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Register a new userBreaking and restoration of rotational symmetry on the lattice for bound state multiplets
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We explore the breaking of rotational symmetry on the lattice for bound state energies and practical methods for suppressing this breaking. We demonstrate the general problems associated with lattice discretization errors and finite-volume errors using an $alpha$ cluster model for $^8$Be and $^{12}$C. We consider the two and three $alpha$-particle systems and focus on the lowest states with non-zero angular momentum which split into multiplets corresponding to different irreducible representations of the cubic group. We examine the dependence of such splittings on the lattice spacing and box size. We find that lattice spacing errors are closely related to the commensurability of the lattice with the intrinsic length scales of the system. We also show that rotational symmetry breaking effects can be significantly reduced by using improved lattice actions, and that the physical energy levels are accurately reproduced by the weighted average of a given spin multiplets.
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We study the breaking of rotational symmetry on the lattice for irreducible tensor operators and practical methods for suppressing this breaking. We illustrate the features of the general problem using an $alpha$ cluster model for $^{8}$Be. We focus on the lowest states with non-zero angular momentum and examine the matrix elements of multipole moment operators. We show that the physical reduced matrix element is well reproduced by averaging over all possible orientations of the quantum state, and this is expressed as a sum of matrix elements weighted by the corresponding Clebsch-Gordan coefficients. For our $alpha$ cluster model we find that the effects of rotational symmetry breaking can be largely eliminated for lattice spacings of $aleq 1.7$ fm, and we expect similar improvement for actual lattice Monte Carlo calculations.
We consider the breaking of Galilean invariance due to different lattice cutoff effects in moving frames and a nonlocal smearing parameter which is used in the construction of the nuclear lattice interaction. The dispersion relation and neutron-proton scattering phase shifts are used to investigate the Galilean invariance breaking effects and ways to restore it. For $S$-wave channels, ${}^1S_0$ and ${}^3S_1$, we present the neutron-proton scattering phase shifts in moving frames calculated using both Luschers formula and the spherical wall method, as well as the dispersion relation. For the $P$ and $D$ waves, we present the neutron-proton scattering phase shifts in moving frames calculated using the spherical wall method. We find that the Galilean invariance breaking effects stemming from the lattice artifacts partially cancel those caused by the nonlocal smearing parameter. Due to this cancellation, the Galilean invariance breaking effect is small, and the Galilean invariance can be restored by introducing Galilean invariance restoration operators.
We extend earlier studies of transverse Ward-Fradkin-Green-Takahashi identities in QED, their usefulness to constrain the transverse fermion-boson vertex and their importance for multiplicative renormalizability, to the equivalent gauge identities in QCD. To this end, we consider transverse Slavnov-Taylor identities that constrain the transverse quark-gluon vertex and derive its eight associated scalar form factors. The complete vertex can be expressed in terms of the quarks mass and wave-renormalization functions, the ghost-dressing function, the quark-ghost scattering amplitude and a set of eight form factors. The latter parametrize the hitherto unknown nonlocal tensor structure in the transverse Slavnov-Taylor identity which arises from the Fourier transform of a four-point function involving a Wilson line in coordinate space. We determine the functional form of these eight form factors with the constraints provided by the Bashir-Bermudez vertex and study the effects of this novel vertex on the quark in the Dyson-Schwinger equation using lattice QCD input for the gluon and ghost propagators. We observe significant dynamical chiral symmetry breaking and a mass gap that leads to a constituent mass of the order of 500 MeV for the light quarks. The flavor dependence of the mass and wave-renormalization functions as well as their analytic behavior on the complex momentum plane is studied and as an application we calculate the quark condensate and the pions weak decay constant in the chiral limit. Both are in very good agreement with their reference values.
We project onto the light-front the pions Poincare-covariant Bethe-Salpeter wave-function, obtained using two different approximations to the kernels of QCDs Dyson-Schwinger equations. At an hadronic scale both computed results are concave and significantly broader than the asymptotic distribution amplitude, phi_pi^{asy}(x)=6 x(1-x); e.g., the integral of phi_pi(x)/phi_pi^{asy}(x) is 1.8 using the simplest kernel and 1.5 with the more sophisticated kernel. Independent of the kernels, the emergent phenomenon of dynamical chiral symmetry breaking is responsible for hardening the amplitude.
The IKKT matrix model is a promising candidate for a nonperturbative formulation of superstring theory, in which spacetime is conjectured to emerge dynamically from the microscopic matrix degrees of freedom in the large-$N$ limit. Indeed in the Lorentzian version, Monte Carlo studies suggested the emergence of (3+1)-dimensional expanding space-time. Here we study the Euclidean version instead, and investigate an alternative scenario for dynamical compactification of extra dimensions via the spontaneous symmetry breaking (SSB) of 10D rotational symmetry. We perform numerical simulations based on the complex Langevin method (CLM) in order to avoid a severe sign problem. Furthermore, in order to avoid the singular-drift problem in the CLM, we deform the model and determine the SSB pattern as we vary the deformation parameter. From these results, we conclude that the original model has an SO(3) symmetric vacuum, which is consistent with previous results obtained by the Gaussian expansion method (GEM). We also apply the GEM to the deformed matrix model and find consistency with the results obtained by the CLM.
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