from the calculation of the Upsilon and B_s energies in lattice QCD including the effect of u, d and s sea quarks. We use an improved NRQCD action for the b-quark. This is combined with the heavy quark energy shift in NRQCD determined using a mixed approach of high-beta simulation and automated lattice perturbation theory. Comparison with experiment enables the quark mass to be extracted: in the MS bar scheme we find m_b(m_b) = 4.166(43) GeV.

Renormalization of heavy-light currents in moving nonrelativistic QCD

Heavy-light decays such as $B \to \pi \ell \nu$, $B \to K^{*} \gamma$ and $B \to K^{(*)} \ell \ell$ can be used to constrain the parameters of the Standard Model and in indirect searches for new physics. While the precision of experimental results has improved over the last years this has still to be matched by equally precise theoretical predictions. The calculation of heavy-light form factors is currently carried out in lattice QCD. Due to its small Compton wavelength we discretize the heavy quark in an effective non-relativistic theory. By formulating the theory in a moving frame of reference discretization errors in the final state are reduced at large recoil. Over the last years the formalism has been improved and tested extensively. Systematic uncertainties are reduced by renormalizing the m(oving)NRQCD action and heavy-light decay operators. The theory differs from QCD only for large loop momenta at the order of the lattice cutoff and the calculation can be carried out in perturbation theory as an expansion in the strong coupling constant. In this paper we calculate the one loop corrections to the heavy-light vector and tensor operator. Due to the complexity of the action the generation of lattice Feynman rules is automated and loop integrals are solved by the adaptive Monte Carlo integrator VEGAS. We discuss the infrared and ultraviolet divergences in the loop integrals both in the continuum and on the lattice. The light quarks are discretized in the ASQTad and highly improved staggered quark (HISQ) action; the formalism is easily extended to other quark actions.

Bottomonium spectrum at orderv6from domain-wall lattice QCD: Precise results for hyperfine splittings

The bottomonium spectrum is computed in dynamical 2+1 flavor lattice QCD, using NRQCD for the b quarks. The main calculations in this work are based on gauge field ensembles generated by the RBC and UKQCD collaborations with the Iwasaki action for the gluons and a domain-wall action for the sea quarks. Lattice spacing values of approximately 0.08 fm and 0.11 fm are used, and simultaneous chiral extrapolations to the physical pion mass are performed. As a test for gluon discretization errors, the calculations are repeated on two ensembles generated by the MILC collaboration with the Luscher-Weisz gauge action. Gluon discretization errors are also studied in a lattice potential model using perturbation theory for four different gauge actions. The nonperturbative lattice QCD results for the radial and orbital bottomonium energy splittings obtained from the RBC/UKQCD ensembles are found to be in excellent agreement with experiment. To get accurate results for spin splittings, the spin-dependent order-v^6 terms are included in the NRQCD action, and suitable ratios are calculated such that most of the unknown radiative corrections cancel. The cancellation of radiative corrections is verified explicitly by repeating the calculations with different values of the couplings in the NRQCD action. Using the lattice ratios of the S-wave hyperfine and the 1P tensor splitting, and the experimental result for the 1P tensor splitting, the 1S hyperfine splitting is found to be 60.3+-5.5(stat)+-5.0(syst)+-2.1(exp) MeV, and the 2S hyperfine splitting is predicted to be 23.5+-4.1(stat)+-2.1(syst)+-0.8(exp) MeV.

Leptonic widths of heavy quarkonia:S-wave QCD/NRQCD matching coefficients for the electromagnetic vector annihilation current atO(αsv2)

We construct the $S$-wave part of the electromagnetic vector annihilation current to $\mathcal{O}({\ensuremath{\alpha}}_{s}{v}^{2})$ on the lattice for heavy quarks whose dynamics are described by the NRQCD action, and $v$ is the nonrelativistic quark velocity inside the meson. The lattice vector current for $Q\overline{Q}$ annihilation is expressed as a linear combination of lattice operators with quantum numbers $L=0$, ${J}^{P}={1}^{\ensuremath{-}}$, and the coefficients are determined by matching this lattice current to the corresponding continuum current in QCD to $O({v}^{2})$ at one-loop. The annihilation channel gives a complex amplitude and a proper choice for the contours of integration is needed; a simple Wick rotation is not possible. In this way, and with a careful choice of subtraction functions in the numerical integration, the Coulomb-exchange and infrared singularities appearing in the amplitudes are successfully treated. The matching coefficients are given as a function of the heavy quark mass $Ma$ in lattice units. An automated vertex generation program written in Python is employed, allowing us to use a realistic NRQCD action and an improved gluon lattice action. A change in the definition of either action is easily accommodated in this procedure. The final result, when combined with lattice simulation results, describes the electromagnetic decays of heavy quarkonia, notably the $\ensuremath{\Upsilon}$ meson.

B→D*lνandB→Dlνform factors in staggered chiral perturbation theory

We calculate the B {yields} D and B {yields} D* form factors at zero recoil in Staggered Chiral Perturbation Theory. We consider heavy-light mesons in which only the light (u, d, or s) quark is staggered; current lattice simulations generally use a highly improved action such as the Fermilab or NRQCD action for the heavy (b or c) quark. We work to lowest order in the heavy quark expansion and to next-to-leading order in the chiral expansion. We present results for a partially quenched theory with three sea quarks in which there are no mass degeneracies (the ''1+1+1'' theory) and for a partially quenched theory in which the u and d sea quark masses are equal (the ''2+1'' theory). We also present results for full (2+1) QCD, along with a numerical estimate of the size of staggered discretization errors. Finally, we calculate the finite volume corrections to the form factors and estimate their numerical size in current lattice simulations.

Matching of the Heavy-Light Currents with NRQCD Heavy and Improved Naive Light Quarks

One-loop matching of heavy-light currents is carried out for a highly improved lattice action, including the effects of mixings with dimension 4 O ( 1 / M ) and O ( a ) operators. We use the NRQCD action for heavy quarks, the Asqtad improved naive action for light quarks, and the Symanzik improved glue action. These results are being used in recent heavy meson decay constant and semileptonic form factor calculations on the MILC dynamical configurations.

Heavy quark expansion parameters from lattice NRQCD

We present a lattice QCD calculation of the heavy quark expansion parameters $\mu_{\pi}^2$ and $\mu_G^2$ for heavy-light mesons and heavy-light-light baryons. The calculation is carried out on a 20$^3\times$48 lattice at $\beta$ = 6.0 in the quenched approximation, using the lattice NRQCD action for heavy quarks. We obtain the parameters $\mu_{\pi}^2$ and $\mu_G^2$ in two different methods: a direct calculation of the matrix elements and an indirect calculation through the mass spectrum, and confirm that the both methods give consistent results. We also discuss an application to the lifetime ratios.

One-loop matching of the heavy-lightA0andV0currents with nonrelativistic QCD heavy and improved naive light quarks

One-loop matching of heavy-light currents is carried out for a highly improved lattice action, including the effects of dimension 4 O(1/M) and O(a) operators. We use the NRQCD action for heavy quarks, the Asqtad improved naive action for light quarks, and the Symanzik improved glue action. As part of the matching procedure we also present results for the NRQCD self energy and for massless Asqtad quark wavefunction renormalization with improved glue.

Progress calculating decay constants with NRQCD and AsgTad actions

Abstract We combine a light AsgTad antiquark with a nonrelativistic heavy quark to compute the decay constants of heavy-light pseudoscalar mesons using the ensemble of 3-flavor gauge field configurations generated by the MILC collaboration. Preliminary results for f B s and f D s are given and status of the chiral extrapolation to f B is reported. We also touch upon results of the perturbative calculation which matches matrix elements in the effective theory to the full theory at 1-loop order.

Precise determination of the Grinstein ratio of heavy-light decay constant in unquenched QCD

We study the chiral behavior of the ratio of heavy-light decay constants on unquenched lattices with two flavor dynamical quarks using the plaquette gauge action and the nonperturbatively O(a)-improved Wilson fermion action at β = 5.2. The heavy quarks are described by the clover action and the NRQCD action. We find that the sea quark mass dependence of the decay constants largely cancels in the double ratio R 1 =f B s /f B d /f D s /f D d and the systematic error in the chiral limit can be drastically reduced. Our preliminary result with heavy clover is R 1 = 1.018 ± 0.006 ± 0.010, where the first and the second errors are the statistical and the systematics errors.

B0−B0mixing in quenched lattice QCD

We present our results of lattice calculations of $B$ parameters, which parameterize $\Delta B$=2 transition amplitudes together with the leptonic decay constant. Calculations are made in the quenched approximation at $\beta$=5.7, 5.9, 6.0 and 6.1, using NRQCD action for heavy quark and the $O(a)$-improved Wilson action for light quark. The operators are perturbatively renormalized including the correction of $O(\alpha_s/(aM)^m)$ ($m\ge$0). We examine the scaling behavior of $B$ parameters, and discuss the systematic uncertainties based on the results with several different truncations of higher order terms in 1/M and $\alpha_s$ expansions. We find $B_{B_d}(m_b)=0.84(3)(5)$, $B_{B_s}/B_{B_d}=1.020(21)(^{+15}_{-16})(^{+5}_{-0})$ and $B_{S_s}(m_b)=0.85(1)(5)(^{+1}_{-0})$ in the quenched approximation. The errors represent statistical and systematic as well as the uncertainty in the determination of strange quark mass.

Heavy quark expansion parameters from lattice NRQCD

Using the lattice NRQCD action for heavy quark, we calculate the heavy quark expansion parameters μ 2 π and μ 2 G for heavy-light mesons and heavy-light-light baryons. The results are compared with the mass differences among heavy hadrons to test the validity of HQET relations on the lattice.

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.

Differential decay rate for B → π lν semileptonic decays

We present our study on B → π lν semileptonic decay form factors using NRQCD action for heavy quark. In the analysis, we use the form factors f 1(υ· k π) and f 2(υ· k π) defined in the context of heavy quark effective theory by Burdman et al.. Since the NRQCD action for heavy quark respects the heavy quark symmetry, our results are described by the HQET form factors most naturally. From a quenched lattice QCD simulation at β=5.9 on a 16 3 × 48 lattice, we obtain the form factors f 1(υ· k π) and f 2(υ· k π), and find that their 1/ m B correction is small. The limit of physical heavy and light quark masses can be reached without introducing any model function, and we obtain a prediction for the differential decay rate dΓ/ dq 2. We also discuss the soft pion limit of the form factors.

Unquenched charmonium with nonrelativistic QCD

We present the results from a series of lattice simulations of the charmonium system using a highly-improved NRQCD action, both in the quenched approximation, and with n_f = 2 light dynamical quarks. The spectra show some evidence for quenching effects of roughly 10% in the S- and P-hyperfine spin splittings---probably too small to account for the severe underestimates in these quantities seen in previous quenched charmonium simulations. We also find estimates for the magnitude of other systematic effects---in particular, the choice of the tadpole factor can alter spin splittings at the 10-20% level, and radiative corrections may be as large as 40% for charmonium. We conclude that quenching is just one of a collection of important effects that require attention in precision heavy-quark simulations.