We investigate the in-medium modifications of heavy quarkonia in the vector channel and the heavy quark diffusion coefficient by comparing Euclidean correlators from the lattice to a perturbative spectral function. On the lattice side, we work with continuum extrapolated data from four different large and fine lattices with Clover-improved Wilson fermions in the quenched approximation at five temperatures (0.75, 1.1, 1.3, 1.5 and 2.25$T_c$). On the perturbative side, we use a combination of pNRQCD and vacuum asymptotics to describe the spectral function. After accounting for systematic errors, we obtain a spectral function that is suited to describe the bound state region. This spectral function describes charmonium well without a resonance peak at any of our analyzed temperatures above $T_c$, while we observe a thermally broadened resonance peak for bottomonium that is only melted at our largest temperature, $2.25T_c$. For the transport contribution we assume a Breit-Wigner shaped peak and find that the drag coefficient of charm quarks is larger than that of bottom quarks.
In this work, we strive to gain insight into thermal modifications of charmonium and bottomonium bound states as well as the heavy quark diffusion coefficient. The desired information is contained in the spectral function which can not be calculated on the lattice directly. Instead, the correlator given by an integration over the spectral function times an integration kernel is obtained. Extracting the spectral function is an ill-posed inversion problem and various different solutions have been proposed. We focus on a comparison to a spectral function obtained from combining perturbative and p…
After discussing results of dilepton rates and electrical conductivity obtained from continuum extrapolated results of light quark correlation functions in quenched QCD I will give a review on recent developments in lattice QCD studies for quarkonia in the quark gluon plasma. Recent progress in the extraction of spectral properties from lattice QCD calculations of hadronic correlation functions will be discussed. Besides medium modifications of bound states and their dissociation in the plasma I will focus on transport coefficients, like heavy quark diffusion constants extracted from different…
The status of determinations of $\al_s$ from quarkonia using lattice QCD is reviewed. We compare the results with those obtained from perturbative QCD.
We investigate the effect of a non-perturbative gluon condensate on the spectrum of bound states of extremely heavy quarks using Non-Relativistic-QCD (NRQCD). Simulations are done with an improved gluon action on small lattices. We compare our numerical results with perturbative calculations to test the consistency.
We report here on recent progress in the determination of S-wave and P-wave heavy-quarkonium states at finite temperature. Our results are based on the combination of effective field theories with numerical lattice QCD simulations. These non-perturbative tools allow us to compute the heavy-quarkonium in-medium spectral functions, from which we in turn determine the melting temperatures of individual states and estimate phenomenologically relevant observables, such as the $ψ^\prime$ to J/$ψ$ ratio in heavy-ion collisions.
Rigorous QCD predictions for decay rates of the P-wave states of heavy quarkonia are presented. They are based on a new factorization theorem which is valid to leading order in the heavy quark velocity and to all orders in the running coupling constant of QCD. The decay rates for all four P states into light hadronic or electromagnetic final states are expressed in terms of two phenomenological parameters, whose coefficients are perturbatively calculable. Logarithms of the binding energy encountered in previous perturbative calculations of P-wave decays are factored into a phenomenological par…
Lattice NRQCD with leading finite lattice spacing errors removed is used to simulate heavy-heavy vector decay constants. Quenched simulations are performed at three values of the coupling and fifteen values of the quark mass. The improved gauge action with plaquettes and rectangles is used. Landau link improvement is used throughout. "Perturbative" and nonperturbative meson masses are compared. One-loop perturbative matching between lattice and continuum heavy-heavy vector currents is performed. The data is consistent with the vector meson decay constant being proportional to the square root o…
The early idea that a non-perturbative gluon condensate affects the spectrum of heavy quarks is revisited in the light of modern simulation techniques. We evaluate the low lying spectrum of bound states of two heavy quarks for large hypothetical quark mass, m_Q > m_b, using non-relativistic QCD and compare with other models to test the consistency.
We report on the status of the determination of the heavy quark masses from our calculation of the quarkonia spectra. All sources of systematic errors that enter the quark mass determination are accounted for. We explicitly keep $ma \neq 0$ in the perturbative calculation relating the bare lattice mass to a renormalized mass. Our results are still preliminary.
We present a new method to study the properties of heavy quarks at finite temperature. It combines non-relativistic QCD with an improved gluonic action on anisotropic lattices. The efficiency of the approach is demonstrated by the first non-perturbative calculation of the temperature dependence of low-lying quarkonium "pole" masses. For ground state meson masses in the region between charmonium and bottomonium we find only very little variation up to our highest temperature which corresponds to T approximately 1.2 Tc while first excited states indicate a large mass shift.
We study quarkonium correlators and spectral functions at zero and finite temperature in QCD with only heavy quarks using potential models combined with perturbative QCD. First, we show that this approach can describe the quarkonium correlation function at zero temperature. Using a class of screened potentials based on lattice calculations of the static quark-antiquark free energy we calculate spectral functions at finite temperature. We find that all quarkonium states, with the exception of the $1S$ bottomonium, dissolve in the deconfined phase at temperatures smaller than $1.5T_c$, in contra…
Since the initial investigation by Matsui and Satz heavy quark bound states at finite temperature have been subject to numerous studies. The derivation of a finite-temperature potential from first principles was attempted only recently however, by generalising the Schroedinger equation which is successfully employed for the description of quarkonia at zero temperature to a thermal setting. In this note the finite-temperature static potential is derived to leading order using resummed perturbation theory. The modification of the heavy quarkonium spectral function by an imaginary part of the pot…
Lattice NRQCD with leading finite lattice spacing errors removed is used to calculate decay constants of mesons made up of heavy quarks. Quenched simulations are done with a tadpole improved gauge action containing plaquette and six-link rectangular terms. The tadpole factor is estimated using the Landau link. For each of the three values of the coupling constant considered, quarkonia are calculated for five masses spanning the range from charmonium through bottomonium, and one set of quark masses is tuned to the B(c). "Perturbative" and nonperturbative meson masses are compared. One-loop pert…
The $Q \bar{Q}$ semirelativistic interaction in QCD can be simply expressed in terms of the Wilson loop and its functional derivatives. In this approach we present the $Q \bar{Q}$ potential up to order $1/m^2$ using the expressions for the Wilson loop given by the Wilson Minimal Area Law (MAL), the Stochastic Vacuum Model (SVM) and Dual QCD (DQCD). We confirm the original results given in the different frameworks and obtain new contributions. In particular we calculate up to order $1/m^2$ the complete velocity dependent potential in the SVM. This allows us to show that the MAL model is entirel…
Precision tests of QCD perturbation theory are not readily available from experimental data. The main reasons are systematic uncertainties due to the confinement of quarks and gluons, as well as kinematical constraints which limit the accessible energy scales. We here show how continuum extrapolated lattice data may overcome such problems and provide excellent probes of renormalized perturbation theory. This work corresponds to an essential step in the ALPHA collaboration's project to determine the $Λ$-parameter in 3-flavour QCD. I explain the basic techniques used in the high energy regime, n…
We present lattice QCD calculations of valence parton distribution function (PDF) of pion employing next-to-next-leading-order (NNLO) perturbative QCD matching. Our calculations are based on three gauge ensembles of 2+1 flavor highly improved staggered quarks and Wilson--Clover valance quarks, corresponding to pion mass $m_π=140$~MeV at a lattice spacing $a=0.076$~fm and $m_π=300$~MeV at $a=0.04, 0.06$~fm. This enables us to present, for the first time, continuum-extrapolated lattice QCD results for NNLO valence PDF of the pion at the physical point. Applying leading-twist expansion for renorm…
The electric dipole transitions $χ_{bJ}(1P)\to γΥ(1S)$ with $J=0,1,2$ and $h_{b}(1P)\to γη_{b}(1S)$ are computed using the weak-coupling version of a low-energy effective field theory named potential non-relativistic QCD (pNRQCD). In order to improve convergence and thus give firm predictions for the studied reactions, the full static potential is incorporated into the leading order Hamiltonian; moreover, we must handle properly renormalon effects and re-summation of large logarithms. The precision we reach is $k_γ^{3}/(mv)^{2} \times \mathcal{O}(v^{2})$, where $k_γ$ is the photon energy, $m$ …
In this Chapter QCD interactions between a quark and an anti-quark are discussed. In the heavy quark limit these potentials can be related to quarkonia and $1/m$ corrections can be systematically determined. Excitations of the ground state potential provide an entry point into the phenomenology of quark-gluon hybrids. The short-distance behaviour of non-perturbative potentials can serve as a test of resummation and convergence of perturbative expansions. Torelons and potentials between non-fundamental colour charges offer a window into the origin of the confinement mechanism and relate to effe…
We review the connection between QCD and potential models of quarkonia in the framework of effective field theories, with an emphasis on non-perturbative methods such as lattice simulations. The static and heavy quark potentials are introduced and the applicability of the non-relativistic and adiabatic approximations are discussed. The roles of so-called hybrid potentials are explored and we comment on the QCD analogue of the QED Lamb shift.
Consideration of the analytic properties of pion-induced baryon self-energies leads to new functional forms for the extrapolation of light baryon masses. These functional forms reproduce the leading non-analytic behavior of chiral perturbation theory, the correct non-analytic behavior at the $N π$ threshold and the appropriate heavy-quark limit. They involve only three unknown parameters, which may be obtained by fitting to lattice data. Recent dynamical fermion results from CP-PACS and UKQCD are extrapolated using these new functional forms. We also use these functions to probe the limit of a…
New incarnations of heavy-ion collision experiments are turning our attention to hard processes and a more fine-grained resolution of the QGP. In this endeavor quarkonia or open heavy flavors turn out to be versatile probes, which are usually described through models based on perturbative QCD, AdS, and effective field theories. The lattice provides nonperturbative input and constraints to such models.\newline In-medium bottomonia, the complex static quark-antiquark potential, and the heavy-quark momentum diffusion coefficient are key quantities where lattice gauge theory has recently achieved …
We present continuum extrapolated results of charmonium and bottomonium correlators in the vector channel at several temperatures below and above $T_c$. The continuum extrapolation jointly performed with the interpolations to have physical values of $J/ψ$ and $Υ$ masses in the confined phase is based on calculations on several large quenched isotropic lattices using clover-improved Wilson valence fermions carrying different quark masses. The extrapolated lattice correlators are confronted with perturbation theory results incorporating resummed thermal effects around the threshold from pNRQCD a…
Building on the experience of [1], we develop a formalism to construct operators for higher derivatives of the pressure in hot QCD with respect to the quark chemical potential $μ$. We provide formulae for the operators up to the sixth derivative, and obtain continuum-extrapolated results from lattice EQCD at zero and finite $μ$ and at six different pairs of temperature T and number of massless quark flavors $n_f$. Our data is benchmarked against full-QCD lattice and perturbative results, allowing to judge the quality of the perturbative series expansion in EQCD and the dimensional reduction pr…
We estimate the momentum diffusion coefficient of a heavy quark within a pure SU(3) plasma at a temperature of about 1.5Tc. Large-scale Monte Carlo simulations on a series of lattices extending up to 192^3*48 permit us to carry out a continuum extrapolation of the so-called colour-electric imaginary-time correlator. The extrapolated correlator is analyzed with the help of theoretically motivated models for the corresponding spectral function. Evidence for a non-zero transport coefficient is found and, incorporating systematic uncertainties reflecting model assumptions, we obtain kappa = (1.8 -…
We present continuum extrapolated lattice QCD results for up to fourth-order diagonal and off-diagonal quark number susceptibilities in the high temperature region of 300-700 MeV. Lattice QCD calculations are performed using 2+1 flavors of highly improved staggered quarks with nearly physical quark masses and at four different lattice spacings. Comparisons of our results with recent weak coupling perturbative calculations yield reasonably good agreements for the entire temperature range.
We confront the thermal NLO vector spectral function (both the transverse and longitudinal channel with respect to spatial momentum, both above and below the light cone) with continuum-extrapolated lattice data (both quenched and with $N_{\rm f} = 2$, at $T \sim 1.2 T_{\rm c}$). The perturbative side incorporates new results, whose main features are summarized. The resolution of the lattice data is good enough to constrain the scale choice of $α_{\rm s}$ on the perturbative side. The comparison supports the previous indication that the true spectral function falls below the resummed NLO one in…
The pseudoscalar correlator is an ideal lattice probe for thermal modifications to quarkonium spectra, given that it is not compromised by a contribution from a large transport peak. We construct a perturbative spectral function incorporating resummed thermal effects around the threshold and vacuum asymptotics above the threshold, and compare the corresponding imaginary-time correlators with continuum-extrapolated lattice data for quenched SU(3) at several temperatures. Modest differences are observed, which may originate from non-perturbative mass shifts or renormalization factors, however no…
We consider spatial correlation lengths ξfor various QCD light quark bilinears at temperatures above a few hundred MeV. Some of the correlation lengths (such as that related to baryon density) coincide with what has been measured earlier on from glueball-like states; others do not couple to glueballs, and have a well-known perturbative leading-order expression as well as a computable next-to-leading-order correction. We determine the latter following analogies with the NRQCD effective theory, used for the study of heavy quarkonia at zero temperature: we find (for the quenched case) ξ^{-1} = 2 …
We use a space-time asymmetric O(a) improved fermion action and fix the asymmetry non-perturbatively to restore the relativistic dispersion relation. We compute spectra and matrix elements of quarkonia and heavy-light mesons and compare with results obtained using a symmetric action with the Fermilab interpretation i.e. that the physics of heavy lattice quarks depends solely on their kinetic mass. We provide additional evidence to support this.
The finite temperature phase transition in the SU(2)-Higgs model at a Higgs boson mass $M_H \simeq 35$ GeV is studied in numerical simulations on four dimensional lattices with time-like extensions up to $L_t=5$. $T_c/M_H$ is extrapolated to the continuum limit and a comparison with the perturbative prediction is made. A one-loop calculation to the coupling anisotropies of the SU(2)-Higgs model on lattices with asymmetric lattice spacings is presented. Our numerical simulations show that the above perturbative result is applicable in the phenomenologically interesting parameter region.
We study the asymptotic behavior of the ghost propagator in the quenched SU(3) lattice gauge theory with Wilson action. The study is performed on lattices with a physical volume fixed around 1.6 fm and different lattice spacings: 0.100 fm, 0.070 fm and 0.055 fm. We implement an efficient algorithm for computing the Faddeev-Popov operator on the lattice. We are able to extrapolate the lattice data for the ghost propagator towards the continuum and to show that the extrapolated data on each lattice can be described up to four-loop perturbation theory from 2.0 GeV to 6.0 GeV. The three-loop value…
We estimate the QCD coupling constant from a lattice calculation of the bottomonium spectrum. The second order perturbative expansion of the plaquette expectation value is employed to determine alpha_S at a scale set by the 2S-1S and 1P-1S level splittings. The latter are computed in NRQCD in a dynamical gauge field background with two degenerate flavours of Wilson quarks at intermediate masses and extrapolated to the chiral limit. Combining the N_f=2 result with the quenched result at equal lattice spacing we extrapolate to the physical number of light flavours to find a value of alpha_{\bar …
Using continuum extrapolated lattice data we trace a family of running couplings in three-flavour QCD over a large range of scales from about 4 to 128 GeV. The scale is set by the finite space time volume so that recursive finite size techniques can be applied, and Schrödinger functional (SF) boundary conditions enable direct simulations in the chiral limit. Compared to earlier studies we have improved on both statistical and systematic errors. Using the SF coupling to implicitly define a reference scale $1/L_0\approx 4$ GeV through $\bar{g}^2(L_0) =2.012$, we quote $L_0 Λ^{N_{\rm f}=3}_{\over…
In the investigation of in-medium modifications of quarkonia and for determining heavy quark diffusion coefficients, correlation functions play a crucial role. For the first time we perform a continuum extrapolation of charmonium and bottomonium correlators in the vector channel based on non-perturbatively clover-improved Wilson fermions in quenched lattice QCD. Calculations were done at 4 different lattice spacings, spatial extents between 96 and 192, aspect ratio from 1/6 to 1/2, for 5 temperatures between $T/T_c = 0.75$ and $T/T_c = 2.2$. We interpolate between different quark masses to mat…
We present the current status of the computation of the form factor $f_+ (q^2)$ for the semi-leptonic decay $\mathrm B_\mathrm s \to \mathrm K \ell ν$ by the ALPHA collaboration. We use gauge configurations which were generated as part of the Coordinated Lattice Simulations (CLS) effort. They have $N_\mathrm f=2$ non-perturbatively $O(a)$ improved Wilson fermions, and pion masses down to $\approx 250 \,\mathrm {MeV}$ with $m_πL \geq 4$. The heavy quark is treated in non-perturbative Heavy Quark Effective Theory (HQET). We discuss how to extract the form factors from the correlation functions a…
This is the second of two papers devoted to the perturbative computation of the ghost and gluon propagators in SU(3) Lattice Gauge Theory. Such a computation should enable a comparison with results from lattice simulations in order to reveal the genuinely non-perturbative content of the latter. The gluon propagator is computed by means of Numerical Stochastic Perturbation Theory: results range from two up to four loops, depending on the different lattice sizes. The non-logarithmic constants for one, two and three loops are extrapolated to the lattice spacing $a \to 0$ continuum and infinite vo…
We compute perturbative QCD corrections to $B \to D$ form factors at leading power in $Λ/m_b$, at large hadronic recoil, from the light-cone sum rules (LCSR) with $B$-meson distribution amplitudes in HQET. QCD factorization for the vacuum-to-$B$-meson correlation function with an interpolating current for the $D$-meson is demonstrated explicitly at one loop with the power counting scheme $m_c \sim {\cal O} \left (\sqrt{Λ\, m_b} \right ) $. The jet functions encoding information of the hard-collinear dynamics in the above-mentioned correlation function are complicated by the appearance of an ad…
We present results for the photon emission rate determined from the transverse channel vector correlator at fixed spatial momentum using two flavors of dynamical Wilson fermions at $T\sim$250 MeV. We estimate the transverse channel spectral function using the continuum extrapolated correlator by applying various fit ansätze with a smooth matching to the NLO perturbative result. We confront our estimate based on this channel with the latest results of our collaboration based on the difference of the transverse and longitudinal channels.
We present the 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 $a\approx\{0.06,0.09,0.12\}$ fm and physical pion and kaon masses, with the meson momenta $P_z = \{1.29,1.72,2.15\}$ GeV. The result is non-perturbatively renormalized in a recently proposed hybrid scheme with self renormalization, and extrapolated 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 …
We calculate charmonium correlators on the lattice with 2+1-flavors of sea quarks and charm valence quark both described by the Möbius domain-wall fermion. Temporal moments of the correlators are calculated and matched to perturbative QCD formulae to extract the charm quark mass $m_c(μ)$ and strong coupling constant $α_s(μ)$. Lattice data at three lattice spacings, 0.044, 0.055, and 0.080~fm, are extrapolated to the continuum limit. The correlators in the vector channel are confirmed to be consistent with the experimental data for $e^+e^-\to c\bar{c}$, while the pseudo-scalar channel is used t…
The universality of the continuum limit and the applicability of renormalized perturbation theory are tested in the SU(2) lattice gauge theory by computing two different non-perturbatively defined running couplings over a large range of energies. The lattice data (which were generated on the powerful APE computers at Rome II and DESY) are extrapolated to the continuum limit by simulating sequences of lattices with decreasing spacings. Our results confirm the expected universality at all energies to a precision of a few percent. We find, however, that perturbation theory must be used with care …
We study the vector and axial-vector current correlators in perturbative and non-perturbative regimes of QCD. The correlators in Euclidean coordinate space are calculated on the lattice using the Möbius domain-wall fermion formulation at three lattice spacings covering 0.044--0.080~fm. The dynamical quark effects of $2+1$ light flavors are included. The sum $V+A$ and the difference $V-A$ of the vector ($V$) and axial-vector ($A$) current correlators calculated on the lattice after extrapolating to the physical point agree with those converted from the ALEPH experimental data of hadronic $τ$ de…
We present a non-perturbative study of the λϕ^{4} model in a three dimensional Euclidean space, where the two spatial coordinates are non-commutative. Our results are obtained from numerical simulations of the lattice model, after its mapping onto a dimensionally reduced, twisted Hermitian matrix model. In this way we first reveal the explicit phase diagram of the non-commutative λϕ^{4} lattice model. We observe that the ordered regime splits into a phase of uniform order and a phase of two stripes of opposite sign, and more complicated patterns. Next we discuss the behavior of the spatial and…
