Momentum Cut-off Regularization Research Articles

Numerical study of the $\mathcal{N}=2$ Landau–Ginzburg model

It is believed that the two-dimensional massless $\mathcal{N}=2$ Wess--Zumino model becomes the $\mathcal{N}=2$ superconformal field theory (SCFT) in the infrared (IR) limit. We examine this theoretical conjecture of the Landau--Ginzburg (LG) description of the $\mathcal{N}=2$ SCFT by numerical simulations on the basis of a supersymmetric-invariant momentum-cutoff regularization. We study a single supermultiplet with cubic and quartic superpotentials. From two-point correlation functions in the IR region, we measure the scaling dimension and the central charge, which are consistent with the conjectured LG description of the $A_2$ and $A_3$ minimal models, respectively. Our result supports the theoretical conjecture and, at the same time, indicates a possible computational method of correlation functions in the $\mathcal{N}=2$ SCFT from the LG description.

Study of X(5568) in a unitary coupled-channel approximation of BK̄ and Bsπ

The potential of the B meson and the pseudoscalar meson is constructed up to the next-to-leading order Lagrangian, and then the BK̄ and Bsπ interaction is studied in the unitary coupled-channel approximation. A resonant state with a mass about 5568 MeV and JP=0+ is generated dynamically, which can be associated with the X(5568) state announced by the D0 Collaboration recently. The mass and the decay width of this resonant state depend on the regularization scale in the dimensional regularization scheme, or the maximum momentum in the momentum cutoff regularization scheme. The scattering amplitude of the vector B meson and the pseudoscalar meson is calculated, and an axial-vector state with a mass near 5620 MeV and JP=1+ is produced. Their partners in the charm sector are also discussed.

About renormalization of the Yang – Mills theory and the approach to calculation of the heat kernel

The quantum theory of Yang – Mills in four-dimensional space – time plays an important role in modern theoretical physics. Currently, this model contains many open problems, therefore, it is of great interest to mathematicians. This work consists of several parts, however, it only offers a new approach and, therefore, it is methodological. First of all, the diagram technique and the mathematical basis will be recalled in the first part. Then the process of renormalization will be explained. It is based on momentum cut-off regularization and described in [1] and [2]. However, this type of the regularization has several problems, as a result, only the first correction is calculated. After common constructions and observations, the first correction will be described in detail. Namely, the heat kernel will be considered since it plays a main role in this formalism. In particular, the method for calculating of coefficients of arbitrary order will be proposed.

Subtractive Renormalization of the NJL model *

In this work, we apply a subtractive renormalization method to the SU(2) Nambu-Jona-Lasinio (NJL) model in Born approximation and compare the results obtained here with those achieved by the standard momentum cutoff regularization. We have computed the dynamical quark mass, the chiral condensate, the pion mass and the quark-antiquark T-matrix as a function of the cutoff Λ and the subtraction scale ξ. We have shown that both approaches give similar results and the dependence of the physical quantities on the subtraction point ξ is much weaker than the dependence on the cutoff Λ.

Solution of the relativistic Schrödinger equation for theδ′-Function potential in one dimension using cutoff regularization

We study the relativistic version of Schr\"odinger equation for a point particle in 1-d with potential of the first derivative of the delta function. The momentum cutoff regularization is used to study the bound state and scattering states. The initial calculations show that the reciprocal of the bare coupling constant is ultra-violet divergent, and the resultant expression cannot be renormalized in the usual sense. Therefore a general procedure has been developed to derive different physical properties of the system. The procedure is used first on the non-relativistic case for the purpose of clarification and comparisons. The results from the relativistic case show that this system behaves exactly like the delta function potential, which means it also shares the same features with quantum field theories, like being asymptotically free, and in the massless limit, it undergoes dimensional transmutation and it possesses an infrared conformal fixed point.

GEOMETRICALLY RELATING MOMENTUM CUT-OFF AND DIMENSIONAL REGULARIZATION

The β function for a scalar field theory describes the dependence of the coupling constant on the renormalization mass scale. This dependence is affected by the choice of regularization scheme. I explicitly relate the β functions of momentum cut-off regularization and dimensional regularization on scalar field theories by a gauge transformation using the Hopf algebras of the Feynman diagrams of the theories.

Numerical simulation of the [formula omitted] Landau–Ginzburg model

The two-dimensional N = ( 2 , 2 ) Wess–Zumino (WZ) model with a cubic superpotential is numerically studied with a momentum-cutoff regularization that preserves supersymmetry. A numerical algorithm based on the Nicolai map is employed and the resulting configurations have no autocorrelation. This system is believed to flow to an N = ( 2 , 2 ) superconformal field theory (SCFT) in the infrared (IR), the A 2 model. From a finite-size scaling analysis of the susceptibility of the scalar field in the WZ model, we determine 1 − h − h ¯ = 0.616 ( 25 ) ( 13 ) for the conformal dimensions h and h ¯ , while 1 − h − h ¯ = 0.666 … for the A 2 model. We also measure the central charge in the IR region from a correlation function between conserved supercurrents and obtain c = 1.09 ( 14 ) ( 31 ) ( c = 1 for the A 2 model). These results are consistent with the conjectured emergence of the A 2 model, and at the same time demonstrate that numerical studies can be complementary to analytical investigations for this two-dimensional supersymmetric field theory.

Gravitational contributions to the running Yang-Mills coupling in large extra-dimensional brane worlds

We study the question of a modification of the running gauge coupling of Yang-Mills theories due to quantum gravitational effects in a compact large extra dimensional brane world scenario with a low energy quantum gravity scale. The ADD scenario is applied for a D=d+\delta dimensional space-time in which gravitons freely propagate, whereas the non-abelian gauge fields are confined to a d-dimensional brane. The extra dimensions are taken to be toroidal and the transverse fluctuation modes (branons) of the brane are taken into account. On this basis we have calculated the one-loop corrections due to virtual Kaluza-Klein graviton and branon modes for the gluon two- and three-point functions in an effective field theory treatment. Applying momentum cut-off regularization we find that for a d=4 brane the leading gravitational divergencies cancel irrespective of the number of extra dimensions \delta, generalizing previous results in the absence of extra-dimensions. Hence, again the Yang-Mills \beta-function receives no gravitational corrections at one-loop. This is no longer true in a `universal' extra dimensional scenario with a d>4 dimensional brane. Moreover, the subleading power-law gravitational divergencies induce higher-dimensional counterterms, which we establish in our scheme. Interestingly, for d=4 these gravitationally induced counterterms are of the form recently considered in non-abelian Lee-Wick extensions of the standard model -- now with a possible mass scale in the TeV range due to the presence of large extra dimensions.

Structure of logarithmically divergent one-loop lattice Feynman integrals

For logarithmically divergent one-loop lattice Feynman integrals $I(p,a)$, subject to mild general conditions, we prove the following expected and crucial structural result: $I(p,a)=f(p)\mathrm{log}(aM)+g(p)+h(p,M)$ up to terms which vanish for lattice spacing $a\ensuremath{\rightarrow}0$. Here $p$ denotes collectively the external momenta and $M$ is a mass scale which may be chosen arbitrarily. The $f(p)$ and $h(p,M)$ are shown to be universal and coincide with analogous quantities in the corresponding continuum integral when the latter is regularized either by momentum cutoff or dimensional regularization. The nonuniversal term $g(p)$ is shown to be a homogeneous polynomial in $p$ of the same degree as $f(p)$. This structure is essential for consistency between renormalized lattice and continuum formulations of QCD at one loop.

Two-loop calculations of the matrix σ-model effective action in the background field formalism

We consider the matrix σ-model in the background field formalism. In the two-loop approximation, we demonstrate the equality of “running coupling constants” in the momentum cutoff regularization and in the dimensional regularization by direct calculation. We verify that the β-function coincides with the previously obtained data.

Momentum-cutoff regularization and gauge invariance in QED

Based on a gauge-invariant form of the electron propagation function, we propose a formalism for QED which preserves its gauge-invariant character when both photon and electron propagators are regularized with a sharp momentum-cutoff procedure. Perturbation calculations of the regularized fermion effective action functional of an external electromagnetic field are given. We study radiative corrections induced by a momentum-cutoff vacuum and derive the corresponding Ward–Takahashi identity. Several problems encountered in an attempt of constructing a momentum-cutoff QED model are discussed.

On the prefactor in false vacuum decay

This paper gives a survey of recent work in collaboration with G Dunne concerning a new method for computing determinants in quantum field-theoretic applications, using angular momentum cut-off regularization and renormalization. This method is generally applicable to the situation of computing the quantum fluctuations about a classical configuration that has a symmetry allowing the fluctuation operator to be radially separable. There are many such cases of interest in quantum field theory. Here I describe the case of the false vacuum decay rate in a self-interacting scalar field theory modelling the process of nucleation in a four-dimensional spacetime. The rate prefactor involves quantum fluctuations about the classical bounce solution, which is O(4) symmetric. The computational method is based on the Gelfand–Yaglom approach to determinants of ordinary differential operators, suitably extended to higher dimensions using angular momentum cut-off regularization. I also present a simple new formula for the zero-mode contribution to the fluctuation prefactor, expressed entirely in terms of the asymptotic behaviour of the classical bounce solution.

Connecting the chiral and heavy quark limits: full mass dependence of fermion determinant in an instanton background

This talk reports work done in collaboration with Jin Hur, Choonkyu Lee and Hyunsoo Min concerning the computation of the precise mass dependence of the fermion determinant for quarks in the presence of an instanton background. The result interpolates smoothly between the previously known chiral and heavy quark limits of extreme small and large mass. The computational method makes use of the fact that the single instanton background has radial symmetry, so that the computation can be reduced to a sum over partial waves of logarithms of radial determinants, each of which can be computed numerically in an efficient manner using a theorem of Gelfand and Yaglom. The bare sum over partial waves is divergent and must be regulated and renormalized. We use the angular momentum cutoff regularization and renormalization scheme. Our results provide an extension of the Gelfand-Yaglom result to higher dimensional separable differential operators. I also comment on the application of this approach to a wide variety of fluctuation determinant computations in quantum field theory.

Beyond the thin-wall approximation: Precise numerical computation of prefactors in false vacuum decay

We present a general numerical method for computing precisely the false vacuum decay rate, including the prefactor due to quantum fluctuations about the classical bounce solution, in a self-interacting scalar field theory modeling the process of nucleation in four-dimensional space-time. This technique does not rely on the thin-wall approximation. The method is based on the Gelfand-Yaglom approach to determinants of differential operators, suitably extended to higher dimensions using angular momentum cutoff regularization. A related approach has been discussed recently by Baacke and Lavrelashvili, but we implement the regularization and renormalization in a different manner, and compare directly with analytic computations made in the thin-wall approximation. We also derive a simple new formula for the zero mode contribution to the fluctuation prefactor, expressed entirely in terms of the asymptotic behavior of the classical bounce solution.

Renormalized energy-momentum tensor of λ Φ4 theory in curved space-time

Divergenceless expression for the energy-momentum tensor of scalar field is obtained using the momentum cut-off regularization technique. We consider a scalar field with quartic self-coupling in a spatially flat (3+1)-dimensional Robertson-Walker space-time, having arbitrary mass and coupled to gravity. As special cases, energy-momentum tensor for conformal and minimal coupling are also obtained. The energy-momentum tensor is observed to exhibit trace anomaly in curved space-time

Top-quark pair production close to threshold: Top-quark mass, width, and momentum distribution

The complete NNLO QCD corrections to the total cross section $\sigma(e^+e^- \to Z*,\gamma*\to t\bar t)$ in the kinematic region close to the top-antitop threshold are calculated by solving the corresponding Schroedinger equations exactly in momentum space in a consistent momentum cutoff regularization scheme. The corrections coming from the same NNLO QCD effects to the top quark three-momentum distribution $d\sigma/d |\vec k_t|$ are determined. We discuss the origin of the large NNLO corrections to the peak position and the normalization of the total cross section observed in previous works and propose a new top mass definition, the 1S mass M_1S, which stabilizes the peak in the total cross section. If the influence of beamstrahlung and initial state radiation on the mass determination is small, a theoretical uncertainty on the 1S top mass measurement of 200 MeV from the total cross section at the linear collider seems possible. We discuss how well the 1S mass can be related to the $\bar{MS}$ mass. We propose a consistent way to implement the top quark width at NNLO by including electroweak effects into the NRQCD matching coefficients, which then can become complex.

A symmetry-preserving cut-off regularization

In this paper I show how Schwinger's proper time regularization can be formulated as a smooth momentum cut-off regularization. This technique preserves all symmetries present in the initial theory. For illustration, I apply the regularization scheme to theφ4 theory at zero and finite temperature.

On the renormalization of the supersymmetric Liouville action and the 3D Ising model critical exponent

Assuming the correspondence between the three-dimensional Ising model and the supersymmetric string theory conjectured by Polyakov, we calculate the critical exponent ν of the Ising model from the renormalization properties of the supersymmetric Liouville action. The result ν = 7 9 is obtained using a momentum-cutoff regularization procedure.