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Equivariant Graph Neural Networks (GNNs) that incorporate E(3) symmetry have achieved significant success in various scientific applications. As one of the most successful models, EGNN leverages a simple scalarization technique to perform equivariant message passing over only Cartesian vectors (i.e., 1st-degree steerable vectors), enjoying greater efficiency and efficacy compared to equivariant GNNs using higher-degree steerable vectors. This success suggests that higher-degree representations might be unnecessary. In this paper, we disprove this hypothesis by exploring the expressivity of equivariant GNNs on symmetric structures, including $k$-fold rotations and regular polyhedra. We theoretically demonstrate that equivariant GNNs will always degenerate to a zero function if the degree of the output representations is fixed to 1 or other specific values. Based on this theoretical insight, we propose HEGNN, a high-degree version of EGNN to increase the expressivity by incorporating high-degree steerable vectors while maintaining EGNN's efficiency through the scalarization trick. Our extensive experiments demonstrate that HEGNN not only aligns with our theoretical analyses on toy datasets consisting of symmetric structures, but also shows substantial improvements on more complicated datasets such as $N$-body and MD17. Our theoretical findings and empirical results potentially open up new possibilities for the research of equivariant GNNs.

The Roper state is extracted with valence overlap fermions on a $2+1$-flavor domain-wall fermion lattice (spacing $a = 0.114$ fm and $m_{\pi} = 330$ MeV) using both the Sequential Empirical Bayes (SEB) method and the variational method. The results are consistent, provided that a large smearing-size interpolation operator is included in the variational calculation to have better overlap with the lowest radial excitation. Similar calculations carried out for an anisotropic clover lattice with similar parameters find the Roper $\approx 280$ MeV higher than that of the overlap fermion. The fact that the prediction of the Roper state by overlap fermions is consistently lower than those of clover fermions, chirally improved fermions, and twisted-mass fermions over a wide range of pion masses has been dubbed a "Roper puzzle." To understand the origin of this difference, we study the hairpin $Z$-diagram in the isovector scalar meson ($a_0$) correlator in the quenched approximation. Comparing the $a_0$ correlators for clover and overlap fermions, at a pion mass of 290 MeV, we find that the spectral weight of the ghost state with clover fermions is smaller than that of the overlap at $a = 0.12$ fm and $0.09$ fm, whereas the whole $a_0$ correlators of clover and overlap at $a = 0.06$ fm coincide within errors. This suggests that chiral symmetry is restored for clover at $a \le 0.06$ fm and that the Roper should come down at and below this $a$. We conclude that this work supports a resolution of the "Roper puzzle" due to $Z$-graph type chiral dynamics. This entails coupling to higher components in the Fock space (e.g. $N\pi$, $N\pi\pi$ states) to induce the effective flavor-spin interaction between quarks as prescribed in the chiral quark model, resulting in the parity-reversal pattern as observed in the experimental excited states of $N, \Delta$ and $\Lambda$., Comment: 15 pages, 16 figures, revised manuscript accepted for publication in Phys. Rev. D

In the calculation of Positron Emission Tomography (PET) image reconstruction, System Response Matrix (SRM) characterizes the numerical relationship between measurement space and image space. Due to a significant amount of calculation and low efficiency of the current SRM algorithm, the application of iterative reconstruction in the medical and industrial fields is limited. The time and space of computing SRM are the most important factors that researchers are concerned about. In this paper, the Straight Line Truncation (hereinafter referred to as SLT) method is applied to calculate the SRM. We verified that it improves the speed of calculating the SRM and reduces its spatial complexity. The experimental results show that the forward projection speed of SLT algorithm is six times more than that of the traditional Ray-Tracing algorithm. Besides, the forward projection speed of the small angle of SLT is about 23 times that of the traditional Ray-Tracing algorithm, and the storage space of SLT is about one sixty-fourth of the storage space of traditional Ray-Tracing algorithm.

Metabolic imaging with PET/CT using $^{18}$F-Fludeoxyglucose ($^{18}$F-FDG) as well as other imaging biomarkers has achieved wide acceptance in oncology, cardiology and neurology not only because of the unique metabolic information generated by this modality, but also because of its ability to quantify biological processes. However, PET quantification is affected by many technical and physiologic factors, and then recognized as an important problem for diagnosis, determination of prognosis, and response monitoring in oncology. In this work, we investigated the effect of reduced PET emission count statistics on the accuracy and precision of tracer quantification, and proposed Inhomogeneous Poisson Process Parameter Estimation (I3PE) method. In I3PE method, we modelled the coincidence event as Inhomogeneous Poisson Process, and estimate its parameter directly from the streaming listmode data. To evaluate the effectiveness, a experiment using $^{18}$F-FDG was implemented on LIGHTNING. The results not only demonstrate I3PE method, but also evaluated the minimal detectable activity of the using PET machine. According $0.3\%$ mean error rate criterion, LIGHTNING can detect several nano-Curie, cooperated with I3PE method.

Si-Photomultiplier (SiPM) has several attracting features, which can be helpful in the communication field, such as high photon detection efficiency, fast transient response, excellent timing resolution, and wide spectral range. In this paper, we compare SiPM with photodiode (PD), Avalanche photodiodes (APD), and Photomultiplier tube (PMT) in terms of the transformation related performance, and then implement Si-Photomultiplier (SiPM) based visible light communication (VLC) system driven with visible light light-emitting diode (LED). The system throughout ability, transmission rate, data reconstruction, and required AD sample rate are evaluated. The encouraging results suggest that the SiPM receiver has great application potentials, such as optical wireless communication systems and light fidelity, in which a wide bandwidth of the sensor response is important to enhance the transfer rate.

We use overlap fermions as valence quarks to calculate meson masses in a wide quark mass range on the $2+1$-flavor domain-wall fermion gauge configurations generated by the RBC and UKQCD Collaborations. The well-defined quark masses in the overlap fermion formalism and the clear valence quark mass dependence of meson masses observed from the calculation facilitate a direct derivation of physical current quark masses through a global fit to the lattice data, which incorporates $O(a^2)$ and $O(m_c^4a^4)$ corrections, chiral extrapolation, and quark mass interpolation. Using the physical masses of $D_s$, $D_s^*$ and $J/\psi$ as inputs, Sommer's scale parameter $r_0$ and the masses of charm quark and strange quark in the $\overline{\rm MS}$ scheme are determined to be $r_0=0.465(4)(9)$ fm, $m_c^{\overline{\rm MS}}(2\,{\rm GeV})=1.118(6)(24)$ GeV (or $m_c^{\overline{\rm MS}}(m_c)=1.304(5)(20)$ GeV), and $m_s^{\overline{\rm MS}}(2\,{\rm GeV})=0.101(3)(6)\,{\rm GeV}$, respectively. Furthermore, we observe that the mass difference of the vector meson and the pseudoscalar meson with the same valence quark content is proportional to the reciprocal of the square root of the valence quark masses. The hyperfine splitting of charmonium, $M_{J/\psi}-M_{\eta_c}$, is determined to be 119(2)(7) MeV, which is in good agreement with the experimental value. We also predict the decay constant of $D_s$ to be $f_{D_s}=254(2)(4)$ MeV. The masses of charmonium $P$-wave states $\chi_{c0}, \chi_{c1}$ and $h_c$ are also in good agreement with experiments., Comment: 19 pages, 15 figures, revised version accepted for publication in PRD

We carried out a calculation of the Roper state with the Sequential Empirical Bayesian (SEB) method with overlap valence fermion on 2+1-flavor domain-wall fermion configurations on the 24^3 x 64 lattice with a^{-1} = 1.73 GeV. The light sea quark mass corresponds to a pion mass of 330 MeV. The mass of the Roper, chirally extrapolated to the physical pion mass, is 1404(112) MeV which is consistent with the experimental value at 1440 MeV. When compared to the Roper state calculation with variational method for Clover and twisted mass fermions, it is found that the Roper states from SEB with overlap fermion are systematically lower by 400 - 800 MeV for all the quark masses ranging from light to the strange mass region. We study the origin of the difference by exploring the size of the interpolation field in relation to the radial wavefunction of the Roper and also the dynamical influence of the higher Fock space., Comment: 7 pages, 6 figures, presented at the 31st International Symposium on Lattice Field Theory (LATTICE 2013), 29 July - 3 August 2013, Mainz, Germany

We present renormalization constants of overlap quark bilinear operators on 2+1-flavor domain wall fermion configurations. This setup is being used by the chiQCD collaboration in calculations of physical quantities such as strangeness in the nucleon and the strange and charm quark masses. The scale independent renormalization constant for the axial vector current is computed using the Ward Identity. The renormalization constants for scalar, pseudoscalar and vector current are calculated in the RI-MOM scheme. Results in the MS-bar scheme are also given. The step scaling function of quark masses in the RI-MOM scheme is computed as well. The analysis uses, in total, six different ensembles of three sea quarks each on two lattices with sizes 24^3x64 and 32^3x64 at spacings a=(1.73 GeV)^{-1} and (2.28 GeV)^{-1}, respectively., Comment: 26 pages, 17 figures. More discussions on O(4) breaking effects, and on the perturbative running and a^2p^2 extrapolation of Zs. A subsection for the calculation of the step scaling function of quark mass is added. References added. Version to appear in PRD

We investigate the phase diagram in the temperature, imaginary chemical potential plane for QCD with three degenerate quark flavors using Wilson type fermions. While more expensive than the staggered fermions used in past studies in this area, Wilson fermions can be used safely to simulate systems with three quark flavors. In this talk, we focus on the (pseudo)critical line that extends from $\mu=0$ in the imaginary chemical potential plane, trace it to the Roberge-Weiss line, and determine its location relative to the Roberge-Weiss transition point. In order to smoothly follow the (pseudo)critical line in this plane we perform a multi-histogram reweighting in both temperature and chemical potential. To perform reweighting in the chemical potential we use the compression formula to compute the determinants exactly. Our results are compatible with the standard scenario., Comment: 7 pages, 5 figures. Proceedings of the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, Germany

We present renormalization constants of overlap quark bilinear operators on 2+1-flavor domain wall fermion configurations. Both overlap and domain wall fermions have chiral symmetry on the lattice. The scale independent renormalization constant for the local axial vector current is computed using a Ward Identity. The renormalization constants for the scalar, pseudoscalar and vector current are calculated in the RI-MOM scheme. Results in the MS-bar scheme are obtained by using perturbative conversion ratios. The analysis uses in total six ensembles with lattice sizes 24^3x64 and 32^3x64., Comment: 7 pages, 10 figures, presented at the 31st International Symposium on Lattice Field Theory (LATTICE 2013), 29 July - 3 August 2013, Mainz, Germany

We study quantum critical behavior in three dimensional lattice Gross-Neveu models containing two massless Dirac fermions. We focus on two models with SU(2) flavor symmetry and either a $Z_2$ or a U(1) chiral symmetry. Both models could not be studied earlier due to sign problems. We use the fermion bag approach which is free of sign problems and compute critical exponents at the phase transitions. We estimate $\nu = 0.83(1)$, $\eta = 0.62(1)$, $\eta_\psi = 0.38(1)$ in the $Z_2$ and $\nu = 0.849(8)$, $\eta = 0.633(8)$, $\eta_\psi = 0.373(3)$ in the U(1) model., Comment: 5 page, 3 figures

Lattice four-fermion models containing $N$ flavors of staggered fermions, that are invariant under $Z_2$ and U(1) chiral symmetries, are known to suffer from sign problems when formulated using the auxiliary field approach. Although these problems have been ignored in previous studies, they can be severe. In this talk, we show that the sign problems disappear when the models are formulated in the fermion bag approach, allowing us to solve them rigorously for the first time., Comment: 10 pages, 9 figures, including the results of U(1) GN model. Proceedings of Extreme QCD (xQCD)2012, August 21 - 23, 2012, The George Washington University, Washington, DC

Lattice four-fermion models containing $N$ flavors of staggered fermions, that are invariant under $Z_2$ and U(1) chiral symmetries, are known to suffer from sign problems when formulated using the auxiliary field approach. Although these problems have been ignored in previous studies, they can be severe. Here we show that the sign problems disappear when the models are formulated in the fermion bag approach, allowing us to solve them rigorously for the first time., Comment: references added

The recently proposed fermion bag approach is a powerful technique to solve some four-fermion lattice field theories. Due to the existence of a duality between strong and weak couplings, the approach leads to efficient Monte Carlo algorithms in both these limits. The new method allows us for the first time to accurately compute quantities close to the quantum critical point in the three dimensional lattice Thirring model with massless fermions on large lattices. The critical exponents at the quantum critical point are found to be $\nu=0.85(1)$, $\eta = 0.65(1)$ and $\eta_\psi = 0.37(1)$., Comment: 5 pages, 2 figures and 1 table, published version

We present a new approach to some four-fermion lattice field theories which we call the generalized fermion bag approach. The basic idea is to identify unpaired fermionic degrees of freedom that cause sign problems and collect them in a bag. Paired fermions usually act like bosons and do not lead to sign problems. A resummation of all unpaired fermion degrees of freedom inside the bag is sufficient to solve the fermion sign problem in a variety of interesting cases. Using a concept of duality we then argue that the size of the fermion bags is small both at strong and weak couplings. This allows us to construct efficient algorithms in both these limits. Using the fermion bag approach, we study the quantum phase transition of the 3D massless lattice Thirrring model which is of interest in the context of Graphene. Using our method we are able to solve the model on lattices as large as $40^3$ with moderate computational resources. We obtain the precise location of the quantum critical point and the values of the critical exponents through this study., Comment: 14 pages, 5 figures, 1 table, contribution to Lattice 2011, July 10-16, 2011, Squaw Valley, Lake Tahoe, California, USA

A canonical ensemble algorithm is employed to study the phase diagram of $N_f = 3$ QCD using lattice simulations. We lock in the desired quark number sector using an exact Fourier transform of the fermion determinant. We scan the phase space below $T_c$ and look for an S-shape structure in the chemical potential, which signals the coexistence phase of a first order phase transition in finite volume. Applying Maxwell construction, we determine the boundaries of the coexistence phase at three temperatures and extrapolate them to locate the critical point. Using an improved gauge action and improved Wilson fermions on lattices with a spatial extent of $1.8 \fm$ and quark masses close to that of the strange, we find the critical point at $T_E = 0.925(5) T_c$ and baryon chemical potential $\mu_B^E = 2.60(8) T_c$., Comment: 5 pages, 7 figures, references added, published version

We study a strongly coupled $Z_2$ lattice gauge theory with two flavors of quarks, invariant under an exact $\mathrm{SU}(2)\times \mathrm{SU}(2) \times \mathrm{U}_A(1) \times \mathrm{U}_B(1)$ symmetry which is the same as QCD with two flavors of quarks without an anomaly. The model also contains a coupling that can be used to break the $\mathrm{U}_A(1)$ symmetry and thus mimic the QCD anomaly. At low temperatures $T$ and small baryon chemical potential $\mu_B$ the model contains massless pions and massive bosonic baryons similar to QCD with an even number of colors. In this work we study the $T-\mu_B$ phase diagram of the model and show that it contains three phases : (1) A chirally broken phase at low $T$ and $\mu_B$, (2) a chirally symmetric baryon superfluid phase at low $T$ and high $\mu_B$, and (3) a symmetric phase at high $T$. We find that the nature of the finite temperature chiral phase transition and in particular the location of the tricritical point that seperates the first order line from the second order line is affected significantly by the anomaly., Comment: 22 pages, 16 figures, 5 tables, references added

We explore the sign problem in strongly coupled lattice QED with one flavor of Wilson fermions in four dimensions using the fermion bag formulation. We construct rules to compute the weight of a fermion bag and show that even though the fermions are confined into bosons, fermion bags with negative weights do exist. By classifying fermion bags as either simple or complex, we find numerical evidence that complex bags with positive and negative weights come with almost equal probabilities and this leads to a severe sign problem. On the other hand simple bags mostly have a positive weight. Since the complex bags almost cancel each other, we suggest that eliminating them from the partition function may be a good approximation. This modified partition function suffers only from a mild sign problem. We also find a simpler model which does not suffer from any sign problem and may still be a good approximation at small and intermediate values of the hopping parameter. We also prove that when the hopping parameter is strictly infinite all fermion bags are non-negative., Comment: 17 pages, 4 figures, 5 tables

In a progress toward searching for the QCD critical point, we study the finite density phase transition of $N_f = 4$ and 2 lattice QCD at finite temperature with the canonical ensemble approach. We develop a winding number expansion method to accurately project out the particle number from the fermion determinant which greatly extends the applicable range of baryon number sectors to make the study feasible. Our lattice simulation was carried out with the clover fermions and improved gauge action. For a given temperature, we calculate the baryon chemical potential from the canonical approach to look for the mixed phase as a signal for the first order phase transition. In the case of $N_f=4$, we observe an "S-shape" structure in the chemical potential-density plane due to the surface tension of the mixed phase in a finite volume which is a signal for the first order phase transition. We use the Maxwell construction to determine the phase boundaries for three temperatures below $T_c$. The intersecting point of the two extrapolated boundaries turns out to be at the expected first order transition point at $T_c$ with $\mu = 0$. This serves as a check for our method of identifying the critical point. We also studied the $N_f =2$ case, but do not see a signal of the mixed phase for temperature as low as 0.83 $T_c$., Comment: 28 pages, 11 figures,references added, final version

The existence of the QCD critical point at non-zero baryon density is not only of great interest for experimental physics but also a challenge for the theory. Any hint of the existence of the first order phase transition and, particularly, its critical point will be valuable towards a full understanding of the QCD phase diagram. We use lattice simulation based on the canonical ensemble method to explore the finite baryon density and finite temperature region and look for the QCD critical point. As a benchmark, we run simulations for the four degenerate flavor QCD where we observe a clear signal of the expected first order phase transition. In the two flavor case, we do not see any signal for temperatures as low as $0.83 \rm{T_c}$. Although our real world contains two light quarks and one heavier quark, three degenerate flavor case shares a lot of similar phase structures as the QCD. We scan the phase diagram using clover fermions with $m_\pi \approx 700{MeV}$ on $6^3\times4$ lattices. The baryon chemical potential is measured as we increase the baryon number and we see the characteristic "S-shape" that signals the first order phase transition. We determine the phase boundaries by Maxwell construction and report our preliminary results for the location of critical point for the present lattice., Comment: 17 pages, 11 figures. Plenary talk at 27th International Symposium on Lattice Field Theory (Lattice 2009), Beijing, China, 25-31 Jul 2009

The existence of the QCD critical point at non-zero baryon density is not only of great interest for experimental physics but also a challenge for the theory. We use lattice simulations based on the canonical ensemble method to explore the finite baryon density region and look for the critical point. We scan the phase diagram of QCD with three degenerate quark flavors using clover fermions with $m_\pi \approx 700{MeV}$ on $6^3\times4$ lattices. We measure the baryon chemical potential as we increase the density and we see the characteristic "S-shape" that signals the first order phase transition. We determine the phase boundary by Maxwell construction and report our preliminary results for the location of critical point., Comment: 2 pages, 2 figures - To appear in the conference proceedings for Quark Matter 2009, March 30 - April 4, Knoxville, Tennessee

The canonical partition function approach was designed to avoid the overlap problem that affects the lattice simulations of nuclear matter at high density. The method employs the projections of the quark determinant on a fix quark number sector. When the quark number is large, the evaluation of the projected determinant becomes numerically unstable. In this paper a different evaluation method based on expanding the determinant in terms of loops winding around the lattice is studied. We show that this method is stable and significantly faster than our original algorithm. This greatly expands the range of quark numbers that we can simulate effectively., Comment: 8 pages, 5 figures, LATTICE 2008, parallel talk

QCD at non-zero baryon density is expected to have a critical point where the zero-density cross-over turns into a first order phase transition. To identify this point we scan the density-temperature space using a canonical ensemble method. For a given temperature, we plot the chemical potential as a function of density looking for an "S-shape" as a signal for a first order transition. We carried out simulations using Wilson fermions with $m_\pi \approx 1{GeV}$ on $6^3\times 4$ lattices. As a benchmark, we ran four flavors simulations where we observe a clear signal. In the two flavors case we do not see any signal for temperatures as low as $0.83 T_c$. Preliminary results for the three flavor case are also presented., Comment: 7 pages, 7 figures, Talk given at the XXVI International Symposium on Lattice Field Theory, July 14 - 19 2008, Williamsburg, Virginia, USA

We present some new results regarding simulations of finite density QCD based on a canonical approach. A previous study has shown that such simulations are feasible, at least on small lattices. In the current study, we investigate some of the issues left open: we study the errors introduced by our approximation of the action and we show how to tune it to reduce the cost of the simulations while keeping the errors under control. To further reduce the cost of the simulations, we check the reliability of reweighting method with respect to the baryon number. Finally, using these optimizations, we carry out the simulations at larger densities than in our previous study to look for signals of a phase transition., Comment: 7 pages, 3 figures, Contribution to Lattice 2007, Regensburg, Germany, 30 July - 4 August 2007

Previous investigations have shown that the canonical approach to simulating QCD at finite density is promising. The algorithm we used in our earlier work employs an exact calculation of the fermionic determinant which limits the size of the lattices we can simulate. Interesting questions can only be answered if we simulate at larger volume. In this paper we explore an algorithm, Hybrid Noisy Monte Carlo, that employs a determinant estimator rather than an exact calculation. We first present the technical aspects of the estimator, check that the algorithm is correct by comparing it with our previous study, and then discuss its merits. We will also discuss the challenges faced when simulating larger lattice volumes., Comment: 7 pages, 4 figures, Contribution to Lattice 2007, Regensburg, Germany, 30 July - 4 August 2007

Finite density simulations require dynamical fermions which are computationally demanding. We employ Ferrenberg and Swendsen reweighting method to reduce the number of ensembles needed. We use their method to do a combined reweighting in both $\beta$ and quark number $k$ on an ensemble generated by the Kentucky group., Comment: PoS(2006) 7 pages, 7 figures