Finite Spatial Volume Research Articles

Temperature and volume dependence of pion-pion scattering lengths* *Supported by the Fostering Program in Disciplines Possessing Novel Features for Natural Science of Sichuan University (2020SCUNL209)

The s-wave pion-pion scattering lengths and are studied at finite temperature and in finite spatial volume under the framework of the Nambu–Jona-Lasinio model. The behavior beyond the pseudo transition temperature is investigated using proper time regularization. The scattering length exhibits singularity near the Mott temperature, and is a continuous but non-monotonic function of temperature. We present the effect of finite volume on the scattering length and find that can be negative and its singularity disappears at small volumes, which may hint at the existence of a chiral phase transition with decreasing volume.

Hybrid static potentials in SU(3) lattice gauge theory at small quark-antiquark separations

We compute the ${\mathrm{\ensuremath{\Pi}}}_{u}$ and ${\mathrm{\ensuremath{\Sigma}}}_{u}^{\ensuremath{-}}$ hybrid static potentials in SU(3) lattice gauge theory using four different lattice spacings ranging from $a=0.040\text{ }\text{ }\mathrm{fm}$ to $a=0.093\text{ }\text{ }\mathrm{fm}$. We provide lattice data points for quark-antiquark separations as small as 0.08 fm, where the $a$-dependent self-energy as well as lattice discretization errors at tree level of perturbation theory and at leading order in ${a}^{2}$ have been removed. We also investigate and exclude possibly present systematic errors from topological freezing, due to the finite spatial lattice volume and from glueball decays. Moreover, we provide corresponding parametrizations of the potentials, which can e.g. be used for Born-Oppenheimer predictions of heavy hybrid mesons.

Analytic nonhomogeneous condensates in the (2+1)-dimensional Yang-Mills-Higgs-Chern-Simons theory at finite density

We construct the first analytic examples of non-homogeneous condensates in the Georgi-Glashow model at finite density in $(2+1)$ dimensions. The non-homogeneous condensates, which live within a cylinder of finite spatial volume, possess a novel topological charge that prevents them from decaying in the trivial vacuum. Also the non-Abelian magnetic flux can be computed explicitly. These solutions exist for constant and non-constant Higgs profile and, depending on the length of the cylinder, finite density transitions occur. In the case in which the Higgs profile is not constant, the full system of coupled field equations reduce to the Lam\'e equation for the gauge field (the Higgs field being an elliptic function). For large values of this length, the energetically favored configuration is the one with a constant Higgs profile, while, for small values, it is the one with non-constant Higgs profile. The non-Abelian Chern-Simons term can also be included without spoiling the integrability properties of these configurations. Finally, we study the stability of the solutions under a particular type of perturbations.

Application of effective field theory to finite-volume effects in aμHVP

One of the more important systematic effects affecting lattice computations of the hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon, $a_\mu^{\rm HVP}$, is the distortion due to a finite spatial volume. In order to reach sub-percent precision, these effects need to be reliably estimated and corrected for, and one of the methods that has been employed for doing this is finite-volume chiral perturbation theory. In this paper, we argue that finite-volume corrections to $a_\mu^{\rm HVP}$ can, in principle, be calculated at any given order in chiral perturbation theory. More precisely, once all low-energy constants needed to define the Effective Field Theory representation of $a_\mu^{\rm HVP}$ in infinite volume are known to a given order, also the finite-volume corrections can be predicted to that order in the chiral expansion.

A gauge fixing procedure for causal fermion systems

Causal fermion systems incorporate local gauge symmetry in the sense that the Lagrangian and all inherent structures are invariant under local phase transformations of the physical wave functions. In the present paper, it is explained and worked out in detail that, despite this local gauge freedom, the structures of a causal fermion system give rise to distinguished gauges where the local gauge freedom is fixed completely up to global gauge transformations. The main method is to use spectral and polar decompositions of operators on Hilbert spaces and on indefinite inner product spaces. We also introduce and make use of a Riemannian metric, which is induced on the manifold of all regular correlation operators by the Hilbert–Schmidt scalar product. Gaussian coordinate systems corresponding to this Riemannian metric are constructed. Moreover, we work with so-called wave charts where the physical wave functions are used as coordinates. Our constructions and results are illustrated in the example of Dirac sea configurations in finite and infinite spatial volume.

Finite-Volume Effects in (g-2)_{μ}^{HVP,LO}.

An analytic expression is derived for the leading-order finite-volume effects arising in lattice QCD calculations of the hadronic-vacuum-polarization contribution to the muon's magnetic moment a_{μ}^{HVP,LO}≡(g-2)_{μ}^{HVP,LO}/2. For calculations in a finite spatial volume with periodicity L, a_{μ}^{HVP,LO}(L) admits a transseries expansion with exponentially suppressed L scaling. Using a Hamiltonian approach, we show that the leading finite-volume correction scales as exp[-M_{π}L] with a prefactor given by the (infinite-volume) Compton amplitude of the pion, integrated with the muon-mass-dependent kernel. To give a complete quantitative expression, we decompose the Compton amplitude into the spacelike pion form factor F_{π}(Q^{2}) and a multiparticle piece. We determine the latter through next-to leading order in chiral perturbation theory and find that it contributes negligibly and through a universal term that depends only on the pion decay constant, with all additional low-energy constants dropping out of the integral.

Bose-Fermi cancellations without supersymmetry

We show that adjoint QCD features very strong Bose-Fermi cancellations in the large $N$ limit, despite the fact that it is manifestly non-supersymmetric. The difference between the bosonic and fermionic densities of states in large $N$ adjoint QCD turns out to have a `two-dimensional' scaling $\sim \exp{(\sqrt{\ell E})}$ for large energies $E$ in finite spatial volume, where $\ell$ is a length scale associated with the curvature of the spatial manifold. In particular, all Hagedorn growth cancels, and so does the growth $\exp{(V^{1/4} E^{3/4})}$ expected in a standard local 4d theory in spatial volume $V$. In these ways, large $N$ adjoint QCD, a manifestly non-supersymmetric theory, acts similarly to supersymmetric theories. We also show that at large $N$, the vacuum energy of multi-flavor adjoint QCD is non-negative and exponentially small compared to the UV cutoff with several natural regulators.

Analytic studies of static and transport properties of (gauged) Skyrmions

We study static and transport properties of Skyrmions living within a finite spatial volume in a flat (3+1)-dimensional spacetime. In particular, we derive an explicit analytic expression for the compression modulus corresponding to these Skyrmions living within a finite box and we show that such expression can produce a reasonable value. The gauged version of these solitons can be also considered. It is possible to analyze the order of magnitude of the contributions to the electrons conductivity associated to the interactions with this Baryonic environment. The typical order of magnitude for these contributions to conductivity can be compared with the experimental values of the conductivity of layers of Baryons.

The I = 1 pion–pion scattering amplitude and timelike pion form factor from Nf = 2 + 1 lattice QCD

The elastic I=1p-wave ππ scattering amplitude is calculated together with the isovector timelike pion form factor using lattice QCD with Nf=2+1 dynamical quark flavors. Wilson clover ensembles generated by the Coordinated Lattice Simulations (CLS) initiative are employed at four lattice spacings down to a=0.05fm, several pion masses down to mπ=200MeV, and spatial volumes of extent L=3.1–5.5fm. The set of measurements on these ensembles, which is publicly available, enables an investigation of systematic errors due to the finite lattice spacing and spatial volume. The ππ scattering amplitude is fit on each ensemble by a Breit–Wigner resonance lineshape, while the form factor is described better by a thrice-subtracted dispersion relation than the Gounaris–Sakurai parametrization.

Scattering processes and resonances from lattice QCD

The vast majority of hadrons observed in nature are not stable under the strong interaction, rather they are resonances whose existence is deduced from enhancements in the energy dependence of scattering amplitudes. The study of hadron resonances offers a window into the workings of quantum chromodynamics (QCD) in the low-energy non-perturbative region, and in addition, many probes of the limits of the electroweak sector of the Standard Model consider processes which feature hadron resonances. From a theoretical standpoint, this is a challenging field: the same dynamics that binds quarks and gluons into hadron resonances also controls their decay into lighter hadrons, so a complete approach to QCD is required. Presently, lattice QCD is the only available tool that provides the required non-perturbative evaluation of hadron observables. In this article, we review progress in the study of few-hadron reactions in which resonances and bound-states appear using lattice QCD techniques. We describe the leading approach which takes advantage of the periodic finite spatial volume used in lattice QCD calculations to extract scattering amplitudes from the discrete spectrum of QCD eigenstates in a box. We explain how from explicit lattice QCD calculations, one can rigorously garner information about a variety of resonance properties, including their masses, widths, decay couplings, and form factors. The challenges which currently limit the field are discussed along with the steps being taken to resolve them.

Penning trap with an inclined magnetic field.

A modified Penning trap with a spatially uniform magnetic field B inclined with respect to the axis of rotational symmetry of the electrodes is considered. The inclination angle can be arbitrary. Canonical transformation of phase variables transforming the Hamiltonian of the considered system into a sum of three uncoupled harmonic oscillators is found. We determine the region of stability in space of two parameters controlling the dynamics: the trapping parameter κ and the squared sine of the inclination angle ϑ0. If the angle ϑ0 is smaller than 54°, a charge occupies a finite spatial volume within the processing chamber. A rigid hierarchy of trapping frequencies is broken if B is inclined at the critical angle: the magnetron frequency reaches the modified cyclotron frequency while the axial frequency exceeds them. Apart from this resonance, we reveal the family of resonant curves in the region of stability. In the relativistic regime, the system is not linear. We show that it is not integrable in the Liouville sense. The averaging over the fast variable allows to reduce the system to two degrees of freedom. An analysis of the Poincaré cross-sections of the averaged systems shows the regions of effective stability of the trap.

Geoids in general relativity: geoid quasilocal frames

We develop, in the context of general relativity, the notion of a geoid—a surface of constant ‘gravitational potential’. In particular, we show how this idea naturally emerges as a specific choice of a previously proposed, more general and operationally useful construction called a quasilocal frame—that is, a choice of a two-parameter family of timelike worldlines comprising the worldtube boundary of the history of a finite spatial volume. We study the geometric properties of these geoid quasilocal frames, and construct solutions for them in some simple spacetimes. We then compare these results—focusing on the computationally tractable scenario of a non-rotating body with a quadrupole perturbation—against their counterparts in Newtonian gravity (the setting for current applications of the geoid), and we compute general-relativistic corrections to some measurable geometric quantities.

Low-energy QCD in the delta regime

We investigate properties of low-energy QCD in a finite spatial volume, but with arbitrary temperature. In the limit of small temperature and small cube size compared to the pion Compton wavelength, Leutwyler has shown that the effective theory describing low-energy QCD reduces to that of quantum mechanics on the coset manifold, which is the so-called delta regime of chiral perturbation theory. We solve this quantum mechanics analytically for the case of a $U(1)_L \times U(1)_R$ subgroup of chiral symmetry, and numerically for the case of $SU(2)_L \times SU(2)_R$. We utilize the quantum mechanical spectrum to compute the mass gap and chiral condensate, and investigate symmetry restoration in a finite spatial volume as a function of temperature. Because we obtain the spectrum for non-zero values of the quark mass, we are able to interpolate between the rigid rotor limit, which emerges at vanishing quark mass, and the harmonic approximation, which is referred to as the p-regime. We find that the applicability of perturbation theory about the rotor limit largely requires lighter-than-physical quarks. As a stringent check of our results, we raise the temperature to that of the inverse cube size. When this condition is met, the quantum mechanics reduces to a matrix model. The condensate we obtain in this limit agrees with that determined analytically in the epsilon regime.

Excited-state contribution to the axial-vector and pseudoscalar correlators with two extra pions

We study multi-particle state contributions to the QCD two-point functions of the axial-vector and pseudo-scalar quark bilinears in a finite spatial volume. For sufficiently small quark masses one expects three-meson states with two additional pions at rest to have the lowest total energy after the ground state. We calculate this three-meson state contribution using chiral perturbation theory. We find it to be strongly suppressed and too small to be seen in present-day lattice simulations.

Nucleon form factors with2+1flavor dynamical domain-wall fermions

We report our numerical lattice QCD calculations of the isovector nucleon form factors for the vector and axial-vector currents: the vector, induced tensor, axial-vector, and induced pseudoscalar form factors. The calculation is carried out with the gauge configurations generated with ${N}_{f}=2+1$ dynamical domain-wall fermions and Iwasaki gauge actions at $\ensuremath{\beta}=2.13$, corresponding to a cutoff ${a}^{\ensuremath{-}1}=1.73\text{ }\text{ }\mathrm{GeV}$, and a spatial volume of $(2.7\text{ }\text{ }\mathrm{fm}{)}^{3}$. The up and down-quark masses are varied so the pion mass lies between 0.33 and 0.67 GeV while the strange quark mass is about 12% heavier than the physical one. We calculate the form factors in the range of momentum transfers, $0.2<{q}^{2}<0.75\text{ }\text{ }{\mathrm{GeV}}^{2}$. The vector and induced tensor form factors are well described by the conventional dipole forms and result in significant underestimation of the Dirac and Pauli mean-squared radii and the anomalous magnetic moment compared to the respective experimental values. We show that the axial-vector form factor is significantly affected by the finite spatial volume of the lattice. In particular in the axial charge, ${g}_{A}/{g}_{V}$, the finite-volume effect scales with a single dimensionless quantity, ${m}_{\ensuremath{\pi}}L$, the product of the calculated pion mass and the spatial lattice extent. Our results indicate that for this quantity, ${m}_{\ensuremath{\pi}}L>6$ is required to ensure that finite-volume effects are below 1%.

Rigid motion revisited: rigid quasilocal frames

In this first of a series of papers we will introduce the notion of a rigid quasilocal frame (RQF) as a geometrically natural way to define a ‘system’ in the context of the dynamical spacetime of general relativity. An RQF is defined as a two-parameter family of timelike worldlines comprising the worldtube boundary (topologically ) of the history of a finite spatial volume, with the rigidity conditions that the congruence of worldlines is expansion-free (the ‘size’ of the system is not changing) and shear-free (the ‘shape’ of the system is not changing). This definition of a system is anticipated to yield simple, exact geometrical insights into the problem of motion in general relativity. It begins by answering, in a precise way, the questions of what is in motion (a rigid two-dimensional system boundary with topology S2, and whatever matter and/or radiation it happens to contain at the moment), and what motions of this rigid boundary are possible. Nearly a century ago Herglotz and Noether showed that a three-parameter family of timelike worldlines in Minkowski space satisfying Born's 1909 rigidity conditions does not have the 6 degrees of freedom we are familiar with from Newtonian mechanics, but a smaller number—essentially only 3. This result curtailed, to a large extent, subsequent study of rigid motion in special and (later) general relativity. We will argue that in fact we can implement Born's notion of rigid motion in both flat spacetime (this paper) and arbitrary curved spacetimes containing sources (subsequent papers)—with precisely the expected 3 translational and 3 rotational degrees of freedom (with arbitrary time dependence)—provided the system is defined quasilocally as the two-dimensional set of points comprising the boundary of a finite spatial volume, rather than the three-dimensional set of points within the volume.

Light front field theory: An advanced Primer

Light front field theory: An advanced PrimerWe present an elementary introduction to quantum field theory formulated in terms of Dirac's light front variables. In addition to general principles and methods, a few more specific topics and approaches based on the author's work will be discussed. Most of the discussion deals with massive two-dimensional models formulated in a finite spatial volume starting with a detailed comparison between quantization of massive free fields in the usual field theory and the light front (LF) quantization. We discuss basic properties such as relativistic invariance and causality. After the LF treatment of the soluble Federbush model, a LF approach to spontaneous symmetry breaking is explained and a simple gauge theory - the massive Schwinger model in various gauges is studied. A LF version of bosonization and the massive Thirring model are also discussed. A special chapter is devoted to the method of discretized light cone quantization and its application to calculations of the properties of quantum solitons. The problem of LF zero modes is illustrated with the example of the two-dimensional Yukawa model. Hamiltonian perturbation theory in the LF formulation is derived and applied to a few simple processes to demonstrate its advantages. As a byproduct, it is shown that the LF theory cannot be obtained as a "light-like" limit of the usual field theory quantized on an initial space-like surface. A simple LF formulation of the Higgs mechanism is then given. Since our intention was to provide a treatment of the light front quantization accessible to postgradual students, an effort was made to discuss most of the topics pedagogically and a number of technical details and derivations are contained in the appendices.

Volume and quark mass dependence of the chiral phase transition

We investigate chiral symmetry restoration in finite spatial volume and at finite temperature by calculating the dependence of the chiral phase transition temperature on the size of the spatial volume and the current-quark mass for the quark-meson model, using the proper-time Renormalization Group approach. We find that the critical temperature is weakly dependent on the size of the spatial volume for large current-quark masses, but depends strongly on it for small current-quark masses. In addition, for small volumes we observe a dependence on the choice of quark boundary conditions.

Wireless communication systems with-spatial diversity: a volumetric model

This paper presents a novel physical-modeling approach to wireless systems with multiple antennas. The fundamental problem of modeling the communication channel is studied, where the channel consists of a finite spatial volume for transmitting, a finite spatial volume for reception, and an arbitrary set of reflective-scattering bodies. The number of communication modes (or degrees of freedom) for such a system is calculated, using the procedure developed. We present a simple model for multipath channels, which allows insight into the development of a correlated multiple-input multiple-output (MIMO) channel model. In particular, the model is independent of transmitter and receiver elements and relies on the physical parameters of the channel involved. Our work explains which physical parameters determine the channel model and its channel capacity.

Light hadron spectroscopy with two flavors ofO(a)-improved dynamical quarks

We present a high statistics study of the light hadron spectrum and quark masses in QCD with two flavors of dynamical quarks. Numerical simulations are carried out using the plaquette gauge action and the O(a)-improved Wilson quark action at $\ensuremath{\beta}=5.2,$ where the lattice spacing is found to be $a=0.0887(11)\mathrm{fm}$ from the $\ensuremath{\rho}$ meson mass, on a ${20}^{3}\ifmmode\times\else\texttimes\fi{}48$ lattice. At each of five sea quark masses corresponding to ${m}_{\mathrm{PS}}{/m}_{\mathrm{V}}\ensuremath{\simeq}0.8--0.6,$ we generate 12 000 trajectories using a symmetrically preconditioned Hybrid Monte Carlo algorithm. Finite spatial volume effects are investigated employing ${12}^{3}\ifmmode\times\else\texttimes\fi{}48,$ ${16}^{3}\ifmmode\times\else\texttimes\fi{}48$ lattices. We also perform a set of simulations in quenched QCD with the same lattice actions at a similar lattice spacing to those for the full QCD runs. In the meson sector we find clear evidence of sea quark effects. The J parameter increases for lighter sea quark masses, and the full QCD meson masses are systematically closer to experiment than in quenched QCD. Careful finite-size studies are made to ascertain that these are not due to finite-size effects. Evidence of sea quark effects is less clear in the baryon sector due to larger finite-size effects. We also calculate light quark masses and find ${m}_{\mathrm{ud}}^{\overline{\mathrm{MS}}}(2\mathrm{}\mathrm{GeV}{)=3.223(}_{\ensuremath{-}0.069}^{+0.046})\mathrm{MeV}$ and ${m}_{s}^{\overline{\mathrm{MS}}}(2\mathrm{}\mathrm{GeV}{)=84.5(}_{\ensuremath{-}1.7}^{+12.0})\mathrm{MeV}$ which are about 20% smaller than in quenched QCD.