We convey an idea of the significant recent progress, which opens up good perspectives for high-precision ab-initio computations in heavy flavour physics based on lattice QCD. Rather than surveying the latest results, this contribution focuses on the concept and the challenges of fully non-perturbative computations in the B-meson sector, where the b-quark is treated within an effective theory. We outline its use to determine the b-quark mass and report on the results obtained in the quenched approximation and on the status in the two dynamical flavour theory.
We simulate a theory with $N_f=2$ heavy quarks of mass $M$. At energies much smaller than $M$ the heavy quarks decouple and the theory can be described by an effective theory which is a pure gauge theory to leading order in $1/M$. We present results for the mass dependence of ratios such as $t_0(M)/t_0(0)$. We compute these ratios from simulations and compare them to the perturbative prediction. The latter relies on a factorisation formula for the ratios which is valid to leading order in $1/M$.
We give an introduction to Heavy Quark Effective Theory (HQET). Our emphasis is on its formulation non-perturbative in the strong coupling, including the non-perturbative determination of the parameters in the HQET Lagrangian. In a second part we review the present status of HQET on the lattice, largely based on work of the ALPHA collaboration in the last few years. We finally discuss opportunities and challenges.
We report on a lattice investigation of heavy quark diffusion within pure SU(3) plasma above the deconfinement transition, with the quarks treated to leading order in the heavy mass expansion. Using a multilevel algorithm, several volumes and lattice spacings, as well as tree-level improvement and perturbative renormalization, we measure the relevant "colour-electric" Euclidean correlator, finding that it clearly exceeds its perturbative counterpart. Even without analytic continuation, this suggests that at temperatures just above the critical one, non-perturbative interactions felt by the hea…
We compute the energy levels of some of the lower-lying heavy quarkonium states perturbatively up to O(alpha_s^5*m) and O(alpha_s^5*m*log[alpha_s]). Stability of the predictions depends crucially on the unknown 4-loop pole-MSbar mass relation. We discuss the current status of the predictions with respect to the observed bottomonium spectrum.
We give an introduction to the special problems encountered in a treatment of HQET beyond perturbation theory in the gauge coupling constant. In particular, we report on a recent test of HQET as an effective theory for QCD and discuss how HQET can be implemented on the lattice including the non-perturbative matching of the effective theory to QCD.
We review a lattice strategy how to non-perturbatively determine the coefficients in the HQET expansion of all components of the heavy-light axial and vector currents, including 1/m_h-corrections. We also discuss recent preliminary results on the form factors parameterizing semi-leptonic B-decays at the leading order in 1/m_h.
Starting from the relativistic gauge-invariant quark-antiquark Green function we obtain the relevant interaction in the one-body limit, which can be interpreted as the kernel of a non-perturbative Dirac equation. We study this kernel in different kinematic regions, reproducing, in particular, for heavy quark the potential case and sum rules results. We discuss the relevance of the result for heavy-light mesons and the relation with the phenomenological Dirac equations used up to now in the literature.
We explain how to perform non-perturbative computations in HQET on the lattice. In particular the problem of the subtraction of power-law divergences is solved by a non-perturbative matching of HQET and QCD. As examples, we present a full calculation of the mass of the b-quark in the combined static and quenched approximation and outline an alternative way to obtain the B-meson decay constant at lowest order. Since no excessively large lattices are required, our strategy can also be applied including dynamical fermions.
We report our final estimate of the b-quark mass from $N_f=2$ lattice QCD simulations using Heavy Quark Effective Theory non-perturbatively matched to QCD at $O(1/m_h)$. Treating systematic and statistical errors in a conservative manner, we obtain $\overline{m}_{\rm b}^{\overline{\rm MS}}(2 {\rm GeV})=4.88(15)$ GeV after an extrapolation to the physical point.
We discuss the renormalisation properties of the complete set of $ΔB = 2$ four-quark operators with the heavy quark treated in the static approximation. We elucidate the role of heavy quark symmetry and other symmetry transformations in constraining their mixing under renormalisation. By employing the Schroedinger functional, a set of non-perturbative renormalisation conditions can be defined in terms of suitable correlation functions. As a first step in a fully non-perturbative determination of the scale-dependent renormalisation factors, we evaluate these conditions in lattice perturbation t…
We present further results for one-loop matching of heavy-light axial and vector currents between continuum QCD and a lattice theory with NRQCD heavy quarks and massless clover quarks.
In the lattice formulation of the Heavy Quark Effective Theory, the value of the classical velocity v, as defined through the separation of the 4-momentum of a heavy quark into a part proportional to the heavy quark mass and a residual part which remains finite in the heavy quark limit (P = Mv + p) is different from its value as it appears in the bare heavy quark propagator (S(p) = 1/vp). The origin of the difference, which is effectively a lattice-induced renormalization, is the reduction of Lorentz (or O(4)) invariance to (hyper)-cubic invariance. The renormalization is finite and depends sp…
We consider QCD with one massless quark and one heavy quark in a finite volume of linear extent L_0 ~ 0.2 fm. In this situation, HQET represents an expansion in terms of 1/z=1/(m L_0), which we test by a non-perturbative computation of quenched current matrix elements and energies, taking the continuum limit of lattice results. These are seen to approach the corresponding renormalization group invariant matrix elements of the static effective theory as the quark mass becomes large. We are able to obtain estimates of the size of the 1/m-corrections to the static theory, which are also of practi…
We present results of a perturbative matching calculation performed at one-loop for heavy-light currents. We use the Fermilab action for the heavy quarks, the Asqtad action for the light quarks, and an improved gluon action. We also present results for heavy-heavy currents with Fermilab heavy quarks and improved glue.
We briefly review the strategy to perform non-perturbative heavy quark effective theory computations and we specialize to the case of the b quark mass which has recently been computed including the 1/m term.
Perturbative QCD has made significant progress over the last few decades. In the first part, we present an introductory overview of perturbative QCD as seen from a modern viewpoint. We explain the relation between purely perturbative predictions and predictions based on Wilsonian effective field theories. We also review progress of modern computational technologies and discuss intersection with frontiers of mathematics. Analyses of singularities in Feynman diagrams play key roles towards developing a unified view. In the second part, we discuss application of perturbative QCD, based on the for…
We study the heavy quark mass dependence of the perturbative renormalization factors for the heavy-light currents involving Wilson, Clover, and NRQCD heavy quarks. We find that the one-loop Z-factor for the axial-vector current depends significantly on the heavy quark mass commonly for all actions we study, while that for the vector current has smaller dependence.
We give an introduction to three topics in lattice gauge theory: I. The Schroedinger Functional and O(a) improvement. O(a) improvement has been reviewed several times. Here we focus on explaining the basic ideas in detail and then proceed directly to an overview of the literature and our personal assessment of what has been achieved and what is missing. II. The computation of the running coupling, running quark masses and the extraction of the renormalization group invariants. We focus on the basic strategy and on the large effort that has been invested in understanding the continuum limit. We…
We apply a recently introduced non-perturbative renormalization method to two types of lattice operators: the $ΔS=2$ four fermion operator and the heavy-light static axial current, which are relevant for the physics of $K$ and $B$ mesons respectively. The results of the non-perturbative calculations of the renormalization constants are compared with the corresponding perturbative ones.
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 study the heavy quark mass dependence of selected observables constructed from heavy-light meson correlation functions in small-volume two-flavour lattice QCD after taking the continuum limit. The light quark mass is tuned to zero, whereas the range of available heavy quark masses $m_h$ covers a region extending from around the charm to beyond the bottom quark mass scale. This allows entering the asymptotic mass-scaling regime as $1/m_h \to 0$ and performing well-controlled extrapolations to the infinite-mass limit. Our results are then compared to predictions obtained in the static limit o…
We present preliminary results of the non-perturbative heavy quark mass dependence of heavy-light meson observables in the continuum limit of finite-volume two-flavour lattice QCD. These observables, which are derived from heavy-light Schroedinger functional correlation functions and computed over a range of renormalization group invariant heavy quark masses from the charm to beyond the bottom region, allow for a quantitative comparison with the predictions of HQET and are of practical relevance for solving renormalization problems in HQET non-perturbatively by a matching to QCD in finite volu…
A one-loop matching calculation between Lattice NRQCD and full QCD for the heavy-light axial current is described and the effects on f_B are discussed.
Recently we have proved factorization of infrared divergences in NRQCD S-wave heavy quarkonium production at high energy colliders at all orders in coupling constant. One of the problem which still exists in the higher order pQCD calculation of color singlet P-wave heavy quarkonium production/anihillation is the appearance of non-canceling infrared divergences due to real soft gluons exchange, although no such infrared divergences are present in the color singlet S-wave heavy quarkonium. In this paper we find that since the non-perturbative matrix element of the color singlet P-wave heavy quar…
We propose to determine the coefficients in the Fermilab heavy quark action by matching the finite-volume, off-shell, gauge-fixed propagator and vertex functions with those determined in the exact, relativistic theory. The matching relativistic amplitudes may be determined either from short-distance perturbation theory or from finite-volume step-scaling recursion. Tree-level matching and non-perturbative off-shell amplitudes are presented.
We show that the Borel resummed perturbative static potential at N_f=0 converges well, and is in a remarkable agreement with the quenched lattice calculation at distances 1/r > 660 MeV. This shows that Borel resummation is very good at handling the renormalon in the static potential (and in the pole mass), and allows one to use the pole mass in perturbative calculation of heavy quark physics.
We determine the mass dependent renormalization as well as improvement coefficients for the heavy-light vector and axial-vector currents consisting of the relativistic heavy and the domain-wall light quarks through the standard matching procedure. The calculation is carried out perturbatively at the one loop level to remove the systematic error of $O(α_s (am_Q)^n a\p)$ as well as $O(α_s (am_Q)^n)$ ($n\ge$0), where $\p$ is a typical momentum scale in the heavy-light system. We point out that renormalization and improvement coefficients of the heavy-light vector current agree with those of the a…
The parametric error on the QCD-coupling can be a dominant source of uncertainty in several important observables. One way to extract the coupling is to compare high order perturbative computations with lattice evaluated moments of heavy quark two-point functions. The truncation of the perturbative series is a sizable systematic uncertainty that needs to be under control. In this contribution we give an update on our study arXiv:hep-lat/2203.07936v1 on this issue. We measure pseudo-scalar two-point functions in volumes of $L=2$ fm with twisted-mass Wilson fermions in the quenched approximation…
The static part of the heavy quark potential has been shown to be closely related to the spectrum of the rectangular Wilson loop. In particular the lowest lying positive frequency peak encodes the late time evolution of the two-body system, characterized by a complex potential. While initial studies assumed a perfect separation of early and late time physics, where a simple Lorentian (Breit-Wigner) shape suffices to describe the spectral peak, we argue that scale decoupling in general is not complete. Thus early time, i.e. non-potential effects, significantly modify the shape of the lowest pea…
Many lattice studies of heavy quark diffusion originate from a colour-electric correlator, obtained as a leading term after an expansion in the inverse of the heavy-quark mass. In view of the fact that the charm quark is not particularly heavy, we consider subleading terms in the expansion. Working out correlators up to $O(1/M^2)$, we argue that the leading corrections are suppressed by $O(T/M)$, and one of them can be extracted from a colour-magnetic correlator. The corresponding transport coefficient is non-perturbative already at leading order in the weak-coupling expansion, and therefore r…
We discuss the renormalization of the lattice formulation of the Heavy Quark Effective Theory (LHQET). In addition to wave function and composite operator renormalizations, on the lattice the classical velocity is also renormalized. The origin of this renormalization is the reduction of Lorentz (or O(4)) invariance to (hyper)cubic invariance. We present results of a new, direct lattice simulation of this finite renormalization, and compare the results to the perturbative (one loop) result. The simulation results are obtained with the use of a variationally optimized heavy-light meson operator,…
We apply the gradient flow on a color-electric two-point function that encodes the heavy quark momentum diffusion coefficient. The simulations are done on fine isotropic lattices in the quenched approximation at $1.5\,T_c$. The continuum extrapolation is performed at fixed flow time followed by a second extrapolation to zero flow time. Perturbative calculations of this correlation function under Wilson flow are used to enhance the extrapolations of the non-perturbative lattice correlator. The final estimate for the continuum correlator at zero flow time largely agrees with one obtained from a …
The QCD-coupling is a necessary input in the computation of many observables, and the parametric error on input parameters can be a dominant source of uncertainty. The coupling can be extracted by comparing high order perturbative computations and lattice evaluated moments of mesonic two-point functions with heavy quarks, which provide a high energy scale for perturbation theory. The truncation of the perturbative series is an important systematic uncertainty. We report on our attempt to study this issue by measuring pseudo-scalar two-point functions in volumes of $L=2\, \text{fm}$ with twiste…
Effects of heavy sea quarks on the low energy physics are described by an effective theory where the expansion parameter is the inverse quark mass, 1/$M$. At leading order in 1/$M$ (and neglecting light quark masses) the dependence of any low energy quantity on $M$ is given in terms of the ratio of $Λ$ parameters of the effective and the fundamental theory. We define a function describing the scaling with the mass $M$. We find that its perturbative expansion is very reliable for the bottom quark and also seems to work very well at the charm quark mass. The same is then true for the ratios of $…
We investigate the phase diagram of QCD with heavy quarks at finite temperature and chemical potential in the context of background field methods. In particular, we use a massive extension of the Landau-DeWitt gauge which is motivated by previous studies of the deconfinement phase transition in pure Yang-Mills theories. We show that a simple one-loop calculation is able to capture the richness of the phase diagram in the heavy quark region, both at real and imaginary chemical potential. Moreover, dimensionless ratios of quantities directly measurable in numerical simulations are in good agreem…
We discuss the perturbative $O(α_s a)$ matching in the static heavy and domain-wall light quark system. The gluon action is the Iwasaki action and the link smearing is performed in the static heavy action. The chiral symmetry of the light quark realized by using the domain-wall fermion formulation does not prohibit the mixing of the operators at $O(a)$. The application of $O(a)$ improvement to the actual data shows that the B meson decay constant $f_B$, the matrix elements ${\cal M}_B$ and the B parameter $B_B$ have non-negligible effects, while the effect on the SU(3) breaking ratio $ξ$ is sm…
We carry out a perturbative determination of mass dependent renormalization factors and $O(a)$ improvement coefficients for the vector and axial vector currents with a relativistic heavy quark action, which we have designed to control $m_Qa$ errors by extending the on-shell $O(a)$ improvement program to the case of $m_Q \gg Λ_{\rm QCD}$ with $m_Q$ the heavy quark mass. We discuss what kind of improvement operators are required for the heavy-heavy and the heavy-light cases under the condition that the Euclidean rotational symmetry is not retained anymore because of the $m_Qa$ corrections. Our c…
Preliminary results are presented in a step scaling determination of the coefficients in the relativistic heavy quark action. By matching finite volume, heavy-heavy and heavy-light meson masses, we attempt to determine the four parameters ($m$, $ζ$, $c_B$ and $c_E$) in the on-shell-improved, heavy quark action. In this report we carry out one step in this program by matching two physically equivalent systems. The first is a fully relativistic calculation using a $24^3\times 48$, $1/a=5.4$ GeV lattice with both the heavy and light quarks treated as domain wall fermions. The second calculation u…
We carry out the non-perturbative renormalization of the chromo-magnetic operator in Heavy Quark Effective Theory. At order 1/m of the expansion, the operator is responsible for the mass splitting between the pseudoscalar and vector B mesons. We obtain its two-loop anomalous dimension in a Schr"odinger functional scheme by successive one-loop conversions to the lattice MS scheme and the MS-bar scheme. We then compute the scale evolution of the operator non-perturbatively in the N_f=0 theory between $μ\approx 0.3$ GeV and $μ\approx 100$ GeV, where contact is made with perturbation theory. The o…
The temporal and spatial components of the heavy-light vector current and the spatial components of the axial current are expressed in terms of lattice-regulated operators suitable for simulations of B and D mesons. The currents are constructed by matching the appropriate scattering amplitudes in continuum QCD and a lattice model to one-loop order in perturbation theory. In the lattice theory, the heavy quarks are treated using the nonrelativistic (NRQCD) formulation and the light quarks are described by the tadpole-improved clover action. The light quarks are treated as massless. Our currents…
We present a study of the heavy-light spectrum and of the D- and B-meson decay constants. The results wer e obtained in the quenched approximation, by using the non-perturbatively improved Clover lattice action at beta=6.2, with a sample of 100 configurations, on a 24^3 x 64 lattice. After a careful analysis of th e systematic errors present in the extraction of the physical results, by assuming quite conservative discretization errors, we find f_Ds=231 +/- 12^{+6}_{-1} MeV, f_D = 211 +/- 14^{+0}_{-12} MeV, f_Ds/f_D=1.10(2), f_Bs = 204 +/- 16^{+28}_{-0} MeV, f_B = 179 +/- 18^{+26}_{-9} MeV, f_…
We present a method to non-perturbatively determine the parameters of the on-shell, $O(a)$-improved relativistic heavy quark action. These three parameters, $m_0$, $ζ$, and $c_B=c_E$ are obtained by matching finite-volume, heavy-heavy and heavy-light meson masses to the exact relativistic spectrum through a finite-volume, step-scaling recursion procedure. We demonstrate that accuracy on the level of a few percent can be achieved by carrying out this matching on a pair of lattices with equal physical spatial volumes but quite different lattice spacings. A fine lattice with inverse lattice spaci…
We extend a previous investigation of the QCD phase diagram with heavy quarks in the context of background field methods by including the two-loop corrections to the background field effective potential. The nonperturbative dynamics in the pure-gauge sector is modeled by a phenomenological gluon mass term in the Landau-DeWitt gauge-fixed action, which results in an improved perturbative expansion. We investigate the phase diagram at nonzero temperature and (real or imaginary) chemical potential. Two-loop corrections yield an improved agreement with lattice data as compared to the leading-order…
We present the results for a perturbative determination of mass dependent improvement coefficients $ν$, $r_s$, $c_E$ and $c_B$ in a relativistic heavy quark action, which we have designed to control $m_Qa$ errors by extending the on-shell $O(a)$ improvement program to the case of $m_Q \gg Λ_{\rm QCD}$, where $m_Q$ is the heavy quark mass. The parameters $ν$ and $r_s$ are determined from the quark propagator and $c_E$ and $c_B$ are from the on-shell quark-quark scattering amplitude. We show that all the parameters, together with the quark wave function and the mass renormalization factors, are …
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 investigate the Symanzik improvement of the Wilson quark action on anisotropic lattices. Taking first a general action with nearest-neighbor and clover interactions, we study the mass dependence of the ratio of the hopping parameters, the clover coefficients, and an improvement coefficient for heavy-light vector and axial vector currents. We show how tree-level improvement can be achieved. For a particular choice of the spatial Wilson coupling, the results simplify, and $O(m_0a_τ)$ improvement is possible. (Here $m_0$ is the bare quark mass and $a_τ$ the temporal lattice spacing.) With this…
We study the operator product expansion of the plaquette (gluon condensate) and the self-energy of an infinitely heavy quark. We first compute their perturbative expansions to order $α^{35}$ and $α^{20}$, respectively, in the lattice scheme. In both cases we reach the asymptotic regime where the renormalon behavior sets in. Subtracting the perturbative series, we obtain the leading non-perturbative corrections of their respective operator product expansions. In the first case we obtain the gluon condensate and in the second the binding energy of the heavy quark in the infinite mass limit. The …
During the last years, the ALPHA collaboration has been developing and implementing a method based on Heavy Quark Effective Theory (HQET) to compute B-mesons observables through lattice simulations. Thanks to a non-perturbative matching to QCD, the theory is renormalizable at any order of the heavy quark mass expansion. In order to extract precisely the relevant matrix elements and masses, we use all-to-all propagators and solve an generalized eigenvalue problem (GEVP). We have shown in the quenched approximation that quantities like the b-quark mass mb, the heavy-light decay constant(s) or th…
We study the anisotropic lattice QCD for precision computations of heavy-light matrix elements. Our previous study in which the lattices are calibrated with a few percent accuracy has already given results comparable to the existing calculations. This suggests that even higher precision may be achieved by a more precise calibration of anisotropic lattices. We describe our strategy to tune the gauge and quark parameters with accuracies much less than 1 % in the quenched approximation.
We report on a non-perturbative determination of the parameters of the lattice Heavy Quark Effective Theory (HQET) Lagrangian and of the time component of the heavy-light axial-vector current with Nf=2 flavors of massless dynamical quarks. The effective theory is considered at the 1/mh order, and the heavy mass mh covers a range from slightly above the charm to beyond the beauty region. These HQET parameters are needed to compute, for example, the b-quark mass, the heavy-light spectrum and decay constants in the static approximation and to order 1/mh in HQET. The determination of the parameter…
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 study effective energies of heavy-light meson correlation functions in lattice QCD and a small volume of (0.2 fm)^4 to non-perturbatively calculate their dependence on the heavy quark mass in the continuum limit. Our quenched results obtained here constitute an essential intermediate step of a first fully non-perturbative computation of the b-quark's mass in the static approximation that has recently been presented as an application of a new proposal to non-perturbatively renormalize the Heavy Quark Effective Theory. The renormalization constant and the improvement coefficients relating the…
The extraction of the finite temperature heavy quark potential from lattice QCD relies on a spectral analysis of the real-time Wilson loop. Through its position and shape, the lowest lying spectral peak encodes the real and imaginary part of this complex potential. We benchmark this extraction strategy using leading order hard-thermal loop (HTL) calculations. I.e. we analytically calculate the Wilson loop and determine the corresponding spectrum. By fitting its lowest lying peak we obtain the real- and imaginary part and confirm that the knowledge of the lowest peak alone is sufficient for obt…
We calculate one-loop renormalization factors for heavy-light bilinears as well as four-fermion operators relevant for $B^{0} - \bar{B}^{0}$ mixing calculations on the lattice. We use the static approximation for heavy quarks and the domain-wall formulation for light quarks. We present results for different choices of improved gauge action.
We compute the B^*Bπcoupling \hat{g}_{\infty} for static heavy-light meson using all-to-all propagators. It is shown that low-mode averaging with 100 low-lying eigenmodes indeed improves the signal for the 2-point and 3-point functions for heavy-light meson significantly. Our study suggests that the all-to-all propagator will be a very efficient method for high precision computation of the B^*Bπcoupling especially in unquenched QCD where the number of configurations is limited.
We discuss the application of the regularization independent (RI) scheme of Rome/Southampton to determine the normalization of heavy quark operators non-perturbatively using the Fermilab action.
