We consider the distribution of the complex phase of the fermion determinant in QCD at nonzero chemical potential and examine the physical conditions under which the distribution takes a Gaussian form. We then calculate the baryon number as a function of the complex phase of the fermion determinant and show 1) that the exponential cancellations produced by the sign problem take the form of total derivatives 2) that the full baryon number is orthogonal to this noise. These insights allow us to define a self-consistency requirement for measurements of the baryon number in lattice simulations.
We investigate how the derivative expansion in the HAL QCD method works to extract physical observables, using a separable potential in quantum mechanics, which is solvable but highly non-local in the coordinate system. We consider three cases for inputs to determine the HAL QCD potential in the derivative expansion, (1) energy eigenfunctions (2) time-dependent wave functions as solutions to the time dependent Schrödinger equation with some boundary conditions (3) time-dependent wave function made by a linear combination of finite number of eigenfunctions at low energy to mimic the finite volu…
By matching Wilson loop ratios on anisotropic lattices we measure the coefficients $\cs$ and $\ct$, which are required for the calculation of the energy density. The results are compared to that of an indirect method of determination. We find similar behaviour, the differences are attributed to different discretization errors.
We study the convergence of the derivative expansion in HAL QCD method from the finite volume analysis. Employing the (2+1)-flavor lattice QCD data obtained at nearly physical light quark masses $(m_π, m_K) \simeq (146, 525)$ MeV and the physical charm quark mass, we study two representative systems, $ΩΩ$ and $Ω_{ccc}Ω_{ccc}$ in the $^1S_0$ channel, where both systems were found to have a shallow bound state in our previous studies. The HAL QCD potentials are determined at the leading-order in the derivative expansion, from which finite-volume eigenmodes are obtained. Utilizing the eigenmode p…
A formalism is given to hermitize the HAL QCD potential, which needs to be non-hermitian except the leading order (LO) local term in the derivative expansion as the Nambu-Bethe-Salpeter (NBS) wave functions for different energies are not orthogonal to each other. It is shown that the non-hermitian potential can be hermitized order by order to all orders in the derivative expansion. In particular, the next-to-leading order (NLO) potential can be exactly hermitized without approximation. The formalism is then applied to a simple case of $ΞΞ(^{1}S_{0}) $ scattering, for which the HAL QCD calculat…
We investigate the behavior of the chiral condensate in lattice QCD at finite temperature and finite chemical potential. The study was done using two flavors of light quarks and with a series of $β$ and $ma$ at the lattice size $24\times12^{2}\times6$. The calculation was done in the Taylar expansion formalism. We are able to calculate the first and second order derivatives of $\langle\barψψ\rangle$ in both isoscalar and isovector channels. With the first derivatives being small, we find that the second derivatives are sizable close to the phase transition and the magnitude of $\barψψ$ decreas…
We examine the properties of the wave-function-equivalent potentials which HAL QCD collaboration has introduced. We generalize the derivative expansion, and then apply it to energy-independent and non-local potentials in a coupled-channel model. We observe that the expansion converges by comparing the scattering phase shifts computed from these potentials with the exact one. We have also found that the convergence can be improved by either varying the choice of interpolating fields or tuning the expansion scale in the generalized derivative expansion. The results will be utilized in future lat…
We present perturbative and non-perturbative results on the renormalization constants of the local and one-derivative vector and axial vector operators. Non-perturbative results are obtained using the twistedmassWilson fermion formulation employing two degenerate dynamical quarks and the tree-level Symanzik improved gluon action for pion masses in the range of about 450-260 MeV and at there values of the lattice spacing, namely 0.055 fm, 0.070 fm and 0.089 fm. Subtraction of O(a^2) terms is carried out by performing the perturbative evaluation of these operators at 1- loop and up to O(a^2). Th…
A Lattice QCD approach to quark orbital angular momentum in the proton based on generalized transverse momentum-dependent parton distributions (GTMDs) is enhanced methodologically by incorporating a direct derivative technique. This improvement removes a significant numerical bias that had been seen to afflict results of a previous study. In particular, the value obtained for Ji quark orbital angular momentum is reconciled with the one obtained independently via Ji's sum rule, validating the GMTD approach. Since GTMDs simultaneously contain information about the quark impact parameter and tran…
In this report, we discuss some theoretical and practical progresses in the HAL QCD potential method. We first clarify the issue of the derivative expansion for the non-local potential in the HAL QCD method. As the non-local potential in the original literature is not uniquely defined, we propose a procedure to define a non-local potential from NBS wave functions in terms of the derivative expansion. We then demonstrate how this definition works by using quantum mechanics with a separable potential. Secondly we discuss an issue of Hermiticity of the HAL QCD potential. Since the NBS wav functio…
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 present a direct calculation for the first derivative of the isovector nucleon form factors with respect to the momentum transfer $q^2$ using the lower moments of the nucleon 3-point function in the coordinate space. Our numerical simulations are performed using the $N_f = 2 + 1$ nonperturbatively $O(a)$-improved Wilson quark action and Iwasaki gauge action near the physical point, corresponding to the pion mass $M_π=138$ MeV, on a (5.5 fm)$^4$ lattice at a single lattice spacing of $a = 0.085$ fm. In the momentum derivative approach, we can directly evaluate the mean square radii for the e…
We discuss the connection between the contributions of large field configurations and the large order behavior of perturbation theory. For quenched QCD, the sensitivity of the average plaquette to a removal of large field configurations has a narrow peak near beta=5.6. Various analysis of the order 10 weak coupling series for the plaquette give robust indications for a singularity in the third derivative of the free energy (second derivative of the plaquette) with respect to beta, near beta = 5.7. We report results of numerical calculations. The peak in the third derivative of the free energy …
We study the topological properties of full QCD with four flavours of dynamical staggered fermions. In particular the topological susceptibility is measured and the problem of the determination of its first derivative is discussed.
We propose a new method to investigate the thermal properties of QCD with a small chemical potential $(μ)$. The derivatives of the phase transition point with respect to $μ$ are computed for 2 flavors of p4-improved staggered fermions. We moreover comment on the complex phase of the fermion determinant in finite-density QCD.
Knowledge of the derivative of the topological susceptibility at zero momentum is important for assessing the validity of the Witten-Veneziano formula for the eta' mass, and likewise for the resolution of the EMC proton spin problem. We investigate the momentum dependence of the topological susceptibility and its derivative at zero momentum using overlap fermions in quenched lattice QCD simulations. We expose the role of the low-lying Dirac eigenmodes for the topological charge density, and find a negative value for the derivative. While the sign of the derivative is consistent with the QCD su…
We study the equation of state in two-flavor QCD at finite temperature and density. Simulations are made with the RG-improved gluon action and the clover-improved Wilson quark action. Along the lines of constant physics for $m_{\rm PS}/m_{\rm V} = 0.65$ and 0.80, we compute the derivatives of the quark determinant with respect to the quark chemical potential $μ_q$ up to the fourth order at $μ_q=0$. We adopt several improvement techniques in the evaluation. We study thermodynamic quantities and quark number susceptibilities at finite $μ_q$ using these derivatives. We find enhancement of the qua…
We describe a procedure for extracting momentum derivatives of nucleon matrix elements on the lattice directly at $Q^2=0$. This is based on the Rome method for computing momentum derivatives of quark propagators. We apply this procedure to extract the nucleon isovector magnetic moment and charge radius as well as the isovector induced pseudoscalar form factor at $Q^2=0$ and the axial radius. For comparison, we also determine these quantities with the traditional approach of computing the corresponding form factors, i.e. $G^v_E(Q^2)$ and $G_M^v(Q^2)$ for the case of the vector current and $G_P^…
The leading order hadronic contribution to the muon magnetic moment anomaly, $a^{HAD}_μ$, is determined entirely in the framework of QCD. The result in the light-quark sector, in units of $10^{-10}$, is $a^{HAD}_μ|_{uds} =686 \pm 26$, and in the heavy-quark sector $a^{HAD}_μ|_{c} =14.4 \pm 0.1$, and $a^{HAD}_μ|_{b} =0.29 \pm 0.01$, resulting in $a^{HAD}_μ= 701 \pm 26$. The main uncertainty is due to the current lattice QCD value of the first and second derivative of the electromagnetic current correlator at the origin. Expected improvement in the precision of these derivatives may render this …
We discuss lattice methods to obtain the derivatives of a lattice meson mass with respect to the bare sea and valence quark masses. Applications are made to quenched and dynamical fermion configurations. We find evidence for significant differences between quenched and dynamical fermion configurations. We discuss how to relate dependence on the bare lattice parameters to more phenomenologically useful quantities.
The Nambu-Bethe-Salpeter (NBS) wave function for two nucleons on the lattice has been shown to yield a non-local and energy-independent nucleon-nucleon (NN) potential, U(r,r'). In practice, the derivative expansion of U(r,r') is currently employed to determine the potential at low energies. In this report, we study the magnitude of non-locality to check the convergence of such a derivative expansion. With quenched lattice QCD at m_π= 530MeV, we compare the NN potentials at the center of mass energy E ~ 0 MeV and at E ~ 45 MeV. We also investigate the angular momentum dependence of the spin sin…
Susskind's continuous-time fermions, with two flavours, can be latticized using a one-sided time derivative. We are presently investigating the interacting case, where we hope to find the onset at finite $μ$ at the right place due to the reduced number of flavours. As for these fermions there is only a discrete chiral symmetry left over, the lightness of pions in the broken phase has to be investigated.
We propose a new approach to finite density QCD based on a histogram method with phase quenched simulations at finite chemical potential. Integrating numerically the derivatives of the logarithm of the quark determinant with respect to the chemical potential, we calculate the reweighting factor and the complex phase of the quark determinant. The complex phase is handled with a cumulant expansion to avoid the sign problem. We examine the applicability of this method.
We give an update of results from two-colour, two-flavour QCD. Using a Wilson fermion action we calculate thermodynamic quantities as a function of chemical potential μ. Calculating the Karsch Coefficients non-perturbatively gives us access to the derivative method. Compared to our previously published results, we have improved our analysis leading to revised and more accurate estimates for the renormalised energy density, pressure and the trace anomaly.
We present results of a simulation of QCD on a 4x16^3 lattice with 2 continuum flavors of p4-improved staggered fermion with mass m/T=0.4. Derivatives of the thermodynamic grand potential with respect to quark chemical potential mu_q up to fourth order are calculated, enabling estimates of the pressure, quark number density and associated susceptibilities as functions of mu_q via Taylor series expansion. Discretisation effects associated with various staggered fermion formulations are discussed in some detail. In addition it is possible to estimate the radius of convergence of the expansion as…
We study the thermal properties of QCD in the presence of a small quark chemical potential $μ$. Derivatives of the phase transition point with respect to $μ$ are computed at $μ=0$ for 2 and 3 flavors of p-4 improved staggered fermions on a $16^3\times4$ lattice. Moreover we contrast the case of isoscalar and isovector chemical potentials, quantify the effect of $μ\not=0$ on the equation of state, and comment on the screening effect by dynamical quarks and the complex phase of the fermion determinant in QCD with $μ\not=0$.
We present direct representations of the scaling functions of the 3d O(4) model which are relevant for comparisons to other models, in particular QCD. This is done in terms of expansions in the scaling variable z=t/h^{1/βδ}. The expansions around z=0 and the corresponding asymptotic ones for z --> +/- infty, overlap such that no interpolation is needed. We explicitly present the expansion coefficients which have been determined numerically from data of a previous high statistics simulation of the O(4) model on a three-dimensional lattice of linear extension L=120. This allows to derive smooth …
We discuss the phase structure and the equation of state for QCD at non-zero temperature and density. Derivatives of $\ln Z$ with respect to quark chemical potential $μ_q$ up to fourth order are calculated for 2-flavor QCD, enabling estimates of the pressure, quark number density and associated susceptibilities as functions of $μ_q$ via a Taylor series expansion. Also, the phase transition line for 2 and 3-flavor QCD and the critical endpoint in the $(T, μ_q)$ plane are investigated in the low density regime.
We study the susceptibilities of quark number, isospin number and electric charge in numerical simulations of lattice QCD at high temperature and density. We discuss the equation of state for 2 flavor QCD at non-zero temperature and density. Derivatives of $\ln Z$ with respect to quark chemical potential $(μ_q)$ are calculated up to sixth order. From this Taylor series, the susceptibilities are estimated as functions of temperature and $μ_q$. Moreover, we comment on the hadron resonance gas model, which explains well our simulation results below $T_c$.
We compute derivatives of thermodynamic quantities with respect to $μ$, at $μ=0$ for 2 and 3 flavors of degenerate quark masses. This allows us to estimate the phase transition line in the $T,μ$ plane and quantify the influence of a non vanishing chemical potential on the equation of state by computing lines of constant energy, pressure and density. Moreover we evaluate the order of the QCD phase transition by measuring the Binder Cumulant of the chiral condensate. This gives access to the chiral critical point on the phase transition line.
The leading order hadronic contribution to the muon g-2, $a_μ^{HAD}$, is determined entirely from theory using an approach based on Cauchy's theorem in the complex squared energy s-plane. This is possible after fitting the integration kernel in $a_μ^{HAD}$ with a simpler function of $s$. The integral determining $a_μ^{HAD}$ in the light-quark region is then split into a low energy and a high energy part, the latter given by perturbative QCD (PQCD). The low energy integral involving the fit function to the integration kernel is determined by derivatives of the vector correlator at the origin, p…
Physical observables, such as the scattering phase shifts and the binding energies, calculated from the non-local HAL QCD potential do not depend on the sink operators used to define the potential. This is called the scheme independence of the HAL QCD method. In practical applications, the derivative expansion of the non-local potential is employed, so that physical observables may receive some scheme dependence at given order of the expansion. In this paper, we compare the $I=2$ $ππ$ scattering phase shifts obtained in the point-sink scheme (the standard scheme in the HAL QCD method) and the …
We study the equation of state at finite temperature and density in two-flavor QCD with the RG-improved gluon action and the clover-improved Wilson quark action on a $ 16^3 \times 4$ lattice. Along the lines of constant physics at $m_{\rm PS}/m_{\rm V} = 0.65$ and 0.80, we compute the second and forth derivatives of the grand canonical partition function with respect to the quark chemical potential $μ_q = (μ_u+μ_d)/2$ and the isospin chemical potential $μ_I = (μ_u-μ_d)/2$ at vanishing chemical potentials, and study the behaviors of thermodynamic quantities at finite $μ_q$ using these derivativ…
In this article, we review the HAL QCD method to investigate baryon-baryon interactions such as nuclear forces in lattice QCD. We first explain our strategy in detail to investigate baryon-baryon interactions by defining potentials in field theories such as QCD. We introduce the Nambu-Bethe-Salpeter (NBS) wave functions in QCD for two baryons below the inelastic threshold. We then define the potential from NBS wave functions in terms of the derivative expansion, which is shown to reproduce the scattering phase shifts correctly below the inelastic threshold. Using this definition, we formulate …
We determine the curvature of the phase transition line in the mu-T plane through an analysis of various observables, including the Polyakov loop, the quark number susceptibilities and the susceptibility of the chiral condensate. The second derivative of these quantities with respect to mu was calculated. The measurements were carried out on N_T = 4,6,8 and 10 lattices generated with a Symanzik improved gauge and stout-link improved 2+1 flavour staggered fermion action using physical quark masses.
We discuss the equation of state for 2 flavor QCD at non-zero temperature and density. Derivatives of $\ln Z$ with respect to quark chemical potential $μ_q$ up to fourth order are calculated, enabling estimates of the pressure, quark number density and associated susceptibilities as functions of $μ_q$ via a Taylor series expansion. It is found that the fluctuations in the quark number density increase in the vicinity of the phase transition temperature and the susceptibilities start to develop a pronounced peak as $μ_q$ is increased. This suggests the presence of a critical endpoint in the $(T…
We discuss universal scaling properties of (2+1)-flavor QCD in the vicinity of the chiral phase transition at vanishing as well as non-vanishing light quark chemical potential (mu_l). We provide evidence for O(N) scaling of the chiral order parameter in (2+1)-flavor QCD and show that the scaling analysis of its derivative with respect to the light quark chemical potential provides a unique approach to the determination of the curvature of the chiral phase transition line in the vicinity of mu_l/T=0.
Thermodynamic properties of the QCD nonperturbative vacuum with two light quarks are studied. It is shown that at low temperatures T<M_πrelativistic massive pions can be treated within the dilute gas approximation. Analytic temperature dependence of the quark condensate is found in perfect agreement with the numerical calculations obtained at the three-loop level of the chiral perturbation theory with non-zero quark mass. The gluon condensate slightly varies with the increase of the temperature. It is shown that the temperature derivatives of the anomalous and normal (quark massive term) contr…
We calculate several diagonal and non-diagonal fluctuations of conserved charges in a system of 2+1+1 quark flavors with physical masses, on a lattice with size $48^3\times12$. Higher order fluctuations at $μ_B=0$ are obtained as derivatives of the lower order ones, simulated at imaginary chemical potential. From these correlations and fluctuations we construct ratios of net-baryon number cumulants as functions of temperature and chemical potential, which satisfy the experimental conditions of strangeness neutrality and proton/baryon ratio. Our results qualitatively explain the behavior of the…
We discuss the equation of state for QCD at non-zero temperature and density. We present results of a simulation for QCD with 2 flavors of p4-improved staggered fermions. Derivatives of $\ln {\cal Z}$ with respect to quark chemical potential $μ_q$ up to fourth order are calculated, enabling estimates of the pressure, quark number density and associated susceptibilities as functions of $μ_q$ via a Taylor series expansion. We also discuss the radius of convergence of the expansion as a function of temperature. It is found that the fluctuations in the quark number density increase in the vicinity…
We analyze the vacuum structure of N=2, SU(2) QCD with massive quark hypermultiplets, once supersymmetry is softly broken down to N=0 with dilaton and mass spurions. We give general expressions for the low energy couplings of the effective potential in terms of elliptic functions to have a complete numerical control of the model. We study in detail the possible phases of the theories with Nf = 1, 2 flavors for different values of the bare quark masses and the supersymmetry breaking parameters and we find a rich structure of first order phase transitions. The chiral symmetry breaking pattern of…
Both direct and HAL QCD methods are currently used to study the hadron interactions in lattice QCD. In the direct method, the eigen-energy of two-particle is measured from the temporal correlation. Due to the contamination of excited states, however, the direct method suffers from the fake eigen-energy problem, which we call the "mirage problem," while the HAL QCD method can extract information from all elastic states by using the spatial correlation. In this work, we further investigate systematic uncertainties of the HAL QCD method such as the quark source operator dependence, the convergenc…
The chiral susceptibility, or the first derivative of the chiral condensate with respect to the quark mass, is often used as a probe for the QCD phase transition since the chiral condensate is an order parameter of $SU(2)_L \times SU(2)_R$ symmetry breaking. However, the chiral condensate also breaks the axial $U(1)$ symmetry, which is usually not paid attention to as it is already broken by anomaly and apparently gives little impact on the transition. We investigate the susceptibilities in the scalar and pseudoscalar channels in order to quantify how much the axial $U(1)$ breaking contributes…
We study scaling behavior of a chiral order parameter performing a simulation of two-flavor QCD with improved Wilson quarks. It has been shown that the scaling behavior of the chiral order parameter defined by a Ward-Takahashi identity agrees with the scaling function of the three-dimensional O(4) spin model at zero chemical potential. We extend the scaling study to finite density QCD. Calculating derivatives of the chiral order parameter with respect to the chemical potential in two-flavor QCD, the scaling property of chiral phase transition is discussed in the low density region. We moreover…
We discuss the equation of state (EoS) for two flavor QCD at non-zero temperature and density. Derivatives of $\ln Z$ with respect to quark chemical potential $μ_q$ are calculated up to sixth order. From this Taylor series, the pressure, quark number density and associated susceptibilities are estimated as functions of temperature and $μ_q$. It is found that the fluctuations in the quark number density increase in the vicinity of the phase transition temperature and the susceptibilities start to develop a pronounced peak as $μ_q$ is increased. This suggests the presence of a critical endpoint …
We consider the renormalisation of composite quark-antiquark operators with one and two lattice covariant derivatives related to the lowest moments of generalised parton distributions (GPDs) and meson distribution amplitudes (DAs). Their matrix elements are calculated in one-loop lattice perturbation theory for non-zero momentum transfer. Using clover and overlap fermions we present the resulting matrices of mixing and renormalisation factors. For overlap fermions we explicitly check the absence of mixing with lower-dimensional operators of different chirality in particular representations of …
We study scaling behavior of a chiral order parameter in the low density region, performing a simulation of two-flavor QCD with improved Wilson quarks. The scaling behavior of the chiral order parameter defined by a Ward-Takahashi identity agrees with the scaling function of the three-dimensional O(4) spin model at zero chemical potential. We extend the scaling study to finite density QCD. Applying the reweighting method and calculating derivatives of the chiral order parameter with respect to the chemical potential, the scaling properties of the chiral phase transition are discussed in the lo…
We investigate the critical endline of the finite temperature phase transition of QCD around the SU(3)-flavor symmetric point at zero chemical potential. We employ the renormalization-group improved Iwasaki gauge action and non-perturbatively $O(a)$-improved Wilson-clover fermion action. The critical endline is determined by using the intersection point of kurtosis, employing the multi-parameter, multi-ensemble reweighting method to calculate observables off the SU(3)-symmetric point, at the temporal size $N_{\rm T}$=6 and lattice spacing as low as $a \approx 0.19$ fm. We confirm that the slop…
The $ΞΞ$ interaction in the $^1$S$_0$ channel is studied to examine the convergence of the derivative expansion of the non-local HAL QCD potential at the next-to-next-to-leading order (N$^2$LO). We find that (i) the leading order potential from the N$^2$LO analysis gives the scattering phase shifts accurately at low energies, (ii) the full N$^2$LO potential gives only small correction to the phase shifts even at higher energies below the inelastic threshold, and (iii) the potential determined from the wall quark source at the leading order analysis agrees with the one at the N$^2$LO analysis e…
We propose a new method to investigate the thermal properties of QCD with a small quark chemical potential $μ$. Derivatives of the phase transition point with respect to $μ$ are computed at $μ=0$ for 2 flavors of p-4 improved staggered fermions with $ma=0.1,0.2$ on a $16^3\times4$ lattice. The resulting Taylor expansion is well behaved for the small values of $μ_{\rm q}/T_c\sim0.1$ relevant for RHIC phenomenology, and predicts a critical curve $T_c(μ)$ in reasonable agreement with estimates obtained using exact reweighting. In addition, we contrast the case of isoscalar and isovector chemical …
