Quantum chromodynamics on lattice: direct minimization of QCD-QED-action with new results

This paper describes a new numerical QCD calculation method (direct minimization of QCD-QED-action) and its results for the first-generation (u,d) hadrons. Here we start with the standard color-Lagrangian LQCD=LDirac+Lgluon , model the quarks q i as parameterized gaussians, and the gluons Ag i as Ritz-Galerkin-series. We minimize the Lagrangian numerically with parameters par=(par(q),{α k },par(Ag)) for first-generation hadrons (nucleons, pseudo-scalar mesons, vector mesons). The resulting parameters yield the correct masses, correct magnetic moments for the nucleons, the gluon-distribution and the quark-distribution with interesting insights into the hadron structure.


Introduction
The Quantum Chromodynamics (QCD) is based on the SU(3)-color interaction, and is in general considered as a theory, which is both mathematically complicated and numerically hard-to-handle.We present here a new calculation method (on-lattice minimization of action), which is numerically simpler than the other methods, and which uses a new ansatz for wavefunctions.We also present calculation results for energy-mass of first-generation hadrons, which agree well with the observed values, and new results for internal component distribution, which give interesting insights into the symmetry and internal structure of these hadrons.In chap.2 we compare the different calculation methods.In chap.3 we describe the ansatz for the component quark and gluon wavefunctions.In chap.4 the numerical algorithm is described.In chap.5 the calculation results for energy-mass and component distribution are presented for the three families of first-generation hadrons: nucleons, pseudo-scalar mesons, and vector mesons.
2. Solution methods in lattice-QCD [8], [9], [12], [13], [14] Basically, there are four solution methods in lattice-QCD (LQCD): Perturbative analytic Feynman solution Here one calculates the reaction cross-sections from Feynman diagrams evaluating the corresponding Feynmanintegrals in analogy to the QED.As the QCD is renormalizable, all Feynman integrals can be made finite.However, this works only for convergent Feynman series, i.e. if the interaction constant gc<1 .This is the case for large energies E>EΛ=220MeV .

Non-perturbative on-lattice Wilson-loop method
Here the expectation value of an operator (e.g.energy=Hamilton operator) is calculated using path integrals the local gauge transformation, with coupling constant g , lattice step size a , gluon field , on closed loops on the lattice.
We get the Wilson action on equidistant lattice   , ( ) ( ) ( ) ( ) The Wilson action   The solution , ( , ) yields the corresponding masses , q i m for fermions and energies i E for gluons.

Non-perturbative on-lattice eom solution
The QCD equations-of-motion (eom) are derived from the minimal-action-principle as the Euler-Lagrangeequations corresponding to the QCD Lagrangian.They are 0    and the quark-wavefunction ) (x a   These are nq+ng partial differential equations (pdeq) first order in  x , for the nq =2 or nq =3 quarks and ng =8 gluons, adding a gauge condition and a boundary condition for ) (x A a  .They must be solved numerically on a lattice as an eigenvalue problem of the Dirac equation, which is very difficult and time-consuming for a one-dimensional lattice of, say, n1=100 points (total number of points n=n1 4 =10 for the quarks i q , QCD-gluons i Ag , QED-photons i Ae , In order to carry out the minimization numerically, we introduce an equidistant 4-dimensional lattice , extract a small random sub-lattice Lsub and approximate the integral by a sum over Lsub : is the elementary integration volume in spherical coordinates, and model the quark wavefunctions as parameterized Gauss functions   ) ( , q par x q q  and the gluon-wavefunctions as Ritz-Galerkin series on a function system , where par0 yields information about the energy (=mass) , the sizes and the inner structure of the considered hadron.Non-perturbative on-lattice minimization of action
, where i aA is the phase angle between the particle and the antiparticle part of the gluon, and with the Ritz-Galerkin-expansion

Quark wavefunction
The first-generation (u,d)-hadrons consist of three quarks (nucleons) or three color-symmetric quark-antiquarkcombinations (vector-mesons) or two quark-antiquark-combinations (pseudo-scalar mesons) for nucleons A Ritz-Galerkin series for quarks would blow up the complexity of calculation, therefore we use here a simpler model, based on the asymptotic-freedom property of quarks: gaussian "blobs" dr , is the position(r,) and its width, k a is the quark-antiquark phase and the antiquark is

The ansatz and the color symmetry
The form of the quark color-wavefunction and the corresponding set of active gluons are enforced by the colorsymmetry and the number of particles equal to the number of combinations.The 8 gluons of the SU(5) form 3 families: the diagonal } , {

Ag Ag Ag
, which exchange color-index with a different anti-color index.The nucleons consist of three quarks with color (r,g,b), and the color wavefunction q is mapped into itself under color-permutations, therefore the full set of 8 gluons Agi is required, and there are only two possibilities for first-generation hadrons: p=uud and n=ddu .The vector mesons consist of quark-antiquark pairs, where the color wavefunction q has three identical components.q is mapped into itself under the corresponding set of 6  (each flips two color indices).It is seen immediately that the three combinations listed above are the only possible ones, which is confirmed by the existence of the three v-mesons omega0, rho0, rho+ .The pseudo-scalar mesons consist of quark-antiquark pairs, where the color wavefunction q has two non-zero components.The corresponding gluon set are the 3 non-diagonal color-anticolor gluons } , , { . So in reality, the wavefunction is a superposition of the three   31 23 12 , , q q q and is mapped by the gluon set into itself.Again, one can see immediately that there are only two possible combinations, which correspond to the two known ps-mesons pi+ and pi0 .

The corrected coupling constant
In the original Callan-Symanzik relation the QCD coupling constant has the (asymptotic) energy dependence

The numerical algorithm
The energy, length, and time are made dimensionsless by using the units: E( ,where is number of points of the sublattice, we set 100 ]) . We impose the gauge condition and the boundary condition for i Ag via penalty-function (imposing exact conditions is possible, but slows down the minimization process enormously).S ~is minimized 8x in parallel with the Mathematica-minimization method "simulated annealing" , the execution time on a 2.7GHz Xeon E5 is 9100s for the proton p=uud , the complexity ]) , for the proton is the number )=8*2=16 .The proper parameters of the quarks and the gluons are: } , , , , { ) (  Criteria for correctness of the ansatz 1.Convergence of minimization As we found out during the computation, a wrong ansatz, e.g.lacking color symmetry, leads to a nonconvergent minimization.We chose a high goal precision of prec=10-4, so there was a high probability that a convergent minimizations hits a real (global) minimum.

2.High relative deviation between solutions
Strongly differing solutions indicate a non-correct ansatz, as we found out e.g. for the nucleons with too many degrees -of-freedom for the gluons: the relative deviation for crucial variables, like energy, should be no more than 2% for the nucleons and 6% for the ps-mesons.

3.Vanishing parameter-derivatives A true minimum must satisfy the derivative-condition
, where i p is one of the minimization parameters, Normally, the parameter-derivatives are close to zero, otherwise the minimum is not genuine, or the ansatz is wrong.

4.Boundary condition and gauge condition
The boundary and gauge condition must have values close to zero, otherwise the weight for the penalty function is too low.

5.Minimum value
The minimum value should be -30,...,30 for the considered parameter range.Very large positive values result in the case of too high penalty weights.Very large negative values may come out, if the Ritz-Galerkin parameters i  are not bounded appropriately.
6.Correct energy scale and number of particles The three types of first-generation hadrons have energy scales: E(nucleon)≈0.98GeV,E(v-meson)≈0.78GeV,E(ps-meson)≈0.14GeV , and these values emerge automatically with 8, 6 and 3 gluons respectively.Furthermore, with the above ansatz, the number of possible particles is 2, 3, 2 respectively.The quark distribution differs largely between the nucleons: the proton is ring-symmetric (no -component), the neutron has two orbitals with an angle of α=π/2 .The small mass difference is due to the electromagnetic contribution, which is about 1% of the total mass.The mass of the nucleons, as is the case for all first-generation-hadrons is generated almost exclusively by the energy of the gluons and the quarks, the rest masses of u and d (mu=2.3MeV, md=4.8MeV) contribute very little to the total mass.The gluon distribution is practically the same for both nucleons, which is to be expected, since the two particles are identical for the color interaction.

The results for first-generation hadrons
The radius of the nucleons can be assessed from the above diagram: r(p)≈0.8fm, r(n)≈1fm gluons Agi The neutron n has two orbitals with an angle of α=π/4 , the u-quark is at the center with low energy E=0.05 , the two d-quarks sit in the orbitals with higher energies E=0.09,0.013 .The "smearing" width is comparable, δr≈0.4 and higher than with the proton.The gluon distribution is practically the same as for the proton, which is to be expected, since the two particles are identical for the color interaction.The electromagnetic correction is positive and much smaller than with the proton, dEem=+0.0017GeV, which is probably the reason for the proton's smaller mass.

magnetic moment of nucleons
The magnetic moment is , for a rotating charge distribution: is the momentum of charge, in analogy to the momentum of inertia dm r For a rotating solid sphere with radius r0 with constant charge density 2 0 5 2 qr I q  .
The magnetic moment of the nucleons is measured in nuclear magnetons , which is the magnetic moment of a rotating solid sphere with constant charge density The actual momentum of charge is therefore: We have to take into account the "smearing" Δri of radius ri We get for the neutron Iqn=−0.1766e , IqNn= 0.106 e , so The calculation does not take into account the orbitals, and there is also the statistical uncertainty of the order 7% , so the results are satisfactory.pseudo-scalar mesons pi+, pi0 quarks(2),gluons(3),spin=0masses m(pi+) m(pi0) exp.
129MeV 155MeV energy for quark-number (n=1,2), gluon-number (n=3,4,5), both sorted with increasing energy The pseudo-scalar mesons are spherically-symmetric, there is no θ-dependence: θ≈0 in the quark-distribution, the gluon-wavefunctions show little θ-dependence, and the gluon amplitudes are much smaller (factor 30) for pi0 than for pi+ .The vector mesons are spin-1 bosons but only rho+ shows an explicit θ-dependence of quark-distribution: it is ellipsoidal.The gluons show explicit θ-dependence and are, as for the nucleons, practically equal for all three particles.For rho0: the quarks u u and d d have identical parameters r=0.5, δr=0.3,E=0.1 For omega0: the quarks u u and d d again have identical parameters, are at center, δr=0.25,E=0.1 For rho+: the heavier quark d has r=0.5 , δr=0.05,E=0.05 , the light quark u has r=0.9 , δr=0.5, E=0.07 , rho+ has two orthogonal orbitals.Its two quarks have completely different width; the d quark closer to the center has a small bandwidth, the light u quark is strongly "smeared" like all the other quarks in the 3 particles.The measured masses of the v-mesons (0.775, 0.775, 0.782) are reproduced correctly by the calculation (0.771±0.0052, 0.779±0.012,0.782±0.007).


, where the action integral is approximated as a summation on a random sublattice of an equidistant lattice.This has the following advantages: ▪ The minimal-action principle is a fundamental principle, from which the equation-of-motion (eom), i.e. the Dirac equation for QCD is derived.The parameters i j q , Ag of quarks i q and gluons j Ag in the Lagrangian   QCD i i L x ,q , Ag  (e.g.energy-mass of a quark ) can be calculated from it in principle exactly without solving the eom .In comparison, the Wilson loop method uses an approximation in order to make the path integral tractable numerically.▪ The calculation is a simple summation, which is very fast numerically, as opposed to the analytical integral calculation of the perturbative analytic Feynman solution.
▪ The calculation is scalable, i.e. the precision can be increased arbitrarily, simply by making the step size of the lattice smaller, or the size of the sublattice larger.
▪ The calculation can be carried-out in parallel by Np processes on Np different sublattices with the same number of points (in this implementation we have Np =8 ).The mean of the Np resulting values is then the calculation result for a parameter, whereas the standard-deviation assesses the calculation error.▪ On-lattice minimization of action uses parameter minimization algorithms instead of solving partial differential equations (as in on-lattice eom solution) or instead of calculation of parameterized integrals (as in Wilson-loop method or in analytic Feynman method).Nowadays, there exists a large selection of powerful algorithms for parameter minimization which can be used for this purpose.▪ On-lattice minimization of action, as opposed to the other solution methods, yields information about radial and axial distribution within hadrons.

2 Ag
exchange a color-quark with a different anti-color-quark.For example, flips color-indices(3 QCD energy nf =3 generations, N=3 QCD charges For energies μ≈Ʌ it must be modified to avoid the singularity with the Callan-Symanzik relation for   2  , as shown below gc(μ) , μ in units distribution quarks (r[fm] ,): independent(θ)=spherical 8