## Abstract

We report the results of the U.S. Department of Energy “Grand Challenge” project on the ab initio calculations of hadron structure (proton, neutron, and other strongly interacting particles) and interaction in the framework of lattice gauge quantum chromodynamics. The electric and magnetic form factors which are deduced from the elastic scattering of electrons off the proton and neutron targets are simulated on a quenched 163 x 24 lattice. The masses of the exotic four-quark mesoniums, the molecules of the strongly interacting mesons, are calcu lated on a 24 x 12 × 12 x 24 lattice. Stable and meta stable configurations are found in certain quantum states. This finding agrees with the observed experi mental trend.

Original language | English |
---|---|

Pages (from-to) | 72-80 |

Number of pages | 9 |

Journal | International Journal of High Performance Computing Applications |

Volume | 4 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1990 |

### Bibliographical note

Funding Information:Liu Keh-Fei UNIVERSITY OF KENTUCKY LEXINGTON, KENTUCKY 40506 09 1990 4 3 72 80 We report the results of the U.S. Department of Energy "Grand Challenge" project on the ab initio calculations of hadron structure (proton, neutron, and other strongly interacting particles) and interaction in the framework of lattice gauge quantum chromodynamics. The electric and magnetic form factors which are deduced from the elastic scattering of electrons off the proton and neutron targets are simulated on a quenched 16 3 x 24 lattice. The masses of the exotic four-quark mesoniums, the molecules of the strongly interacting mesons, are calcu lated on a 24 x 12 × 12 x 24 lattice. Stable and meta stable configurations are found in certain quantum states. This finding agrees with the observed experi mental trend. sagemeta-type Journal Article search-text 72 Hadron Structure and Interaction From Lattice Quantum Chromodynamics Calculations SAGE Publications, Inc.1990DOI: 10.1177/109434209000400308 Keh-Fei Liu UNIVERSITY OF KENTUCKY LEXINGTON, KENTUCKY 40506 Summary We report the results of the U.S. Department of Energy "Grand Challenge" project on the ab initio calculations of hadron structure (proton, neutron, and other strongly interacting particles) and interaction in the framework of lattice gauge quantum chromodynamics. The electric and magnetic form factors which are deduced from the elastic scattering of electrons off the proton and neutron targets are simulated on a quenched 163 x 24 lattice. The masses of the exotic four-quark mesoniums, the molecules of the strongly interacting mesons, are calcu lated on a 24 x 12 × 12 x 24 lattice. Stable and meta stable configurations are found in certain quantum states. This finding agrees with the observed experi mental trend. Introduction The advent of quantum chromodynamics (QCD), a nonabelian gauge relativistic quantum field theory of quarks and gluons (the constituents of protons and neutrons), in the early 1970s has offered the most promising description of strong interaction dynamics and internal structure of hadrons both in logical consistency and in scope ('t Hooft, 1981). It is widely accepted that it is the correct theory of strong interaction. This acceptance is based on substantial experimental data accumulated and refined over the last two decades. It is particularly true for the inclusive scattering processes at large momentum transfer where the perturbative QCD is applicable. On the other hand, confirmation of QCD as the fundamental theory of strong interaction has been somewhat hampered by the lack of analytical calculations of the low-frequency modes due to the inherent nonperturbative nature of the theory. The invention of lattice-regularized QCD (Wilson, 1974) with Monte Carlo methods (Creutz, Jacobs, and Rebbi, 1979a, 1979b) holds the promise of circumventing the difficulty of obtaining analytic results via numerical simulation. In the lattice-regularized gauge theory, the continuous spacetime is discretized by a finite lattice with a periodic boundary condition. As a consequence, the infinite dynamic degrees of freedom associated with spacetime is reduced to a finite number, which makes numerical analysis feasible. The recent advances in supercomputer technology and a DOE "Grand Challenge" computer allocation have made it possible to carry out large-scale Monte Carlo simulations to calculate the hadronic structure and interaction in the framework of lattice QCD. This allows us to compare the lattice results with the experimental data in a qualitative and semiquantitative manner, which is very valuable in understanding physics from first principles, free from the uncertainty of models. We report the "Grand Challenge" project on the electromagnetic form factors of the nucleon that determine the charge and magnetic radii of the proton and neutron, the masses of mesoniums which may imply the molecular structure of mesons and their interactions. We hope to learn from lattice simulations the origin of the experimental dipole forms of the electric and mag- 7573 netic form factors of the proton, the negative charge radius of the neutron. Various hadron models give different realizations of the form factors. We believe lattice calculations should be able to tell us which realization is the correct physical picture. There is abundant tantalizing experimental evidence for the existence of mesoniums. These are multiquark hadrons made of two quarks and two antiquarks. However, the existing quark models and experiments are not able to distinguish metastable states from threshold effects. We study these mesoniums on the lattice to fmd out whether the experimental systematics, in quark masses and in spin and isospin quantum numbers can be understood, if not the details. We first present the formalism for the lattice gauge Monte Carlo simulation and the numerical steps involved in the calculation. We then report the calculation of the electric and magnetic form factors of the proton and the neutron, followed by the mesonium mass calculation. In the final section the computational aspects of the calculations and the conclusion are presented. Formalism and Numerical Steps The Euclidean path integral formulation of quantum field theory is used to calculate Green's function. In this way, the quantum field theory becomes a problem in classical statistical mechanics. Physical observables are then extracted from the statistical correlation functions of composite operators built on the fundamental dynamical variables in quark and gluon fields. Interested readers are referred to the article by Bitar et al. in this issue for an introduction to the theory of QCD and its simulation on the lattice. Here we present only a synopsis of the formalism and related numerical steps. The correlation functions have the following generic form where Z is the partition function. ~[~7] is the action of the gauge link variable U, and S F[ U, 1.\1, 1.\1] is the fermion action, which is bilinear in the quark field variable 4; and can be written in the form SF = ~M[O~. Since both the fermion action SF and the operator product 0102 ... are bilinear in ~ and ~, the fermion integral over the anti- commuting Grassmann variables ~ and ~ can be done analytically to give The path integral in Eq. (2) is an ordinary multiple integral over the group manifold, in this case the SU(3) group. One might think of approximating the integral by summing over a sufficiently dense set of mesh points, say four per variable. However, this "modest" goal is well beyond the capacity of existing supercomputers: for a (small) 104 lattice, it would require a calculation on 432°,°°° = 10192,659 points. Instead, one may use the important sampling technique to select a comparatively small subset of gauge configurations (Ul, U2, - - -, UN) among the "important" ones, such that the probability of occurrence of a given configuration Ui in this ensemble approaches the desired distribution for N ~ 00. Then the quantum averages are given (for sufficiently large N) by averages taken over the sample; that is, ... I . , There are three numerical steps to lattice gauge calculations. The first step is to prepare an ensemble of gluon configurations {6~ in the vacuum. This is done using the Cabibbo-Marinari (1982) pseudoheatbath algorithm, a Monte Carlo method that generates sets of link matrices (U) according to the distribution in Eq. (3). In principle, the probability distribution includes a factor of det(M[~71) (see Eq. 3) which represents the effect of quark-antiquark polarization in the vacuum. In the present simulations, we omit this factor as an approximation since it would require more computation than we can do. We will discuss the validity of this approximation when we come to the results of the calculations. Omitting this determinant is referred to as the quenched approximation. 7674 The second step is to calculate the quark propagator matrix M-1[U] in the prepared gluon (o background. This is done with the conjugate-gradient algorithm (Hestenes and Stiefel, 1952). This is an iterative scheme that is capable of giving the desired solution to arbitrary accuracy. We typically find a few hundred iterations to be sufficient. The last step is to assemble the quark propagator matrices M-1[th] and the matrix 7JUJ defmed in Eqs. (1) and (2) to evaluate the correlation function in Eq. (4) from which we extract the physical quantities with statistical analysis. Electromagnetic Form Factors of Nucleon FORMALISM Elastic electron scattering off the nucleon target is a useful probe of the nucleon internal structure by measuring its electromagnetic form factors. These form factors can be simulated in the lattice gauge calculation and compared with experiments directly. In addition, they represent an ongoing challenge for developing efficient computer algorithms. Previous nucleon studies include the proton electric form factor (Martinelli, 1989; Martinelli and Sachrajda, 1989) as well as a more complete study of the electric and magnetic form factors for both the proton and neutron (Draper, Woloshyn, and Liu, 1990; Draper et al., 1989a) and the electric form factor of the pion (Draper et al., 1989b). The present work is aimed at investigating the mass and momentum dependence of these same quantities and thus complements the low-momentum results. We are especially interested in trying to understand the origin of the approximate dipole dependence of the nucleon form factors. (A good, although somewhat dated, review of nucleon form factor properties can be found in Gourdin, 1974.) The result of the present work on the electromagnetic form factors on the nucleon was reported at the Lattice '89 Conference (Wilcox et al., 1990). The lattice two- and three-point functions that we measure are where N represents the proton or neutron interpolating field (see Draper, Woloshyn, and Liu, 1990; Draper et al., 1989a), r is some 4 x 4 matrix, andi, represents the conserved lattice vector current, which is nonlocal in time. Of course, the right-hand side of Eq. (6) is also a function of t2, but in the present application t2 will be fixed. By inserting complete sets of states and setting t2 > tl > 1, we obtain the electric and magnetic form factors GE( q2) and GJq2) from the ratios of the respective three-point functions to two-point functions. NUMERICAL DETAILS Our quenched gluon configurations are on a 163 x 24 lattice and were calculated using the Monte Carlo Cabibbo-Marinari (1982) pseudoheatbath algorithm. The SU(3) fundamental Wilson action was used with periodic boundary conditions and the coupling constant was set at @ = 6.0. The gauge field was thermalized for 5,000 sweeps from a cold start, and 12 configurations separated by at least 1,000 sweeps were saved. For the quarks we used periodic boundary conditions in the spatial directions and "f~ed" time boundary conditions, which consist of setting the quark couplings across the time edge to zero. The origin of all quark propagators was chosen to be at lattice time site 5; the secondary zero momentum nucleon source was fixed at time site 20. We expect that these positions are sufficiently far from the lattice time boundaries to avoid nonvacuum contaminations. We used the conditioned conjugate-gradient technique for quark propagator evaluation described by De- Grand (1988). For our convergence criterion we demanded that the absolute sum of the squares of the quark propagators be less than 5 x 10-5 over five iterations. As one check of the nucleon secondary source, we verified current conservation for t2 > t1 > 0 to 0 (10-4). 7775 RESULTS The dimensionless masses we find at K = 0.154 are 0.38 (0.01) for the pion, 0.47 (0.01) for the p, and 0.75 (0.03) for the nucleon. The form factor results of this study are presented in Figures 1 through 4. Figure 1 shows the proton electric form factor compared with the monopole, (1 + q2/ MM) -1, and dipole ( 1 + q2/MB)-2, fits. The dipole form is favored, but the monopole is not statistically ruled out. Experimentally the proton electric form factor is reasonably well fitted by the dipole form with MD = 0.84 GeV, giving MDIMN = 0.89, whereas we find MDa = 0.57 (0.02), giving MDIMN = 0.76 (0.04). For the fitted monopole mass, we find Mma = 0.34 (0.02), significantly different from the p mass. Figures 2 and 3 show the proton and neutron magnetic fonn factors along with single-parameter monopole and dipole fits, using the fitted Mma and MDa values from Figure 1. Figure 4 compares the neutron electric form factor data to the phenomenological form Fig. 'f The proton electric form factor versus dimensionless four momentum transfer squared The solid and dashed lines show dipole and monopole fits, respectively. where we use the dipole (solid line) or monopole (dashed line) fit from Figure 3 for GM(q2). Since we have only one quark mass at K = 0.154, we are not able to distinguish a monopole from a dipole fit to the electric and magnetic form factors, as is the case of the pion with different K values. Therefore we think it is essential to have results from at least two more K values in order to understand the origin of the experimental dipole form. Even though the quenched approximation does not include the vacuum polarization due to quark loops, it can still generate quark-antiquark pairs polarized by the coupling to photons. This is because quarks can propagate backward in time, which corresponds to antiquarks. This is evidenced in the study of the electric form factor of the pion (Draper et al., 1989b). There we find that the monopole form Fig. 2 The proton magnetic form factor versus dimensionless four momentum transfer squared The solid and dashed lines show single-parameter dipole and monopole fits, respectively, using the fitted M,4a and Mva values from Figure 1. 7876 Fig. 3 Same as Figure 2, but for the neutron magnetic form factor factor is indeed dominated by the p-meson propagator, consistent with the vector-meson dominance idea. Therefore we expect the quenched approximation to be valid for the nucleons as well. We should be able to verify this when we have results from two more quark masses. Meson-Meson Interactions Lattice gauge calculations can be utilized to study ha- dron-hadron interactions and the exotic stable bound states and metastable resonances in the composite systems. pp MESONIUMS The hadron spectroscopy of QQ mesons like the pion and the p meson and Q3 baryons like the proton and neutron are well studied and understood both experimentally and theoretically. The next group in the hierarchy of multiquark hadrons which may form color- neutral states, and therefore could exist in nature, is the Pig. 4 The neutron electric form factor versus dimensionless four momentum transfer squared compared with phenomenological forms described in the text Q~Q_2 type of mesonium. The spectroscopy of the s-wave Q2Q2 states has been studied in the MIT bag model ( Jaffe, 1977; Jaffe and Johnson, 1976) and the potential models (Lipkin, 1977; liu and Wong, 1981; Weinstein and Isgur, 1983). Experimentally, there are now several candidates for these mesoniums; for example, pp (Brandelik et al., 1980), pow (ARGUS Collab., 1987; Ronan, 1988), K*K* (Nilsson, 1988) observed in yy reactions, K*K* (Armstrong et al., 1987), ~~ (Yamanouchi et al., 1981; Green, 1981; Daum, 198I), and J/gJJ/gJ (Badier et al., 1982) from hadron collisions, and S* and 8, which are believed to be the KK molecules (Weinstein and Isgur, 1982, 1983). One of the strongest candidates among these is the pp mesonium observed in my reactions (Brandelik et al., 1980) and Pn annihilations ~ .~ pOp°'7T- (Bi..idges et al., 1986). The predicted (Li and Liu, 1982; Achasov, Devyanin, and Shestakov, 7977 1982) suppression of yy - p+p- ( JADE Collab., 1983) was interpreted as due to the interference of the isospin 0 and isospin 2 amplitudes. If this is true, then the requirement of an isospin 2 state would signal the existence of the exotic Q2Q2 nature of the mesoniums. The spin parity analysis in 1'1' reactions (Althoff et al., 1982; Burke et al., 1981; Behrend et al., 1984; Aihara et al., 1988) and pn annihilation (Bettini et al., 1986) favors a 0+ at 1,480 MeV and a 2+ at 1.6 - 1.7 GeV. (The 0+ assignment of pp at 1,480 MeV is not unambiguously accepted. For example, the angular analysis of the pOpO decay of Xo [1,480] observed in the difference spectrum in pn annihilation at rest [Bridges, Daftari, and Kaloger- opoulos, 1986] indicates that a 2 + + assignment is preferred over the 0++.) Even though the experimental findings of the mesonium candidates with predominant vector (spin 1)- meson pair decays match well with our understanding of the flavor systematics, the decay pattern and the spin- color structure based on the bag model (Achasov, De- vyanin, and Shestakov, 1982; Li and Liu, 1982, 1983, 1984; Liu and Li, 1987), one still has reservations. From the experimental point of view, all these W (vector-meson pair) mesoniums are very near (either below or above) the mass threshold. They could simply be threshold effects (Maor and Williams, 1982; Alexander, Levy, and Maor, 1986; Karl, 1985). On the theoretical side, the MIT bag model and potential models are successful in delineating the spectroscopy of QQ and Hadrons. However, their ability to deal with hadronic interactions, hence multiquark structures, is at best dubious (Jaffe and Low, 1979; Lipkin, 1982; Liu, 1983; Greenberg and Lipkin, 1981; Liu, 1984). In view of the above theoretical quandary, which renders the existing models inadequate to interpret the volumes of data, we turn to the lattice QCD calculation in an attempt to clear up the theoretical situation. Specifically, we shall address the issues regarding mesoniums with dominant vector-meson pair decay modes to see if we can understand the trend of the experimental findings. The experimental findings that we would like to verify (or disprove) can be summarized as follows: 1. All the known 2+ vector-vector (VV) mesoniums (e.g., pp, pw, K~K~, andJ/'¥J/~). The only known p + bound mesoniums are in the light quark system (i.e., pp). 3. The isosinglet (I = 0) and isononet (I = 2) 0+ (and 2 +) pp mesoniums are nearby, so that yy - pop0 is enhanced and ''''1''1 ~ P + P - suppressed. FORMALISM AND MONTE CARLO RESULTS The masses of hadrons have been calculated in the lattice gauge Monte Carlo simulations of QCD from the propagators . for large t. O(ta = 2:x O(x, t) is the zero momentum interpolation quark field operator for the hadron H. The naive operator to generate two vector-meson C~2~2 mesoniums would seem to be >Gr~y~,~r~~y"~r. However, when the Fierz transform is considered, this operator could create a pair of pseudoscalar (0-) mesons and others which may have a lower mass than the vector-meson pairs that we intend to study. To ensure that the Q2Q2 mesoniums we calculate have their lowest masses near the threshold of a pair of vector mesons, we use the following Fierz-invariant operators: where v'L = t~rywe~, tw" = i\f1[')'IL, -yv]t~/2, and s = \f1$. These Fierz-invariant operators are chosen so that aIL = igJy5y>gJ and _ ~')'5$, which would create pseudoscalar mesons with lower masses than the vector meson, are absent. For the flavor part, we use the following Fierz-invariant combinations: Isasinglet: Isononet: With the above operators, we calculated (Liang et al., 1990) the VV mesoniums in the 0+ isosinglet (SS), 0+ 8078 isononet (ST), 2 ~ isosinglet (TS), and 2 + isononet (T7) channels. They were done on a 24 x 12 x 12 x 24 lattice at 13 = 5.9. Three K values (0.152, 0.154, and 0.156) were used for different quark masses. Twenty-eight quenched configurations were generated by the pseudoheatbath method. After 5,000 thermalization sweeps from a cold start, configurations were selected every 1,000 sweeps. Periodic boundary conditions in the spatial directions and fixed boundary conditions in the time direction were used for the quark fields. Correlations involving disconnected quark loops in the SS and TS channels which allow mixing between Q2Q2 and QQ states were neglected. There are no such disconnected diagrams in the ST and TT channels. We plot the ratio R = C(~2~2)~C2(p) 1-1 exp(2mp - mQ2-a2)t as a function of t, where C(p) is the p-meson propagator. If there is a bound state, the ratio R will have a positive slope; for a resonance, the slope is negative. The results for the ratio R are plotted in Figure 5. The binding energies (2m~ - mQ2Q ) are given in Figure 6. We see from Figures 5 and 6 that as the quark masses are lowered (K ranges from 0.152 to O.I56), bound states appear to show up in the 0+(SS and S7) channels. For the isosinglet 2 ~ (TS) channel, there appears to be a resonance for all the three K values we examined. The signal is about 1 a away from the pp threshold, whereas in the TT channel the calculated Q~~2 masses show up at the pp threshold. Although this pattern of bound states in the 0+ channel for lower quark masses and the resonances in the 2+ channel for the range of quark masses we studied seem to confirm the experimental trend we mentioned earlier, we should point out that the ~2~2 masses calculated in the isosinglet 0+ channel are sitting right on the 2'IT threshold. Even though we eliminated the 1r'iT channel with the Fierz-invariant operator, we suspect it may be due to the fact that the V8V" operator (W is the color octet vector Q~ shows up after the Fierz transform, and it could project to Tr7r when the two pseudoscalars are overlapping spatially. Therefore the "bound state" we see in the SS channel may well be the 2'iT threshold. We cannot verify this at this stage, but we speculate that the reason we see this "threshold effect" in the SS and not the ST channel has to do with the 1T'IT dynamics. It is known that the interaction between two pions is attrac- Fig. 5 The ratio R mentioned in the text plotted as a function of t for the isosinglet 0+ (SS), the isononet 0+- (ST), the isosinglet 2+(TS), and the isononet 2+ (TT) C#Cfl states The square data points are for the case of K = 0.152, the circles for K = 0.154, and the triangles for w = 0.156. tive in the SS channel yet repulsive in the ST channel. As a result, the wave function overlap between the two pions will be enhanced or diminished in the SS or ST channel. This should increase the 7T'1i' signal in the SS channel relative to the ST channel in the present calculation. In any event, the "bound state" we see in the ST channel, be it a '1i'1T resonance or a pp bound state or a mixture of both, and the resonance in the TS channel are definitely exotic mesoniums. In the future, we plan to use spatially separated t~y~,~r interpolation fields at time t to avoid the projection to 'IT1T states. This should be feasible as long as the spatial separation is large enough to reduce substantially the V80 component due to the 1T1T overlap and yet can still project on to the pp 8179 Fig. 6 The binding energies mrz7õ.2 - 2m~ plotted for the SS, ST, TS, and TT channels In each channel, the masses are fitted with different fime-slice separations to reflect the sensitivity of the fit. Different quark masses with K = 0.152, 0.154, and D.156 are represented with triangular, circular, and diamond points, tespectivehl. bound state. Since the mesonium dynamics is mainly due to the valence quarks, much like the quarkoniums, we think the quenched approximation should be adequate. It appears, however, that the largest uncertainty in the present calculation is the neglect of the disconnected quark loops, which allows mixing with the QQ mesons. We will attempt to address this issue in the future. Computational Aspects and Conclusion We have demonstrated in this paper that it is feasible to perform ab initio calculations of the hadronic structure and interaction directly from the fundamental field theory-quantum chromodynamics-without having to rely on models. In spite of the fact that the present lattice size is modest (e.g., 163 x 24) and the quark masses used are still heavy compared to the physical situation, and that the quenched approximation is used, present calculations nevertheless required considerable supercomputer time to perform. The project on the electromagnetic form factors of the proton and neutron with only one quark mass took 600 CRAY-2 hours and we are currently running two other quark masses to study the quark mass effect and the origin of the experimental dipole form of these form factors. The project on the pp mesoniums used quark propagators generated on the IBM 3090/300E at the University of Kentucky, which took more than 4,000 CPU hours. The memory required for the matrix inversion to obtain the quark propagator M ~ 1 is the largest in the calculation. A 16s x 24 lattice requires 14 million words of memory on the CRAY-2. For the statistical errors in these calculations, we used a single-elimination jackknife method (Efron, 1979). It is encouraging to learn that our preliminary results, however crude, have already had an impact on confirming the qualitative features of the exotic mesons in various spin and isospin channels and showing that the electromagnetic form factors of the nucleon are not far from the physical results. In order to test QCD as the fundamental theory and make predictions to compare with experiments beyond doubt, we need to push the calculations to a much larger lattice and smaller quark mass with full description of the dynamic fermions (nonquenched approximation). Each of these directions may require an order of magnitude or more time on the present supercomputers. ACKNOWLEDGMENT This work is partially supported by U.S. Department of Energy grant DE-FG05- 84ER40154 and National Science Foundation grant RII-8610671. The author is indebted to P. de For- crand, T. Draper, Y. G. Liang, B. A. Li, W. Wilcox, R. M. Woloshyn, and C. M. Wu, who also collaborated on the projects presented here. He is grateful to DOE for the generous "Grand Chal- lenge" computer allocation which made the calculations possible. He would also like to thank the Center for Computational Sciences of the University of Kentucky for providing time on the IBM 3090/300E. BIOGRAPHY Keh-Fei (Frank) Liu has been professor of physics at the University of Ken- 8280 tucky from 1986 to the present. He received his B.A. from Tunghai University (1968) and his M.S. (1971) and Ph.D. (1975) from the State University of New York at Stony Brook. 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