Renormalized Coupling Constant Research Articles

Renormalization properties of the mass operator [formula omitted] in three-dimensional Yang-Mills theories in the Landau gauge

Massive renormalizable Yang-Mills theories in three dimensions are analyzed within the algebraic renormalization in the Landau gauge. In analogy with the four-dimensional case, the renormalization of the mass operator A μ a A μ a turns out to be expressed in terms of the fields and coupling constant renormalization factors. We verify the relation we obtain for the operator anomalous dimension by explicit calculations in the large N f expansion. The generalization to other gauges such as the non-linear Curci–Ferrari gauge is briefly outlined.

Ghost busting:PT-symmetric interpretation of the Lee model

The Lee model was introduced in the 1950s as an elementary quantum field theory in which mass, wave function, and charge renormalization could be carried out exactly. In early studies of this model it was found that there is a critical value of g^2, the square of the renormalized coupling constant, above which g_0^2, the square of the unrenormalized coupling constant, is negative. Thus, for g^2 larger than this critical value, the Hamiltonian of the Lee model becomes non-Hermitian. It was also discovered that in this non-Hermitian regime a new state appears whose norm is negative. This state is called a ghost state. It has always been assumed that in this ghost regime the Lee model is an unacceptable quantum theory because unitarity appears to be violated. However, in this regime while the Hamiltonian is not Hermitian, it does possess PT symmetry. It has recently been discovered that a non-Hermitian Hamiltonian having PT symmetry may define a quantum theory that is unitary. The proof of unitarity requires the construction of a new time-independent operator called C. In terms of C one can define a new inner product with respect to which the norms of the states in the Hilbert space are positive. Furthermore, it has been shown that time evolution in such a theory is unitary. In this paper the C operator for the Lee model in the ghost regime is constructed exactly in the V/N-theta sector. It is then shown that the ghost state has a positive norm and that the Lee model is an acceptable unitary quantum field theory for all values of g^2.

Higgs coupling constants as a probe of new physics

We study new physics effects on the couplings of weak gauge bosons with the lightest CP-even Higgs boson ($h$), $hZZ$, and the trilinear coupling of the lightest Higgs boson, $hhh$, at the one-loop order, as predicted by the two Higgs doublet model. Those renormalized coupling constants can deviate from the standard model (SM) predictions due to two distinct origins: the tree level mixing effect of Higgs bosons and the quantum effect of additional particles in loop diagrams. The latter can be enhanced in the renormalized $hhh$ coupling constant when the additional particles show the nondecoupling property. Therefore, even in the case where the $hZZ$ coupling is close to the SM value, deviation in the $hhh$ coupling from the SM value can become as large as plus 100%, while that in the $hZZ$ coupling is at most minus 1% level. Such large quantum effect on the Higgs trilinear coupling is distinguishable from the tree level mixing effect, and is expected to be detectable at a future linear collider.

Two-loop scalar self-mass during inflation

We work in the locally de Sitter background of an inflating universe and consider a massless, minimally coupled scalar with a quartic self-interaction. We use dimensional regularization to compute the fully renormalized scalar self-mass-squared at one- and two-loop order for a state which is released in Bunch–Davies vacuum at t = 0. Although the field strength and coupling constant renormalizations are identical to those of flat space, the geometry induces a non-zero mass renormalization. The finite part also shows a sort of growing mass that competes with the classical force in eventually turning off this system's super-acceleration.

Multi-particle Schrödinger operators with point interactions in the plane

We study a system of N bosons in the plane interacting with delta function potentials. After a coupling constant renormalization we show that the Hamiltonian defines a self-adjoint operator and obtain a lower bound for the energy. The same results hold if one includes a regular inter-particle potential.

Universality in nontrivial continuum limits: A model calculation

We study numerically the continuum limit corresponding to the non-trivial fixed point of Dyson's hierarchical model. We discuss the possibility of using the critical amplitudes as input parameters. We determine numerically the leading and subleading critical amplitudes of the zero-momentum $2l$-point functions in the symmetric phase up to l=10 for randomly chosen local measures. In the infinite cutoff limit, the dimensionless renormalized coupling constants are in very good approximation universal (independent of the choice of the local measure). In addition, ratios of subleading amplitudes also appear to be universal. If we neglect very small log-periodic corrections, the non-universal features of the 2l-point functions appear to depend only the non-universal features of the 2-point function. We infer that when 2l becomes large, the dimensionless renormalized couplings grow as (2l)! despite the non-perturbative nature of our calculation, while the universal ratios of subleading amplitudes grow linearly.

Quantum Monte Carlo study of quasi-one-dimensional Bose gases

We study the behaviour of quasi-one-dimensional (quasi-1D) Bose gases by Monte Carlo techniques, i.e. by the variational Monte Carlo, the diffusion Monte Carlo and the fixed-node diffusion Monte Carlo techniques. Our calculations confirm and extend our results of an earlier study (Astrakharchik et al 2003 Preprint cond-mat/0308585). We find that a quasi-1D Bose gas (i) is well described by a 1D model Hamiltonian with contact interactions and renormalized coupling constant; (ii) reaches the Tonks–Girardeau regime for a critical value of the 3D scattering length a3D; (iii) enters a unitary regime for |a3D| → ∞, where the properties of the gas are independent of a3D and are similar to those of a 1D gas of hard-rods and (iv) becomes unstable against cluster formation for a critical value of the 1D gas parameter. The accuracy and implications of our results are discussed in detail.

A random matrix approach to the crossover of energy-level statistics from Wigner to Poisson

We analyze a class of parametrized random matrix models, introduced by Rosenzweig and Porter, which is expected to describe the energy level statistics of quantum systems whose classical dynamics varies from regular to chaotic as a function of a parameter. We compute the generating function for the correlations of energy levels, in the limit of infinite matrix size. The crossover between Poisson and Wigner statistics is measured by a renormalized coupling constant. The model is exactly solved in the sense that, in the limit of infinite matrix size, the energy-level correlation functions and their generating function are given in terms of a finite set of integrals.

The Functional Renormalization Group Treatment of Disordered Systems, a Review

We review current progress in the functional renormalization group treatment of disordered systems. After an elementary introduction into the phenomenology, we show why in the context of disordered systems a functional renormalization group treatment is necessary, contrary to pure systems, where renormalization of a single coupling constant is sufficient. This leads to a disorder distribution, which after a finite renomalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We discuss, how a renormalizable field theory can be constructed, even beyond 1-loop order. We then discuss an elastic manifold imbedded in N dimensions, and give the exact solution for N -> oo. This is compared to predictions of the Gaussian replica variational ansatz, using replica symmetry breaking. We finally discuss depinning, both isotropic and anisotropic, and the scaling function for the width distribution of an interface.

VANISHING OF THE BARE COUPLING IN FOUR DIMENSIONS

We examine two restructurings of the series relationship between the bare and renormalized coupling constant in dimensional regularization. In one of these restructurings, we are able to demonstrate via all-orders summation of leading and successive ∊ = 0 (dimensionality = 4) poles that the bare coupling vanishes in the dimension-4 limit.

Theory of domain patterns in systems with long-range interactions of Coulomb type.

We develop a theory of the domain patterns in systems with competing short-range attractive interactions and long-range repulsive Coulomb interactions. We take an energetic approach, in which patterns are considered as critical points of a mean-field free energy functional. Close to the microphase separation transition, this functional takes on a universal form, allowing us to treat a number of diverse physical situations within a unified framework. We use asymptotic analysis to study domain patterns with sharp interfaces. We derive an interfacial representation of the pattern's free energy which remains valid in the fluctuating system, with a suitable renormalization of the Coulomb interaction's coupling constant. We also derive integro-differential equations describing stationary domain patterns of arbitrary shapes and their thermodynamic stability, coming from the first and second variations of the interfacial free energy. We show that the length scale of a stable domain pattern must obey a certain scaling law with the strength of the Coulomb interaction. We analyzed the existence and stability of localized (spots, stripes, annuli) and periodic (lamellar, hexagonal) patterns in two dimensions. We show that these patterns are metastable in certain ranges of the parameters and that they can undergo morphological instabilities leading to the formation of more complex patterns. We discuss nucleation of the domain patterns by thermal fluctuations and pattern formation scenarios for various thermal quenches. We argue that self-induced disorder is an intrinsic property of the domain patterns in the systems under consideration.

Magnetic-field effects in energy relaxation mediated by Kondo impurities in mesoscopic wires

We study the energy distribution function of quasiparticles in short voltage biased mesoscopic wires in the presence of magnetic impurities and applied magnetic field. The system is described by a Boltzmann equation where the collision integral is determined by coupling to spin-$\frac{1}{2}$ impurities. We develop a theory of the coupling of nonequilibrium electrons to dissipative spins. This theory is valid as long as the characteristic smearing of the steps in the energy distribution function, which depends both on the bias voltage and the location of the probe, exceeds the Kondo temperature. We further address the renormalization of coupling constants by nonequilibrium electrons. Magnetic-field dependence of the energy relaxation rate turns out to be nonmonotonic. For low magnetic field an enhancement of energy relaxation is found, whereas for larger magnetic fields the energy relaxation decreases again meeting qualitatively the experimental findings by Anthore et al. (cond-mat/0109297). This gives a strong indication that magnetic impurities are in fact responsible for the enhanced energy relaxation in copper wires. Our theoretical results are in good agreement with the experiment at large bias voltages where the theory is applicable. At the same time, at small bias voltages there are substantial quantitative deviations. Furthermore, the concentration of the spins, which follows from the energy relaxation for Cu, seems to to be substantially higher than the concentration estimated from weak localization (dephasing rate) measurements. Since the approach presented is valid only above Kondo temperature, it does not apply to the related problem of weak localization at low temperature in equilibrium.

Inclusive quasielastic ν reaction in 16O

We calculate the inclusive quasielastic ν μ cross sections in 16O. The calculations are done in the Local Density Approximation(LDA) and take into account Pauli blocking, Fermi motion and the effect of renormalization of weak coupling constants in the nuclear medium.

Singular quantum mechanical viewpoint of localized gravity in brane-world scenario

The graviton localized on the 3-brane is examined in Randall–Sundrum brane-world scenario from the viewpoint of one-dimensional singular quantum mechanics. For the Randall–Sundrum single brane scenario the one-parameter family of the fixed-energy amplitude is explicitly computed where the free parameter ξ parametrizes the various boundary conditions at the brane. The general criterion for the localized graviton to be massless is derived when ξ is arbitrary but non-zero. When ξ=0, the massless graviton is obtained via a coupling constant renormalization. For the two branes picture the fixed-energy amplitude is in general dependent on the two free parameters. The numerical test indicates that there is no massless graviton in this picture. For the positive-tension brane, however, the localized graviton becomes massless when the distance between branes are infinitely large, which is essentially identical to the single brane picture. For the negative-tension brane there is no massless graviton regardless of the distance between branes and choice of boundary conditions.

Thermal effects in the Ginzburg–Landau model in a magnetic field

We consider the Ginzburg–Landau N-component model in the large N limit, concurrently with the Matsubara thermal field formalism, the system being embedded in an external constant magnetic field. In this double framework, we investigate the behaviour of the coupling constant and the mass as functions of temperature. Closed exact formulas for the thermal renormalized coupling constant can be obtained in even spacial dimensions, in particular, d=4. We are also able, after performing a mass renormalization, to obtain in spacial dimension d=3, a critical equation relating temperature to the external magnetic field. This equation is compatible with a second-order transition.

Compromise of localized graviton with a small cosmological constant in Randall–Sundrum scenario

A new mechanism which leads to a linearized massless graviton localized on the brane is found in the $AdS$/CFT setting, {\it i.e.} in a single copy of $AdS_5$ spacetime with a singular brane on the boundary, within the Randall-Sundrum brane-world scenario. With an help of a recent development in path-integral techniques, a one-parameter family of propagators for linearized gravity is obtained analytically, in which a parameter $\xi$ reflects various kinds of boundary conditions that arise as a result of the half-line constraint. In the case of a Dirichlet boundary condition ($\xi = 0$) the graviton localized on the brane can be massless {\it via} coupling constant renormalization. Our result supports a conjecture that the usual Randall-Sundrum scenario is a regularized version of a certain underlying theory.

Boundary dependence of the coupling constant and the mass in the vectorN-component (λφ4)Dtheory

Using the Matsubara formalism, we consider the massive (λ4)D vector N-component model in the large-N limit, the system being confined between two infinite parallel planes. We investigate the behaviour of the coupling constant as a function of the separation L between the planes. For the Wick-ordered model in D = 3 we are able to give an exact formula to the L dependence of the coupling constant. For the non-Wick-ordered model we indicate how expressions for the coupling constant and the mass can be obtained for arbitrary dimension D in the small-L regime. Closed exact formulae for the L-dependent renormalized coupling constant and mass are obtained in D = 3 and their behaviours as functions of L are displayed. We are also able to obtain, in generic dimension D, an equation for the critical value of L corresponding to a second-order phase transition in terms of the Riemann zeta-function. In D = 3 an implicit formula for the critical L is given.

Analytic calculation of the 1-loop effective action for the O( [formula omitted])-symmetric 2-dimensional nonlinear σ-model

Starting from the 2-dimensional nonlinear σ-model living on a lattice Λ of lattice spacing a with action S[φ]=− 1 2 β∫ zφΔφ , φ(z)∈S N we compute a manifestly covariant closed form expression for the Wilson effective action S eff[ Φ] on a lattice of lattice spacing ã in a 1-loop approximation for a Gaussian choice of blockspin, where Cφ(x)≡ Cφ(x)/| Cφ(x)| fluctuates around Φ( x). C is averaging of φ( z) over a block x. The limiting case of a δ-function is also considered. The result extends Polyakov which had furnished those contributions to the effective action which are of order ln a ̃ /a . The additional terms which remain finite as a↦0 include corrections other than coupling constant renormalization: a current–current interaction and a contribution from an augmented Jacobian which has a field dependence of a different kind than S has. Particular attention is paid to S eff's domain of validity in field space. It turns out that Hasenfratz and Niedermayer's choice of a low value of the parameter κ which governs the width of the Gaussian is optimal also in this respect.

Renormalization of parameters of a soft breakdown of supersymmetry in the regime of strong yukawa coupling within a nonminimal supersymmetric standard model

Within a nonminimal supersymmetric (SuSy) model, the renormalization of trilinear coupling constants A i(t) for scalar fields and of specific combinations \(\mathfrak{M}_i^2 (t)\) of the scalar-particle masses is investigated in the regime of strong Yukawa coupling. The dependence of these parameters on their initial values at the Grand Unification scale disappears as solutions to the renormalization-group equations approach infrared quasifixed points with increasing Y i(0). In the vicinities of quasifixed points for \(\tilde \alpha _{GUT} \ll Y_i (0) \ll 1\), all solutions A i(t) and \(\mathfrak{M}_i^2 (t)\) are concentrated near some straight lines or planes in the space of parameters of a soft breakdown of supersymmetry. This behavior of the solutions in question is explained by a sufficiently slow disappearance of the A i(0) and \(\mathfrak{M}_i^2 (t)\) dependence of the trilinear coupling constants and combinations of the scalar-particle masses. A method is proposed for deriving equations describing the aforementioned straight lines and planes, and the process of their formation is discussed by considering the example of exact and approximate solutions to the renormalization-group equations within a nonminimal supersymmetric standard model.

Granular superconductors: From the nonlinearσmodel to the Bose-Hubbard description

We modify a nonlinear $\ensuremath{\sigma}$ model $(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ for the description of a granular disordered system in the presence of both the Coulomb repulsion and the Cooper pairing. We show that under certain controlled approximations the action of this model is reduced to the Ambegaokar-Eckern-Sch\"on (AES) action, which is further reduced to the Bose-Hubbard (or ``dirty-boson'') model with renormalized coupling constants. We obtain an effective action which is more general than the AES one but still simpler than the full $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ action. This action can be applied in the region of parameters where the reduction to the AES or the Bose-Hubbard model is not justified. This action may lead to a different picture of the superconductor-insulator transition in two-dimensional systems.