Self-consistent calculation of the optical potential and ground-state properties of nuclear matter

On the basis of the Green-function formalism, we performed a self-consistent calculation of the self-energy ∑(k, ω) of a particle interacting with the infinite nuclear medium. The function ∑(k, ω) was mapped out in the energy-momentum plane, and the single-particle energy ω(k), momentum distribution ?(k) and the “on-shell” part of the self-energy, ∑(k, ω(k)), were defined, from which all physical properties followed. In particular we investigated the ground-state properties of nuclear matter in two Λ-approximations of the T-matrix. In one, the intermediate two-particle propagator, Λ₀₀, represented free-particle propagation; in the other, called Λ₁₁, intermediate states included both interacting particles and holes. Pauli principle effects were included in both approximations. The second approximation was expected to be conserving because it included a large part of the rearrangement effects which, we found, contributed ～6 MeV per particle to the average energy and ～28 MeV to the singleparticle energy at zero momentum. The Hugenholtz-van Hove theorem was nearly satisfied, with only 1 MeV separating the chemical potential from the average energy. We also studied, in the Λ₀₀-approximation, the optical potential for the scattering of a particle by a large nucleus; it was directly related to the “on-shell” part of the self-energy. It was found that, below 100 MeV, the real part varied as (?90 + 0.584E) [MeV], and the imaginary part as (2.4 + 0.009 E) [MeV].     

The non-perturbative renormalization group in the ordered phase

We study some analytical properties of the solutions of the non-perturbative renormalization group flow equations for a scalar field theory with Z₂ symmetry in the ordered phase, i.e. at temperatures below the critical temperature. The study is made in the framework of the local potential approximation. We show that the required physical discontinuity of the magnetic susceptibility χ(M) at M=±M₀ (M₀ spontaneous magnetization) is reproduced only if the cut-off function which separates high and low energy modes satisfies to some restrictive explicit mathematical conditions; we stress that these conditions are not satisfied by a sharp cut-off in dimensions of space d<4.By generalizing a method proposed earlier by Bonanno and Lacagnina [Nucl. Phys. B 693 (2004) 36] to any kind of cut-off we propose to solve numerically the renormalization group flow equations for the threshold functions rather than for the local potential. It yields an algorithm sufficiently robust and precise to extract universal as well as non-universal quantities from numerical experiments at any temperature, in particular at sub-critical temperatures in the ordered phase. Numerical results obtained for the φ⁴ potential with three different cut-off functions are reported and compared. The data confirm our theoretical predictions concerning the analytical behavior of χ(M) at M=±M₀.Fixed point solutions of the adimensioned renormalization group flow equations are also obtained in the same vein, that is by solving the fixed points equations and the associated eigenvalue problem for the threshold functions rather than for the potential. We report high precision data for the odd and even spectra of critical exponents for different cut-offs obtained in this way.     

Momentum scale expansion of sharp cutoff flow equations

We show how the exact renormalization group for the effective action with a sharp momentum cutoff, may be organized by expanding one-particle irreducible parts in terms of homogeneous functions of momenta of integer degree (Taylor expansions not being possible). A systematic series of approximations - the O(p^M) approximations - result from discarding from these parts, all terms of higher than the M^th degree. These approximations preserve a field reparametrization invariance, ensuring that the field's anomalous dimension is unambiguously determined. The lowest order approximation coincides with the local potential approximation to the Wegner-Houghton equations. We discuss the practical difficulties with extending the approximation beyond O(p⁰).     

On an elaboration of M. Kac's theorem concerning eigenvalues of the Laplacian in a region with randomly distributed small obstacles

We removem-balls of centersw ₁,...,w _m with the same radius α/m from a bounded domain Ω inR ³ with smooth boundary γ. Let μ _k (α/m;w(m)) denote thek-th eigenvalue of the Laplacian in Ω/m-balls under the Dirichlet condition. We consider μ _k (α/m;w(m)) as a random variable on a probability space (w ₁,...,w _m)∈Ω × ... × Ω and we examine a precise behaviour of μ _k (α/m;w(m)) asm → ∞. We give an elaboration of. M. Kac's theorem.     

Second-order corrections to correlations in muon capture

Muon capture by a nucleus with an arbitrary spin is considered. Second-order terms in 1/M in the effective weak-interaction Hamiltonian are taken into account. New terms in the Hamiltonian associated with the nucleon-nucleus potential are found. A general expression for the angular distribution of neutrinos (recoil nuclei) is derived for polarized muons and oriented target nuclei. Second-order contributions to the amplitudes M _u (k) are obtained. This allows one to calculate second-order corrections to any integral and correlation characteristics in muon capture that are expressed in terms of M _u (k).     

On the limit cycle for the 1/r potential in momentum space

The renormalization of the attractive 1/r² potential has recently been studied using a variety of regulators. In particular, it was shown that renormalization with a square well in position space allows multiple solutions for the depth of the square well, including, but not requiring a renormalization group limit cycle. Here, we consider the renormalization of the 1/r² potential in momentum space. We regulate the problem with a momentum cutoff and absorb the cutoff dependence using a momentum-independent counterterm potential. The strength of this counterterm is uniquely determined and runs on a limit cycle. We also calculate the bound state spectrum and scattering observables, emphasizing the manifestation of the limit cycle in these observables.     

Intermediate-range potential and the 1S0neutron-proton transition matrix

The half-shell transition matrix t(p, k) for the singlet s-wave neutron-proton interaction has been studied for a class of partly non-local potentials. The potentials have been generated from the experimental phase shifts by using inverse scattering theory. The non-uniqueness of the inversion solution has been exploited to construct potentials with different short-range behaviour. It is shown that the intermediate-range potential essentially determines t(p, k) for momenta p and k both less than 2fm^?. Different short-range behaviour is reflected in t(p, k) for larger momenta. The unknown high-energy phase shifts imply, however, comparable variations in that region even if the on-shell momentum k is small. The implications for nuclear structure calculations are discussed.     

Pseudogap in the spin-polaron approach for the carrier spectrum of a two-dimensional doped antiferromagnet

In the Kondo model of the two-dimensional lattice with a strong spin-hole antiferromagnetic exchange, the pseudogap behavior of the carrier spectral function A(k, ω) is considered in the optimal and almost dielectric doping limits. A distinctive feature of the analysis is the introduction of the spin polaron even in the mean-field approximation that leads to the formation of two bands (the analogs of the upper and lower Hubbard bands) and makes it possible to immediately take into account the main rearrangement of A(k, ω). The inclusion of the scattering of the mean-field polaron (within the irreducible Green’s functions) describes the further rearrangement of A(k, ω), in particular, the unusual appearance of the pseudogap near the points N = (±π/2, ±π/2).     

Properties of quark matter governed by quantum chromodynamics (II). Renormalization schemes in quark matter

Renormalization schemes are examined (in the Coulomb gauge) for quantum chromodynamics in the presence of quark matter. We demand that the effective coupling constant for all schemes become congruent with the vacuum QCD running coupling constant as the matter chemical potential, μ, goes to zero. Also, to enable us to standardize with the vacuum QCD running coupling constant at some asymptotic momentum transfer, $\text{|}\text{p}{}_0\text{|,}\text{we keep}\text{μ ? ¦}\text{p}{}_0\text{¦}$, to ensure that the matter contribution is negligible at this point. This means all schemes merge with vacuum QCD at $\text{|}\text{p}{}_0\text{|}$ and beyond. Two renormalization group invariants are shown to emerge: (i) the effective or invariant charge, g_inv², which is, however, scheme dependent and (ii) g²(M)/S(M), where S(M)^?1 is the Coulomb propagator, which is scheme independent. The only scheme in which g_inv² is scheme independent and identical to g²(M)/S(M) is the screened charged scheme (previous paper) characterised by the normalization of the entire Green function, S^?1, to unity. We conclude that this is the scheme to be used if one wants to identify with the experimental effective coupling in perturbation theory. However, if we do not restrict to perturbation theory all schemes should be allowed. Although we discuss matter QCD in the Coulomb gauge, the above considerations are quite general to gauge theories in the presence of matter.     

On the asymptotic behaviour of the mass operator near the Fermi energy

By taking due account of momentum conservation, it is shown that, when ω is near the Fermi energy ω_F, the imaginary part of the mass operator $\text{M(}\text{k}\text{, ω)}$ for an infinite Fermi system behaves like (ω ? ω_F)^p(k) where the exponent p(k) ? 2 depends on the interval in which $\text{|}\text{k}\text{|}$ is lying. In particular, the commonly asserted quadratic behaviour (ω ? ω_F² is shown to be true only for $\text{|}\text{k}\text{| ? 3k}{}_{\mathrm{<mtext>F</mtext>}}$. It is explicity assumed that the Fermi system admits a perturbative type treatment.     

Quenching of the electron scattering form factor of the 1+ state at 3.48 MeV in88Sr and the influence of theΔ(1232)-isobar

The form factor for excitation of the 1⁺ state at 3.48 MeV in⁸⁸Sr by inelastic electron scattering has been measured for momentum transfersq=0.24–0.62 fm^?1. Neither its magnitude nor shape can be described employing the best available nuclear wave functions. We demonstrate with a schematic model that the observed reduction of the form factor may be understood by taking into account a renormalization of theM1-operator due to virtualΔ-hole excitations.     

More on the renormalization of the horizon function of the Gribov–Zwanziger action and the Kugo–Ojima Green function(s)

In this paper we provide strong evidence that there is no ambiguity in the choice of the horizon function underlying the Gribov–Zwanziger action. We show that there is only one correct possibility which is determined by the requirement of multiplicative renormalizability. As a consequence, this means that relations derived from other horizon functions cannot be given a consistent interpretation in terms of a local and renormalizable quantum field theory. In addition, we also discuss that the Kugo–Ojima functions u(p ²) and w(p ²) can only be defined after renormalization of the underlying Green function(s).     

How to compare the SU(5) value of sin2θw with experiments?

We establish the relation between κsin²θ_w to be found from neutral-current experiments and sin²θ_w(Q) for Q=M_W predicted by grand unified theories. We then calculate sin²θ_W(M_W) in the minimal SU(5) model taking the M_W as well as M_x threshold effects into account. We find that these two threshold effects on sin²θ_W(M_W) cancel with each other and sin²θ_W(M_W)=0.211± 0.005.     

On Lagrangian formulations for arbitrary bosonic HS fields on Minkowski backgrounds

We review the details of unconstrained Lagrangian formulations for Bose particles propagated on an arbitrary dimensional flat space-time and described by the unitary irreducible integer higher-spin representations of the Poincare group subject to Young tableaux Y(s ₁, ..., s _k ) with k rows. The procedure is based on the construction of scalar auxiliary oscillator realizations for the symplectic sp(2k) algebra which encodes the second-class operator constraints subsystem in the HS symmetry algebra. Application of an universal BRST approach reproduces gauge-invariant Lagrangians with reducible gauge symmetries describing the free dynamics of both massless and massive bosonic fields of any spin with appropriate number of auxiliary fields.     

Leading logarithmic corrections to the weak leptonic and semi-leptonic low-energy hamiltonian

If one defines the parameters of the Weinberg-Salam theory at a momentum scale M = O(M_W, M_Z), the weak effective hamiltonian at a momentum scale μ ? M has logarithmically enhanced corrections, of order αln(M²/μ²). We present a computation of these corrections, for that part of the hamiltonian which leads to detectable weak-electromagnetic interference effects. The largest correction can be absorbed into a running sin²θ(μ). Other, smaller, corrections are estimated, taking into account the effect of strong interactions.An estimate of the non-logarithmically enhanced corrections is also given, by evaluating them in the limit sin²θ → 0. From the SLAC e - d asymmetry it was found sin²θ = 0.224 ± 0.020 at μ² ? 2 GeV². In correspondence, we find sin²θ(M) = 0.217 ± 0.020. This value, however, is subject to uncertainties deriving from the effect of the strange and of the antiquark parton sea.     

Electromagnetic corrections to neutrino processes in the leading logarithmic approximation,the value of sin θ and the W and Z masses

The parameters of the Weinberg-Salam model can be defined by amplitudes at a momentum scale M = O(M_W, M_Z). We derive the leading logarithmic e.m. correction to the relations giving the neutrino amplitudes at a momentum scale μ ? M in terms of sin²θ(M), α(M), M_W and M_Z. For leptonic processes, the Fermi constant is not corrected, but a running, universal, sin²θ(μ) > sin²θ(M) should be used. The Fermi constant for semileptonic processes is renormalized by a factor $\text{?(μ) > 1}$, for charged currents, and is not renormalized, for neutral current processes. The latter are described by the same sin²θ(μ) as the leptonic ones. We estimate that sin²θ(M) is about 0.013 smaller than the value of sin²θ obtained from semileptonic data with no correction, thereby improving the agreement with grand unified theories. The prediction for W (Z) masses and widths in terms of the low energy parameters are discussed. Using previous calculations at order α, we obtain predictions for the masses which are accurate up to and including terms of order (αlnM²)².     

Study of two-phonon excitations in superfluid 4He

We explain the second branch of excitations in superfluid ⁴He observed by Cowley and Woods, by accounting for two-phonon contributions to the dynamic structure function, S(k, ω). Our theory gives a good fit with the experimental data in the high energy region for several values of momentum transfer. It is observed that the contribution to S(k, ω) due to two-phonon excitations is of the order of k² as against its k⁴ dependence reported in earlier theories.     

Equivalence of Two Definitions of the Effective Mass of a Polaron

Two definitions of the effective mass of a particle interacting with a quantum field, such as a polaron, are considered and shown to be equal in models similar to the Fröhlich polaron model. These are: 1. the mass defined by the low momentum energy E(P)≈E(0)+P ²/2M of the translation invariant system constrained to have momentum P and 2. the mass M of a simple particle in an arbitrary slowly varying external potential, V, described by the nonrelativistic Schrödinger equation, whose ground state energy equals that of the combined particle/field system in a bound state in the same V.     

Extremes of N Vicious Walkers for Large N: Application to the Directed Polymer and KPZ Interfaces

We compute the joint probability density function (jpdf) P _N (M,?? _M ) of the maximum M and its position ?? _M for N non-intersecting Brownian excursions, on the unit time interval, in the large N limit. For N????, this jpdf is peaked around $M = \sqrt{2N}$ and ?? _M =1/2, while the typical fluctuations behave for large N like $M - \sqrt{2N} \propto s N^{-1/6}$ and ?? _M ?1/2??wN ^?1/3 where s and w are correlated random variables. One obtains an explicit expression of the limiting jpdf P(s,w) in terms of the Tracy-Widom distribution for the Gaussian Orthogonal Ensemble (GOE) of Random Matrix Theory and a psi-function for the Hastings-McLeod solution to the Painlevé II equation. Our result yields, up to a rescaling of the random variables s and w, an expression for the jpdf of the maximum and its position for the Airy₂ process minus a parabola. This latter describes the fluctuations in many different physical systems belonging to the Kardar-Parisi-Zhang (KPZ) universality class in 1+1 dimensions. In particular, the marginal probability density function (pdf) P(w) yields, up to a model dependent length scale, the distribution of the endpoint of the directed polymer in a random medium with one free end, at zero temperature. In the large w limit one shows the asymptotic behavior logP(w)???w ³/12.