Spectroscopy and decay properties of B and Bs mesons

Lattice simulations of light nuclei necessarily take place in finite volumes, thus affecting their infrared properties. These effects can be addressed in a model-independent manner using Effective Field Theories. We study the model case of three identical bosons (mass m with resonant two-body interactions in a cubic box with periodic boundary conditions, which can also be generalized to the three-nucleon system in a straightforward manner. Our results allow for the removal of finite-volume effects from lattice results as well as the determination of infinite-volume scattering parameters from the volume dependence of the spectrum. We study the volume dependence of several states below the break-up threshold, spanning one order of magnitude in the binding energy in the infinite volume, for box side lengths L between the two-body scattering length a and L = 0.25a. For example, a state with a three-body energy of ?3/(ma ²) in the infinite volume has been shifted to ?10/(ma ²) at L = a. Special emphasis is put on the consequences of the breakdown of spherical symmetry and several ways to perturbatively treat the ensuing partial-wave admixtures. We find their contributions to be on the sub-percent level compared to the strong volume dependence of the S-wave component. For shallow bound states, we find a transition to boson-diboson scattering behavior when decreasing the size of the finite volume.     

A Wilson-Yukawa Model with undoubled chiral fermions in 2D

We consider the fermion mass spectrum in the strong coupling vortex phase (VXS) of a lattice fermion-scalar model with a global U(1)_L × U(1)_R, in two dimensions, in the context of a recently proposed two-cutoff lattice formulation. The fermion doublers are made massive by a strong Wilson-Yukawa coupling, but in contrast with the standard formulation of these type of models, in which the light fermion spectrum was found to be vector-like, we find massless fermions with chiral quantum numbers at finite lattice spacing. When the global symmetry is gauged, this model is expected to give rise to a lattice chiral gauge theory.     

The large-N deconfinement phase transition in a partially reduced Eguchi-Kawal model

We construct generalized twisted Eguchi-Kawai models which for large-N reduce space-time to a lattice of arbitrary size. Large-N lattice gauge theory at finite temperature is investigated in a model on a lattice with L₀ time slices and two lattice points in very time slice. We observe the large-N deconfinement phase transition in the weak coupling region. Assuming asymptotic scaling we find a transition temperature T_c = (101±4)Λ_L.     

Unified approach to Thermodynamic Bethe Ansatz and finite size corrections for lattice models and field theories

We present a unified approach to the Thermodynamic Bethe Ansatz (TBA) for magnetic chains and field theories that includes the finite size (and zero-temperature) calculations for lattice BA models. In all cases, the free energy follows by quadratures from the solution of a single nonlinear integral equation (NLIE) (a system of NLIE appears for nested BA). We derive the NLIE for: (a) the six-vertex model with twisted boundary conditions, (b) the XXZ chain in an external magnetic field h_z and (c) the massive Thirring sine-Gordon model (mT-sG) in a periodic box of size β ≡ T^-1 using the light-cone approach. This NLIE is solved by iteration in one regime (high T in the XXZ chain and low T in the sG-mT model). In the opposite (conformal) regime, the leading behaviors are obtained in closed form. Higher corrections can be derived from the Riemann-Hilbert form of the NLIE that we present.     

Zeroing in on SU(3)

We present an improved numerical method for calculating the density of states for lattice field theories. We use it to study the SU(3) pure gauge theory at both zero and finite temperature. We also compute strong and weak coupling expansions for the density of states and find excellent agreement with our data. Using a specially developed algorithm for solving high-order polynomials, we find the zeroes of the partition function. For lattices with L_t = 2, we test the finite-size scaling prediction for the rounding of the transition by following the motion of these zeroes for L_s=6, 8, 10, and 12. We find that the correlation length exponent is 1/v = 3.02 ± 0.05, in excellent agreement with the value d=3 expected for a first-order deconfinement transition.     

Twist and finite size effects for the source method

We analyze the finite size effects for the source method in pure lattice gauge theory at weak coupling. They are found to be strongly suppressed by twisting the boundary conditions, forSU(3) by typically an order of magnitude.     

Three-Body Physics in a Finite Volume

Three particles with large two-body scattering lengths display universal properties including a spectrum of three-body bound states called “Efimov trimers”. I calculate the spectrum of three identical bosons inside a finite cubic box below the three-body breakup threshold. The dependence of the spectrum on the box size and the effects of the breakdown of spherical symmetry are investigated using effective field theory. The renormalization of the effective field theory in the finite volume is explicitly verified. The study of the three-nucleon system inside a finite cubic volume provides a tool for the understanding of Lattice QCD results. I study the triton in a finite volume at physical and unphysical pion masses.     

Analytic approximations to hamiltonian lattice field theories: (I). Periodic gaussian model for U(1) theories

Periodic gaussian models are introduced for local and global U(1) invariant hamiltonian lattice field theories. The models coincide with standard lattice theories at weak coupling, but the leading non-perturbative contributions to wave functions and physical quantities are exactly calculable. Electric charges are confined and the mass gap is finite if correlations of an integer-valued magnetic field are of infinite range (d = 2 + 1 gauge model). Otherwise, for short-range correlations, the mass gap and the string tension vanish at weak coupling (QED, XY model, etc.)     

Lyapunov Bounds for Lattice Gauge Dynamics

We study the classical Hamiltonian dynamics of the Kogut–Susskind model for lattice gauge theories on a finite box in a d-dimensional integer lattice. The coupling constant for the plaquette interaction is denoted λ². When the gauge group is a real or a complex subgroup of a unitary matrix group U(N), N≥ 1, we show that the maximal Lyapunov exponent is bounded by , uniformly in the size of the lattice, the energy of the system as well as the order, N, of the gauge group. Received: 20 December 1997 / Accepted: 21 July 1998     

Finite size effects on the helical edge states on the Lieb lattice

For a two-dimensional Lieb lattice,that is,a line-centered square lattice,the inclusion of the intrinsic spin–orbit(ISO)coupling opens a topologically nontrivial gap,and gives rise to the quantum spin Hall(QSH) effect characterized by two pairs of gapless helical edge states within the bulk gap.Generally,due to the finite size effect in QSH systems,the edge states on the two sides of a strip of finite width can couple together to open a gap in the spectrum.In this paper,we investigate the finite size effect of helical edge states on the Lieb lattice with ISO coupling under three different kinds of boundary conditions,i.e.,the straight,bearded and asymmetry edges.The spectrum and wave function of edge modes are derived analytically for a tight-binding model on the Lieb lattice.For a strip Lieb lattice with two straight edges,the ISO coupling induces the Dirac-like bulk states to localize at the edges to become the helical edge states with the same Dirac-like spectrum.Moreover,it is found that in the case with two straight edges the gapless Dirac-like spectrum remains unchanged with decreasing the width of the strip Lieb lattice,and no gap is opened in the edge band.It is concluded that the finite size effect of QSH states is absent in the case with the straight edges.However,in the other two cases with the bearded and asymmetry edges,the energy gap induced by the finite size effect is still opened with decreasing the width of the strip.It is also proposed that the edge band dispersion can be controlled by applying an on-site potential energy on the outermost atoms.     

Phase transition in four-dimensional compact QED

The energy and the specific heat of the four-dimensional U(1) lattice gauge model is evaluated by Monte Carlo simulations on lattices of size L⁴, where L = 4, 5 and 6, evidence is presented for the occurence of a second-order phase transition. A finite size scaling analysis of our results gives the critical value of the coupling constant e²_c = 0.995 and a correlation length exponent $\text{v ≈}\text{1}\text{3}$.     

Violation of finite-size scaling in three dimensions

We reexamine the range of validity of finite-size scaling in the lattice model and the field theory below four dimensions. We show that general renormalization-group arguments based on the renormalizability of the theory do not rule out the possibility of a violation of finite-size scaling due to a finite lattice constant and a finite cutoff. For a confined geometry of linear size L with periodic boundary conditions we analyze the approach towards bulk critical behavior as at fixed for where is the bulk correlation length. We show that for this analysis ordinary renormalized perturbation theory is sufficient. On the basis of one-loop results and of exact results in the spherical limit we find that finite-size scaling is violated for both the lattice model and the field theory in the region . The non-scaling effects in the field theory and in the lattice model differ significantly from each other. Received 5 February 1999     

Λ-parameters for asymmetric lattice gauge theories with fermions

Dimensional quantities obtained from Monte Carlo simulations on the lattice depend on the lattice mass parameter, Λ_L. To make a connection with continuum physics, a relationship is needed between Λ_L and the Λ-parameters of the continuum theory. This has been done for the euclidean symmetric lattice by others. However, in order to incorporate finite temperature into Monte Carlo studies, or to study the transition from the euclidean formulation to the hamiltonian formulation of gauge theories, asymmetric lattices (a_s≠a_t) may be used. In this paper, the assymetric calculations are extended and the ratio $\text{Λ}{}_{\mathrm{<mtext>min</mtext>}}\text{Λ}{}_{\mathrm{<mtext>L</mtext>}}$, where Λ_min is the continuum mass parameter in the minimal subtraction scheme, is given to one loop for n_f flavors of Wilson and Susskind massless fermions on an asymmetric four-dimensional lattice for two different asymmetric lattice actions.     

Strong-coupling limit for one-dimensional polarons in a finite box

The strong coupling limit is studied for a Pekar-Fröhlich polaron confined to a one-dimensional (1D) structure. The non-linear effective Schrödinger equation is solved exactly in the case of two different external potentials which imitate a finite size 1D sample: an infinite and a finite deep rectangular well. The ground state and excited states are calculated. We found that taking the limit of a finite size box to an infinitely large box leads to additional solutions which are not found in a treatment on an infinite axis. The additional solutions, which have a 1/n ² discrete spectrum, correspond to polaron states in which the wave function is split up in identical parts which are infinitely apart from each other.     

Strong-coupling limit for one-dimensional polarons in a finite box

The strong coupling limit is studied for a Pekar-Fröhlich polaron confined to a one-dimensional (1D) structure. The non-linear effective Schrödinger equation is solved exactly in the case of two different external potentials which imitate a finite size 1D sample: an infinite and a finite deep rectangular well. The ground state and excited states are calculated. We found that taking the limit of a finite size box to an infinitely large box leads to additional solutions which are not found in a treatment on an infinite axis. The additional solutions, which have a 1/n ² discrete spectrum, correspond to polaron states in which the wave function is split up in identical parts which are infinitely apart from each other.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

Pion mass dependence of the <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">l</Emphasis>3</Subscript> semileptonic scalar form factor within finite volume

We calculate the scalar semileptonic kaon decay in finite volume at the momentum transfer t _m =(m _K −m _π )², using chiral perturbation theory. At first we obtain the hadronic matrix element to be calculated in finite volume. We then evaluate the finite size effects for two volumes with L=1.83 fm and L=2.73 fm and find that the difference between the finite volume corrections of the two volumes are larger than the difference as quoted in Boyle et al. (Phys. Rev. Lett. 100:141601, 2008). It appears then that the pion masses used for the scalar form factor in ChPT are large which result in large finite volume corrections. If appropriate values for pion mass are used, we believe that the finite size effects estimated in this paper can be useful for lattice data to extrapolate at large lattice size.