Renormalized Coupling Constant Research Articles

Algebraic structure of the renormalization group in the renormalizable QFT theories

In this paper, we consider the group formed by finite renormalizations as an infinite-dimensional Lie group. It is demonstrated that for the finite renormalization of the gauge coupling constant, its generators [Formula: see text] with [Formula: see text] satisfy the commutation relations of the Witt algebra and, therefore, form its subalgebra. The commutation relations are also written for the more general case when finite renormalizations are made for both the coupling constant and matter fields. We also construct the generator of the Abelian subgroup corresponding to the changes of the renormalization scale. The explicit expressions for the renormalization group generators are written in the case when they act on the [Formula: see text]-function and the anomalous dimension. It is explained how the finite changes of these functions under the finite renormalizations can be obtained with the help of the exponential map.

Evidence of a second-order phase transition in the six-dimensional Ising spin glass in a field.

The very existence of a phase transition for spin glasses in an external magnetic field is controversial, even in high dimensions. We carry out massive simulations of the Ising spin-glass in a field, in six dimensions (which, according to classical-but not generally accepted-field-theoretical studies, is the upper critical dimension). We obtain results compatible with a second-order phase transition and estimate its critical exponents for the simulated lattice sizes. The detailed analysis performed by other authors of the replica symmetric Hamiltonian, under the hypothesis of critical behavior, predicts that the ratio of the renormalized coupling constants remain bounded as the correlation length grows. Our numerical results are in agreement with this expectation.

Hamiltonian truncation crafted for UV-divergent QFTs

We develop the theory of Hamiltonian Truncation (HT) to systematically study RG flows that require the renormalization of coupling constants. This is a necessary step towards making HT a fully general method for QFT calculations. We apply this theory to a number of QFTs defined as relevant deformations of d=1+1 CFTs. We investigated three examples of increasing complexity: The deformed Ising, Tricritical-Ising, and non-unitary minimal model M(3,7). The first two examples provide a crosscheck of our methodologies against well established characteristics of these theories. The M(3,7) CFT deformed by its \mathbb{Z}_2ℤ2-even operators shows an intricate phase diagram that we clarify. At a boundary of this phase diagram we show that this theory flows, in the IR, to the M(3,5) CFT.

A model for time-evolution of coupling constants

A general model is proposed for time-varying coupling constants in field theory, assuming the ultraviolet cutoff is a varying entity in the expanding universe. It is assumed that the cutoff depends on the scale factor of the universe and all bare couplings remain constant. This leads to varying renormalized coupling constants that evolve in proportion to the Hubble parameter. The evolution of the standard model constants is discussed.

Coefficients at powers of logarithms in the higher-derivatives and minimal-subtractions-of-logarithms renormalization scheme

For renormalizable theories with a single coupling constant regularized by higher derivatives we investigate the coefficients at powers of logarithms present in the renormalization constants assuming that divergences are removed by minimal subtractions of logarithms. According to this higher-derivatives and minimal-subtractions-of-logarithms ($\mathrm{HD}+\mathrm{MSL}$) renormalization prescription, the renormalization constants include only powers of $\mathrm{ln}\mathrm{\ensuremath{\Lambda}}/\ensuremath{\mu}$, where $\mathrm{\ensuremath{\Lambda}}$ and $\ensuremath{\mu}$ are the dimensionful regularization parameter and the renormalization point, respectively. We construct general explicit expressions for arbitrary coefficients at powers of this logarithm present in the coupling constant renormalization and in the field renormalization constant which relate them to the $\ensuremath{\beta}$-function and (in the latter case) to the corresponding anomalous dimension. To check the correctness, we compare the results with the explicit four-loop calculation made earlier for $\mathcal{N}=1$ supersymmetric quantum electrodynamics and (for the supersymmetric case) rederive a relation between the renormalization constants following from the Novikov, Shifman, Vainshtein, and Zakharov equation.

Gravitational corrections to a non-Abelian gauge theory

This paper is part of a series of papers exploring the renormalization of field theories coupled to gravity using the effective field theory framework. In previous works, we studied the universality of the electric charge and the two-loops beta function in the Einstein-QED system. Now, we use this framework to study the non-Abelian gauge theory with fermions coupled to gravity. We show that even though some of the counterterms are dependent on the gravitational coupling, the renormalized coupling constant, and therefore the beta function, do not receive any gravitational correction at one-loop order. Also, we explicitly show that, at this order, the Slavnov-Taylor identities are respected. Finally, we end the paper by putting into perspective the results of the present and the past papers.

Three-Body Hamiltonian with Regularized Zero-Range Interactions in Dimension Three

We study the Hamiltonian for a system of three identical bosons in dimension three interacting via zero-range forces. In order to avoid the fall to the center phenomenon emerging in the standard Ter-Martirosyan–Skornyakov (TMS) Hamiltonian, known as Thomas effect, we develop in detail a suggestion given in a seminal paper of Minlos and Faddeev in 1962 and we construct a regularized version of the TMS Hamiltonian which is self-adjoint and bounded from below. The regularization is given by an effective three-body force, acting only at short distance, that reduces to zero the strength of the interactions when the positions of the three particles coincide. The analysis is based on the construction of a suitable quadratic form which is shown to be closed and bounded from below. Then, domain and action of the corresponding Hamiltonian are completely characterized and a regularity result for the elements of the domain is given. Furthermore, we show that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled non-local interactions, also called separable potentials, with a suitably renormalized coupling constant.

Vortex lattices in binary Bose–Einstein condensates: collective modes, quantum fluctuations, and intercomponent entanglement

We study binary Bose–Einstein condensates subject to synthetic magnetic fields in mutually parallel or antiparallel directions. Within the mean-field theory, the two types of fields have been shown to give the same vortex-lattice phase diagram. We develop an improved effective field theory to study properties of collective modes and ground-state intercomponent entanglement. Here, we point out the need to introduce renormalized coupling constants for coarse-grained densities. We show that the low-energy excitation spectra for the two types of fields are related to each other by suitable rescaling with the renormalized coupling constants. By calculating the entanglement entropy, we find that for an intercomponent repulsion (attraction), the two components are more strongly entangled in the case of parallel (antiparallel) fields, in qualitative agreement with recent studies for a quantum (spin) Hall regime. We also find that the entanglement spectrum exhibits an anomalous square-root dispersion relation, which leads to a subleading logarithmic term in the entanglement entropy. All of these are confirmed by numerical calculations based on the Bogoliubov theory with the lowest-Landau-level approximation. Finally, we investigate the effects of quantum fluctuations on the phase diagrams by calculating the correction to the ground-state energy due to zero-point fluctuations in the Bogoliubov theory. We find that the boundaries between rhombic-, square-, and rectangular-lattice phases shift appreciably with a decrease in the filling factor.

The three-loop anomalous dimension and the four-loop β-function for mathcal{N} = 1 SQED regularized by higher derivatives

For mathcal{N} = 1 SQED with Nf flavors regularized by higher derivatives in the general ξ-gauge we calculate the three-loop anomalous dimension of the matter superfields defined in terms of the bare coupling constant and demonstrate its gauge independence. After this the four-loop β-function defined in terms of the bare coupling constant is obtained with the help of the NSVZ equation, which is valid for these renormalization group functions in all loops. Next, we calculate the three-loop anomalous dimension and the four-loop β-function defined in terms of the renormalized coupling constant for an arbitrary subtraction scheme supplementing the higher derivative regularization. Then we construct a renormalization prescription for which the results coincide with the ones in the overline{mathrm{DR}} -scheme and describe all NSVZ schemes in the considered approximation. Also we demonstrate the existence of a subtraction scheme in which the anomalous dimension does not depend on Nf, while the β-function contains only terms of the first order in Nf. This scheme is obtained with the help of a finite renormalization compatible with a structure of quantum corrections and is NSVZ. The existence of such an NSVZ scheme is also proved in all loops.

Universality of gauge coupling constant in the Einstein-QED system

In this paper, we discuss the universality of the renormalization of the gauge coupling constant in the quantum electrodynamics coupled to the Einstein's gravity in the framework of effective field theory in an arbitrary gauge. We observe that the renormalization of the three-point functions with different particles receive different contributions from the gravitational sector. Despite that, the universality of the gauge coupling constants is guaranteed by the Ward identity that makes the renormalized electric charge receive no contribution from the gravitational sector at one-loop order.

Two-loop mass anomalous dimension in reduced quantum electrodynamics and application to dynamical fermion mass generation

We consider reduced quantum electrodynamics ( {mathrm{RQED}}_{d_{gamma },{d}_e} ) a model describing fermions in a de-dimensional space-time and interacting via the exchange of massless bosons in dγ-dimensions (de ≤ dγ). We compute the two-loop mass anomalous dimension, γm, in general {mathrm{RQED}}_{4,{d}_e} with applications to RQED4,3 and QED4. We then proceed on studying dynamical (parity-even) fermion mass generation in {mathrm{RQED}}_{4,{d}_e} by constructing a fully gauge-invariant gap equation for {mathrm{RQED}}_{4,{d}_e} with γm as the only input. This equation allows for a straightforward analytic computation of the gauge-invariant critical coupling constant, αc, which is such that a dynamical mass is generated for αr> αc, where αr is the renormalized coupling constant, as well as the gauge-invariant critical number of fermion flavours, Nc, which is such that αc → ∞ and a dynamical mass is generated for N < Nc. For RQED4,3, our results are in perfect agreement with the more elaborate analysis based on the resolution of truncated Schwinger-Dyson equations at two-loop order. In the case of QED4, our analytical results (that use state of the art five-loop expression for γm) are in good quantitative agreement with those obtained from numerical approaches.

Reweighting in Monte-Carlo calculation Renormalized Coupling Constant for the Three-Dimensional Ising Model

We investigated the thermodynamic behaviour of the three-dimensional Ising model near the critical temperature region using Monte Carlo calculations. The magnetic susceptibility, the correlation-length and renormalized coupling constant were calculated using the reweighting method.

The renormalization structure of [formula omitted] supersymmetric higher-derivative gauge theory

We consider the harmonic superspace formulation of higher-derivative 6D,N=(1,0) supersymmetric gauge theory and its minimal coupling to a hypermultiplet. In components, the kinetic term for the gauge field in such a theory involves four space-time derivatives. The theory is quantized in the framework of the superfield background method ensuring manifest 6D,N=(1,0) supersymmetry and the classical gauge invariance of the quantum effective action. We evaluate the superficial degree of divergence and prove it to be independent of the number of loops. Using the regularization by dimensional reduction, we find possible counterterms and show that they can be removed by the coupling constant renormalization for any number of loops, while the divergences in the hypermultiplet sector are absent at all. Assuming that the deviation of the gauge-fixing term from that in the Feynman gauge is small, we explicitly calculate the divergent part of the one-loop effective action in the lowest order in this deviation. In the approximation considered, the result is independent of the gauge-fixing parameter and agrees with the earlier calculation for the theory without a hypermultiplet.

Renormalization of the electron g factor in the degenerate two-dimensional electron gas of ZnSe- and CdTe-based quantum wells

The effective electron $g$ factor, ${g}_{\text{eff}}$, is measured in a two-dimensional electron gas (2DEG) in modulation-doped ZnSe- and CdTe-based quantum wells by means of time-resolved pump-probe Kerr rotation. The measurements are performed in magnetic fields applied in the Voigt geometry, i.e., normal to the optical axis parallel to the quantum well plane, in the field range $0.05--6$ T at temperatures $1.8--50\phantom{\rule{0.28em}{0ex}}\mathrm{K}$. The ${g}_{\text{eff}}$ absolute value considerably increases with increasing electron density ${n}_{e}$. ${g}_{\text{eff}}$ changes in the ZnSe-based QWs from $+1.1$ to $+1.9$ in the ${n}_{e}$ range $3\ifmmode\times\else\texttimes\fi{}{10}^{10}\ensuremath{-}1.4\ifmmode\times\else\texttimes\fi{}{10}^{12}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}2}$ and in the CdTe-based QWs from $\ensuremath{-}1.55$ down to $\ensuremath{-}1.76$ in the ${n}_{e}$ range $5\ifmmode\times\else\texttimes\fi{}{10}^{9}\ensuremath{-}3\ifmmode\times\else\texttimes\fi{}{10}^{11}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}2}$. The modification of ${g}_{\text{eff}}$ reduces with increasing magnetic field, increasing temperature of lattice and 2DEG, the latter achieved by a higher photoexcitation density. A theoretical model is developed that considers the renormalization of the spin-orbit coupling constant of the two-dimensional electrons by the electron-electron interaction and takes into account corrections to the electron-electron interaction in the Hubbard form. The model results are in good agreement with experimental data.

Exact β-Function in Abelian and non-Abelian $${\cal N} = 1$$ Supersymmetric Gauge Models and Its Analogy with the QCD β-Function in the C-scheme

For $${\cal N} = 1$$ supersymmetric Yang—Mills theory without matter it is demonstrated that there is a class of renormalization schemes, in which the exact Novikov, Shifman, Vainshtein, and Zakharov (NSVZ) formula for the renormalization group β-function, defined in terms of the renormalized coupling constant, is valid. These schemes are related with each other by finite renormalizations forming a one-parameter commutative subgroup of general renormalization group transformations. The analogy between the exact β-function in $${\cal N} = 1$$ supersymmetric Yang-Mills theory without matter and the β-function of quantum chromodynamics in the C-scheme is discussed.

Three-loop verification of a new algorithm for the calculation of a β-function in supersymmetric theories regularized by higher derivatives for the case of [formula omitted] SQED

We verify a recently proposed method for obtaining a β-function of N=1 supersymmetric gauge theories regularized by higher derivatives by an explicit calculation. According to this method, a β-function can be found by calculating specially modified vacuum supergraphs instead of a much larger number of the two-point superdiagrams. The result is produced in the form of a certain integral of double total derivatives with respect to the loop momenta. Here we compare the results obtained for the three-loop β-function of N=1 SQED in the general ξ-gauge with the help of this method and with the help of the standard calculation. Their coincidence confirms the correctness of the new method and the general argumentation used for its derivation. Also we verify that in the considered approximation the NSVZ relation is valid for the renormalization group functions defined in terms of the bare coupling constant and for the ones defined in terms of the renormalized coupling constant in the HD+MSL scheme, both its sides being gauge-independent.

Wilson-Fisher fixed points for any dimension

The critical behavior of a non-local scalar field theory is studied. This theory has a non-local quartic interaction term which involves a real power -\beta of the Laplacian. The parameter \beta can be tuned so as to make that interaction marginal for any dimension. The lowest order Feynman diagrams corresponding to coupling constant renormalization, mass renormalization, and field renormalization are computed. In all cases a non-trivial IR fixed point is obtained. Remarkably, for dimensions different from 4, field renormalization is required at the one-loop level. For d=4, the theory reduces to the usual local \phi^{4} field theory and field renormalization is required starting at the the two-loop level. The critical exponents \nu and \eta are computed for dimensions 2,3,4 and 5. For dimensions greater than four, the critical exponent \eta turns out to be negative for \epsilon>0, which indicates a violation of the unitarity bounds.

Universal effective couplings of the three-dimensional n-vector model and field theory

We calculate the universal ratios R2k of renormalized coupling constants g2k entering the critical equation of state for the generalized Heisenberg (three-dimensional n-vector) model. Renormalization group (RG) expansions of R8 and R10 for arbitrary n are found in the four-loop and three-loop approximations respectively. Universal octic coupling R8⁎ is estimated for physical values of spin dimensionality n=0,1,2,3 and for n=4,...64 to get an idea about asymptotic behavior of R8⁎. Its numerical values are obtained by means of the resummation of the RG series and within the pseudo-ε expansion approach. Regarding R10 our calculations show that three-loop RG and pseudo-ε expansions possess big and rapidly growing coefficients for physical values of n what prevents getting fair numerical estimates.

On Itzykson-Zuber Ansatz

We apply the renormalized coupling constants and Virasoro constraints to derive the Itzykson-Zuber Ansatz on the form of the free energy in 2D topological gravity. We also treat the 1D topological gravity and the Hermitian one-matrix models in the same fashion. Some uniform behaviors are discovered in this approach.

Light front Hamiltonian for boson form of QED (1 + 1) in Pauli–Villars regularization

Light front (LF) Hamiltonian for QED in [Formula: see text] dimensions is constructed using the boson form of this model with additional Pauli–Villars-type ultraviolet regularization. Perturbation theory, generated by this LF Hamiltonian, is proved to be equivalent to usual covariant chiral perturbation theory. The obtained LF Hamiltonian depends explicitly on chiral condensate parameters which enter in a form of some renormalization of coupling constants. The obtained results can be useful when one attempts to apply LF Hamiltonian approach for [Formula: see text]-dimensional models like QCD.