Bare Coupling Constant Research Articles

Theory of superconductivity mediated by Rashba coupling in incipient ferroelectrics

Experimental evidence suggests that superconductivity in ${\mathrm{SrTiO}}_{3}$ is mediated by a soft transverse ferroelectric mode which, according to conventional theories, has negligible coupling with electrons. A phenomenological Rashba-type coupling has been proposed on symmetry arguments but a microscopic derivation is lacking. Here, we fill this gap and obtain a linear coupling directly from a minimal microscopic model of the electronic structure. We find that the effective electron-electron pairing interaction has a strong momentum dependence. This yields an unusual situation in which the leading $s$-wave channel is followed by a subleading $p$-wave state which shows a stronger pairing instability than the $d$-wave state. The bare Rashba coupling constant is estimated for the lowest band of doped ${\mathrm{SrTiO}}_{3}$ with the aid of first-principles computations. Extrapolating the estimation to the multiband regime we show that it can explain bulk superconductivity. For densities where the Meissner effect has not been observed an extra softening due to inhomogeneities is needed to explain the zero resistance state.

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.

Background-field method and initial-time singularity for coherent states

The background field method is adopted for studying the dynamics of coherent states within an interacting scalar field theory. Focusing on a coherent state that corresponds to the homogeneous condensate, the quantum depletion of the expectation value of the field-operator is demonstrated to be due to the annihilation of the condensate constituents into relativistic quanta. Moreover, due to the fact that the initial field acceleration and energy for the non-squeezed coherent states are determined in terms of bare coupling constant, instead of the renormalized one, the appearance of perturbative singularities is shown to be inevitable. In other words, consistency of these states requires the finiteness of the bare coupling constant, through the resummation.

Evolution of coherent states as quantum counterpart of classical dynamics

Quantum dynamics of coherent states is studied within quantum field theory using two complementary methods: by organizing the evolution as a Taylor series in elapsed time and by perturbative expansion in coupling within the interaction-picture formalism. One of the important aspects of our analysis consists in utilizing the operators and the vacuum of interacting theory in constructing the states, without invoking asymptotic particles. Focusing on a coherent state describing a spatially homogeneous field configuration, it is demonstrated that both adopted methods successfully account for nonlinear classical dynamics, giving distinguishable quantum effects. In particular, according to the time-expansion analysis the initial field acceleration, with which the field departs from its initial expectation value, is governed by the tree-level potential with renormalized mass and bare coupling constant. The interaction-picture computation, instead, can be manipulated to give the nonlinear dynamics, determined in terms of renormalized coupling and mass. However, it results in a logarithmic initial-time singularity in the field acceleration, reminiscent of the similar behavior encountered within semiclassical formalism, for certain choices of the initial state for fluctuations. Within our coherent-state analysis, the above-mentioned peculiarities are artifacts of an expansion: in the first case over infinitesimal time, while in the second case in the coupling constant. Despite this, we show that the evolution obtained within the interaction-picture analysis is valid for an extended period of time. Moreover, on top of the desired classical dynamics, it serves us with interesting quantum corrections, previously proposed by Dvali-Gomez-Zell.

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.

Scaling theory of magnetism in frustrated Kondo lattices

A scaling theory of the Kondo lattices with frustrated exchange interactions is developed, criterium of antiferromagnetic ordering and quantum-disordered state being investigated. The calculations taking into account magnon and incoherent spin dynamics are performed. Depending on the bare model parameters, one or two quantum phase transitions into non-magnetic spin-liquid and Kondo Fermi-liquid ground states can occur with increasing the bare coupling constant. Whereas the renormalization of the magnetic moment in the ordered phase can reach orders of magnitude, spin fluctuation frequency and coupling constant are moderately renormalized in the spin-liquid phase. This justifies application of the scaling approach. Possibility of a non-Fermi-liquid behavior is treated.

On the scaling theory of antiferromagnetic ordering in frustrated Kondo lattices

A scaling theory of the Kondo lattices with frustrated exchange interactions is developed, criterium of antiferromagnetic ordering being investigated. Depending on the bare model parameters, one or two quantum phase transitions into non-magnetic spin-liquid and Kondo Fermi-liquid ground states can occur with increasing the bare coupling constant. Whereas the renormalization of the magnetic moment in the ordered phase can reach orders of magnitude, spin fluctuation frequency and coupling constant are moderately renormalized in the spin-liquid phase. This justifies application of the scaling approach.

Renormalization of the 4PI effective action using the functional renormalization group

Techniques based on $n$-particle irreducible effective actions can be used to study systems where perturbation theory does not apply. The main advantage, relative to other non-perturbative continuum methods, is that the hierarchy of integral equations that must be solved truncates at the level of the action, and no additional approximations are needed. The main problem with the method is renormalization, which until now could only be done at the lowest ($n$=2) level. In this paper we show how to obtain renormalized results from an $n$-particle irreducible effective action at any order. We consider a symmetric scalar theory with quartic coupling in four dimensions and show that the 4 loop 4-particle-irreducible calculation can be renormalized using a renormalization group method. The calculation involves one bare mass and one bare coupling constant which are introduced at the level of the Lagrangian, and cannot be done using any known method by introducing counterterms.

On Dimensional Regularization in the Yang–Mills Theory

We suggest an asymptotic approach to renormalization in the case of dimensional regularization. As an example, the quantum Yang–Mills theory in the four-dimensional space-time is considered. A formula for the renormalized effective action is derived by using the asymptotic behavior of the bare coupling constant. Then we discuss the dimensional transmutation, the process of renormalization, and the properties of the coupling constant.

Four Loop Scalar ϕ4 Theory Using the Functional Renormalization Group

We work with a symmetric scalar theory with quartic coupling in 4-dimensions. Using a 2PI effective theory and working at 4 loop order, we renormalize with a renormalization group method. All divergences are absorbed by one bare coupling constant and one bare mass which are introduced at the level of the Lagrangian. The method is much simpler than counterterm renormalization, and can be generalized to higher order nPI effective theories.

NSVZ Relation and the Dimensional Reduction in $$\mathcal{N}$$ = 1 SQED

It is known that factorization of the $\beta$-function loop integrals into integrals of double total derivatives is an important ingredient needed for deriving the NSVZ relation by direct perturbative calculations in ${\cal N}=1$ SQED regularized by the higher derivatives. It allows to relate the $\beta$-function and the anomalous dimension of the matter superfields defined in terms of the bare coupling constant. In this work we find the analog of this result in the case of using dimensional reduction regularization in the lowest orders. However, we demonstrate that in this case the NSVZ relation is not satisfied for the RG functions defined in terms of the bare coupling constant. Nevertheless, it is possible to impose boundary conditions to the renormalization constants determining the NSVZ scheme in the three-loop order for the RG functions defined in terms of the renormalized coupling constant.

Two-loop renormalization of the Faddeev-Popov ghosts in mathcal{N}=1 supersymmetric gauge theories regularized by higher derivatives

For the general renormalizable mathcal{N}=1 supersymmetric gauge theory we investigate renormalization of the Faddeev-Popov ghosts using the higher covariant derivative regularization. First, we find the two-loop anomalous dimension defined in terms of the bare coupling constant in the general ξ-gauge. It is demonstrated that for doing this calculation one should take into account that the quantum gauge superfield is renormalized in a nonlinear way. Next, we obtain the two-loop anomalous dimension of the Faddeev-Popov ghosts defined in terms of the renormalized coupling constant and examine its dependence on the subtraction scheme.

Asymptotic freedom and the thermal Lee model

Dimensional regularization is used to investigate the renormalization of the fermion mass, wave function, and coupling constant in the V-sector of a thermal Lee model in arbitrary space–time dimensions D and temperature T. A closed expression of the fermion mass renormalization shows that it diverges at the high-T limit and replicates a familiar form at the zero-T limit. Corresponding expressions of the wave function renormalization and the renormalized coupling constant vanish at the high-T limit, and resume respective customary forms at the zero-T limit. Likewise, the intrinsic scattering amplitude vanishes at the high-T limit, and reduces to the original amplitude at the zero-T limit. The 1D theory is especially useful for schematically graphing expressions that show thermal effects on the probability, the bare coupling constant, and the bare mass. Bifurcation of the bare parameters is a prominent feature in these graphs for particular choice of input. The β and γ coefficients of the Callan–Symanzik equation are calculated in closed form, and correlations between the β coefficient and the probability are in evidence for dimensions less than, or greater than 4. Schematic graphs incorporating Laurent expansions of the β coefficient, the probability, and the fermion-mass show the dependency of these functions upon the physical parameters in the theory for specific values of D. The thermal model is found to be asymptotically free for D < 4, and for odd D ≥ 5 at the high-T limit.

2PI effective theory at next-to-leading order using the functional renormalization group

We consider a symmetric scalar theory with quartic coupling in 4-dimensions. We show that the 4 loop 2PI calculation can be done using a renormalization group method. The calculation involves one bare coupling constant which is introduced at the level of the Lagrangian and is therefore conceptually simpler than a standard 2PI calculation, which requires multiple counterterms. We explain how our method can be used to do the corresponding calculation at the 4PI level, which can not be done using any known method by introducing counterterms.

Diagrammatic Monte Carlo for the weak-coupling expansion of non-Abelian lattice field theories: Large- N U(N)×U(N) principal chiral model

We develop numerical tools for Diagrammatic Monte-Carlo simulations of non-Abelian lattice field theories in the t'Hooft large-N limit based on the weak-coupling expansion. First we note that the path integral measure of such theories contributes a bare mass term in the effective action which is proportional to the bare coupling constant. This mass term renders the perturbative expansion infrared-finite and allows to study it directly in the large-N and infinite-volume limits using the Diagrammatic Monte-Carlo approach. On the exactly solvable example of a large-N O(N) sigma model in D=2 dimensions we show that this infrared-finite weak-coupling expansion contains, in addition to powers of bare coupling, also powers of its logarithm, reminiscent of re-summed perturbation theory in thermal field theory and resurgent trans-series without exponential terms. We numerically demonstrate the convergence of these double series to the manifestly non-perturbative dynamical mass gap. We then develop a Diagrammatic Monte-Carlo algorithm for sampling planar diagrams in the large-N matrix field theory, and apply it to study this infrared-finite weak-coupling expansion for large-N U(N)xU(N) nonlinear sigma model (principal chiral model) in D=2. We sample up to 12 leading orders of the weak-coupling expansion, which is the practical limit set by the increasingly strong sign problem at high orders. Comparing Diagrammatic Monte-Carlo with conventional Monte-Carlo simulations extrapolated to infinite N, we find a good agreement for the energy density as well as for the critical temperature of the "deconfinement" transition. Finally, we comment on the applicability of our approach to planar QCD at zero and finite density.

The Adler D-function for [formula omitted] SQCD regularized by higher covariant derivatives in the three-loop approximation

We calculate the Adler D-function for N=1 SQCD in the three-loop approximation using the higher covariant derivative regularization and the NSVZ-like subtraction scheme. The recently formulated all-order relation between the Adler function and the anomalous dimension of the matter superfields defined in terms of the bare coupling constant is first considered and generalized to the case of an arbitrary representation for the chiral matter superfields. The correctness of this all-order relation is explicitly verified at the three-loop level. The special renormalization scheme in which this all-order relation remains valid for the D-function and the anomalous dimension defined in terms of the renormalized coupling constant is constructed in the case of using the higher derivative regularization. The analytic expression for the Adler function for N=1 SQCD is found in this scheme to the order O(αs2). The problem of scheme-dependence of the D-function and the NSVZ-like equation is briefly discussed.

Exact renormalization of the photino mass in softly broken mathcal{N} = 1 SQED with Nf flavors regularized by higher derivatives

We consider the softly broken mathcal{N} = 1 supersymmetric electrodynamics, regularized by higher derivatives. For this theory we demonstrate that the renormalization of the photino mass is determined by integrals of double total derivatives in the momentum space in all orders. Consequently, it is possible to derive the NSVZ-like exact relation between the photino mass anomalous dimension and the anomalous dimension of the matter superfields in the rigid theory by direct summation of supergraphs. It is important that both these renormalization group functions are defined in terms of the bare coupling constant, so that the considered NSVZ-like relation is valid independently of the subtraction scheme in the case of using the higher derivative regularization. The factorization of integrals defining the photino mass renormalization into integrals of double total derivatives is verified by an explicit two-loop calculation.

The NSVZ scheme forN=1SQED with N f flavors, regularized by the dimensional reduction, in the three-loop approximation

At the three-loop level we analyze, how the NSVZ relation appears for ${\cal N}=1$ SQED regularized by the dimensional reduction. This is done by the method analogous to the one which was earlier used for the theories regularized by higher derivatives. Within the dimensional technique, the loop integrals cannot be written as integrals of double total derivatives. However, similar structures can be written in the considered approximation and are taken as a starting point. Then we demonstrate that, unlike the higher derivative regularization, the NSVZ relation is not valid for the renormalization group functions defined in terms of the bare coupling constant. However, for the renormalization group functions defined in terms of the renormalized coupling constant, it is possible to impose boundary conditions to the renormalization constants giving the NSVZ scheme in the three-loop order. They are similar to the all-loop ones defining the NSVZ scheme obtained with the higher derivative regularization, but are more complicated. The NSVZ schemes constructed with the dimensional reduction and with the higher derivative regularization are related by a finite renormalization in the considered approximation.

Structure of three-loop contributions to the β-function of N = 1 supersymmetric QED with N f flavors regularized by the dimensional reduction

In the case of using the higher derivative regularization for $N=1$ SQED with $N_f$ flavors the loop integrals giving the $\beta$-function are integrals of double total derivatives in the momentum space. This feature allows to reduce one of the loop integrals to an integral of the $\delta$-function and to derive the NSVZ relation for the renormalization group functions defined in terms of the bare coupling constant. In this paper we consider $N=1$ SQED with $N_f$ flavors regularized by the dimensional reduction in the $\overline{\mbox{DR}}$-scheme. Evaluating the scheme-dependent three-loop contribution to the $\beta$-function proportional to $(N_f)^2$ we find the structures analogous to integrals of the $\delta$-singularities. After adding the scheme-independent terms proportional to $(N_f)^1$ we obtain the known result for the three-loop $\beta$-function.

Asymptotic scaling and continuum limit of pure SU(3) lattice gauge theory

Recently the Yang-Mills gradient flow of pure SU(3) lattice gauge theory has been calculated in the range from $\beta=6/g_0^2=6.3$ to~7.5 (Asakawa et al.), where $g_0^2$ is the bare coupling constant of the SU(3) Wilson action. Estimates of the deconfining phase transition are available from $\beta =5.7$ to~6.8 (Francis et al.). Here it is shown that the entire range from 5.7 to 7.5 is well described by a power series of the lattice spacing $a$ times the lambda lattice mass scale $\Lambda_L$, using asymptotic scaling in the 2-loop and 3-loop approximations for $a\Lambda_L$. In both cases identical ratios for gradient flows versus deconfinement observables are obtained. Differences in the normalization constants with respect to $\Lambda_L$ give a handle on their systematic errors.