Renormalization Procedure Research Articles

A variational approach to S1-harmonic maps and applications

We present a renormalization procedure for the Dirichlet Lagrangian for maps from surfaces with or without boundary into S1, whose finite energy critical points are the S1-harmonic maps with isolated singularities. We give some applications of this renormalization scheme in two different frameworks. The first application has to do with the renormalization of the Willmore energy for Lagrangian singular immersions into Kähler-Einstein surfaces while the second application is dealing with frame energies for surfaces immersions into Euclidian spaces.

Running vacuum in QFT in FLRW spacetime: the dynamics of rho _{textrm{vac}}(H) from the quantized matter fields

Phenomenological work in the last few years has provided significant support to the idea that the vacuum energy density (VED) is a running quantity with the cosmological evolution and that this running helps to alleviate the cosmological tensions afflicting the Lambda CDM. On the theoretical side, recent devoted studies have shown that the properly renormalized rho _{textrm{vac}} in QFT in FLRW spacetime adopts the ‘running vacuum model’ (RVM) form. While in three previous studies by two of us (CMP and JSP) such computations focused solely on scalar fields non-minimally coupled to gravity, in the present work we compute the spin-1/2 fermionic contributions and combine them both. The calculation is performed using a new version of the adiabatic renormalization procedure based on subtracting the UV divergences at an off-shell renormalization point M. The quantum scaling of rho _{textrm{vac}} with M turns into cosmic evolution with the Hubble rate, H. As a result the ‘cosmological constant’ Lambda appears in our framework as the nearly sustained value of 8pi G(H)rho _{textrm{vac}}(H) around (any) given epoch H, where G(H) is the gravitational coupling, which is also running, although very mildly (logarithmically). We find that the VED evolution at present reads delta rho _textrm{vac}(H)sim nu _{textrm{eff}}, m_{textrm{Pl}}^2 left( H^2-H_0^2 right) (|nu _{textrm{eff}}|ll 1). The coefficient nu _{textrm{eff}} receives contributions from all the quantized fields, bosons and fermions, which we compute here for an arbitrary number of matter fields. Remarkably, there also exist higher powers mathcal{O}(H^{6}) which can trigger inflation in the early universe. Finally, the equation of state (EoS) of the vacuum receives also quantum corrections from bosons and fermion fields, shifting its value from − 1. The striking consequence is that the EoS of the quantum vacuum may nowadays effectively appears as quintessence.

Broken scale invariance and the regularization of a conformal sector in gravity with Wess-Zumino actions

We elaborate on anomaly induced actions of the Wess-Zumino (WZ) form and their relation to the renormalized effective action, which is defined by an ordinary path integral over a conformal sector, in an external gravitational background. In anomaly - induced actions, the issue of scale breaking is usually not addressed, since these actions are obtained only by solving the trace anomaly constraint and are determined by scale invariant functionals. We investigate the changes induced in the structure of such actions once identified in dimensional renormalization (DR) when the ϵ=d−4→0 limit is accompanied by the dimensional reduction (DRed) of the field dependencies. We show that operatorial nonlocal modifications (∼□ϵ) of the counterterms are unnecessary to justify a scale anomaly. In this case, only the ordinary finite subtractions play a critical role in the determination of the scale breaking. This is illustrated for the WZ form of the effective action and its WZ consistency condition, as seen from a renormalization procedure. Logarithmic corrections from finite subtractions are also illustrated in a pure d=4 (cutoff) scheme. The interplay between two renormalization schemes, one based on dimensional regularization (DR) and the second on a cutoff in d=4, illustrates the ambiguities of DR in handling the quantum corrections in a curved background. Therefore, using DR in a curved background, the scale and trace anomalies can both be obtained by counterterms that are Weyl invariant only at d=4.

Four-point functions of gravitons and conserved currents of CFT in momentum space: testing the nonlocal action with the TTJJ

We present an analysis of the perturbative realization of the TTJJ correlator, with two stress energy tensors and two conserved currents, using free field theory, integrating out conformal sectors in the quantum corrections. This allows to define, around flat space, an exact perturbative expansion of the complete anomaly effective action – up to 4-point functions – whose predictions can be compared against those of the anomaly induced action. The latter is a variational solution of the conformal anomaly constraint at d=4 in the form of a nonlocal Wess–Zumino action. The renormalization procedure and the degeneracies of the tensor structures of this correlator are discussed, valid for a generic conformal field theory, deriving its anomalous trace Ward identities (WIs). In this application, we also illustrate a general procedure that identifies the minimal number of tensorial structures and corresponding form factors for the TTJJ and any 4-point function. The approach is implemented for three, four and five dimensions, addressing the tensor degeneracies of the expansion in momentum space. We show that the renormalized TTJJ can be split into two contributions, a non anomalous and an anomalous part, each separately conserved. The first satisfies ordinary trace WIs, while the second satisfies anomalous trace WIs. The result of the direct computation is compared against the expression of the same 4-point function derived from the nonlocal anomaly induced action. We show that the prediction for the anomalous part of the TTJJ derived from such action, evaluated in two different conformal decompositions, the Riegert and Fradkin–Vilkovisky (FV) choices, differ from the anomaly part identified in the perturbative TTJJ, in the flat spacetime limit. The anomaly part of the correlator computed with the Riegert choice is affected by double poles, while the one computed with the FV choice does not satisfy the conservation WIs. We present the correct form of the expansion of the anomaly induced action at the second order in the metric perturbations around flat space that reproduces the perturbative result.

Enhancing Concept Inventory Analysis by Using Indexes, Optimal Histogram Idea, and the Likert Analysis

Since the introduction of the Force Concept Inventory (FCI) in 1992, the CI tests have been widely used for measuring conceptual knowledge and for studying teaching issues in almost all disciplines and levels of study. A standard concept inventory analysis includes the design of a qualitative test, adequate realization of testing, calibration procedure, and comprehensive analysis of its findings. Usually, the CI test calibration is carried out through the Rasch sociometric technique, which is also used for calculating crucial indicators of knowledge such as item difficulties, students’ abilities, and many more. Whereas the quality of the tests’ design can be guaranteed by using certified and professional CI tests, the statistical adequacy of the testing merits critical attention before going on to the final step of the analysis. Also, the analysis of CI outcomes can be advanced by contemplating auxiliary tools and complementary techniques. In this framework, we propose to enforce the test index validity requirement for qualifying the CI outcomes as local or global. Specifically, the conclusions of CI analysis are acceptable for the whole population from which the sample has been extracted if the test's indexes comply with the validity requirements provided by the index theory. In the case when test indexes are out of validity range and re-conducting them is impractical for some objective circumstances or research specifics, we suggest injecting some new records into the existing one or mixing the data gathered from different samples until the new indexes are in the desired range. Using this methodology, we have reviewed our previous FCI tests, which were initially intended to demonstrate the impairment of learning in the physics discipline triggered by online learning during the pandemic closure. Through this renormalization procedure, we obtained a credible assessment of the understanding of mechanics and electromagnetism in high school students who followed online lectures during the pandemic closure. Also, by using indexes’ validity as an auxiliary tool, we identified that for measuring the knowledge of electromagnetism in students enrolled in branches where physics is a basic discipline, a shortened version of the BEMA test was a better instrument than the corresponding shortened EMCI test. Next, we used the optimal histogram idea borrowed from distribution fitting procedures to identify the natural levels of students’ abilities for solving a certain CI test. Another intriguing proposal presented in this work consists of combining an ad-hoc Likert scale assignment for usual errors in physics exams with the FCI designation of the basic commonsense confusion in mechanics for identifying their pairing features in common exams. We believe that the methods proposed herein can improve CI analysis in more general senses. Doi: 10.28991/HEF-2023-04-01-08 Full Text: PDF

Introduction to Renormalization Theory and Chiral Gauge Theories in Dimensional Regularization with Non-Anticommuting γ5

This review provides a detailed introduction to chiral gauge theories, renormalization theory, and the application of dimensional regularization with the non-anticommuting BMHV scheme for γ5. One goal was to show how chiral gauge theories can be renormalized despite the spurious breaking of gauge invariance and how to obtain the required symmetry-restoring counterterms. A second goal was to familiarize the reader with the theoretical basis of the renormalization of chiral gauge theories, the theorems that guarantee the existence of renormalized chiral gauge theories at all orders as consistent quantum theories. Relevant topics include BPHZ renormalization, Slavnov–Taylor identities, the BRST formalism, and algebraic renormalization, as well as the theorems guaranteeing that dimensional regularization is a consistent regularization/renormalization scheme. All of these, including their proofs and interconnections, are explained and discussed in detail. Further, these theoretical concepts are illustrated in practical applications with the example of an Abelian and a non-Abelian chiral gauge theory. Not only the renormalization procedure for such chiral gauge theories is explained step by step, but also the results of all counterterms, including the symmetry-restoring ones, necessary for the consistent renormalization, are explicitly provided.

Phases of a 10-D holographic hard wall model

In this article, we study the finite temperature properties of a 10-D version of a hardwall model for QCD. Introducing fundamental matter via probe D7-branes and separate cutoffs rm and rg for the branes and the bulk, we present a detailed exploration of the phases for varying temperature and quark mass. Finite thermodynamic quantities are calculated using the procedure of holographic renormalization and used to characterize the phases. Finally, by fitting glueball and vector meson masses, we show how a unique phase diagram can be isolated.

Revisiting the Cox and Munk wave-slope statistics using IASI observations of the sea surface

We use radiances collected from space by the Infrared Atmospheric Sounder Interferometer (IASI) when looking down at ocean surfaces during the day to remotely determine the probability distribution of wave slopes. This is achieved by using about 300 channels between 3.6 and 4.0μm and a physically-based approach which properly takes the contribution of the reflected solar radiation into account. Based on about 150 millions observations, the same number of wave-slope probabilities are retrieved for wind speeds (at 10 m) up to 15 m/s. We revisit and discuss the methodology proposed by Cox and Munk (CM) to derive their celebrated wave-slope probability distribution function (pdf) from photographs of the sun glitter. We propose an original and robust approach for accurate retrievals of the 7 parameters appearing in the Gram–Charlier representation of the pdf. Our results for the mean square slopes (MSSs) are fully compatible with those of CM, and with the more recent results by Bréon and Henriot, but our lower uncertainties enable to point out departures from the linear wind-speed dependencies and a slight overestimation of the upwind MSS described by the linear fit of CM at moderate wind speed. Our skewness and kurtosis coefficients show clear influences of the wind speed, with a steady decrease of the former and the alongwind kurtosis coefficient being maximal at moderate wind speeds, features that CM could not point out due to the limitations of their measurements. We revisit the renormalization procedure employed by CM to obtain the complete variances from truncated pdfs and show that it imposes stringent conditions on the kurtosis coefficients that allow to determine them accurately, with wind-dependent values specific to the local sea state. We also provide measurements of the shifted position of the most probable slope as well as a demonstration of a qualitative change of regime in the updown wind asymmetry of the wave-slope probability when the wind speed increases.

Nonperturbative renormalization of the supercurrent in N=1 supersymmetric Yang-Mills theory

In this work, we study the nonperturbative renormalization of the supercurrent operator in $\mathcal{N}=1$ supersymmetric Yang-Mills theory, using a gauge-invariant renormalization scheme (GIRS). The proposed prescription addresses successfully the unwanted mixing of the supercurrent with other operators of equal or lower dimension, which respect the same global symmetries. This mixing is introduced by the unavoidable breaking of supersymmetry on the lattice. In GIRS all gauge-noninvariant operators, which mix with the supercurrent, are excluded from the renormalization procedure. The one remaining mixing operator is accessible by numerical simulations. We present results for the renormalization of the supercurrent using GIRS. We also compute at one-loop order the conversion matrix which relates the nonperturbative renormalization factors in GIRS to the reference scheme $\overline{\mathrm{MS}}$.

Calculations of vacuum mean values of spinor field current and energy–momentum tensor in a constant electric background

In the framework of strong-field QED with x-steps, we study vacuum mean values of the current density and energy–momentum tensor of the quantized spinor field placed in the so-called L-constant electric background. The latter background can be, for example, understood as the electric field confined between capacitor plates, which are separated by a sufficiently large distance L. First, we reveal peculiarities of nonperturbative calculating of mean values in strong-field QED with x-steps in general and, in the L-constant electric field, in particular. We propose a new renormalization and volume regularization procedures that are adequate for these calculations. We find necessary representations for singular spinor functions in the background under consideration. With their help, we calculate the above mentioned vacuum means. In the obtained expressions, we show how to separate global contributions due to the particle creation and local ones due to the vacuum polarization. We demonstrate how these contributions can be related to the renormalized effective Heisenberg–Euler Lagrangian.

Bulk renormalization and the AdS/CFT correspondence

We develop a systematic renormalization procedure for QFT in anti-de Sitter spacetime. UV infinities are regulated using a geodesic point-splitting method, which respects AdS isometries, while IR infinities are regulated by cutting off the radial direction (as in holographic renormalization). The renormalized theory is defined by introducing $Z$ factors for all parameters in the Lagrangian and the boundary conditions of bulk fields (sources of dual operators), and a boundary counterterm action, ${S}_{\mathrm{ct}}$, such that the limit of removing the UV and IR regulators exists. The results are in general scheme dependent (mirroring the analogous result in flat space) and require renormalization conditions. These may be provided by the dual CFT (or by string theory in AdS). Our analysis amounts also to a first principles derivation of the Feynman rules regarding Witten diagrams. The presence and treatment of IR divergences is essential for correctly accounting for anomalous dimensions of dual operators. We apply the method to scalar ${\mathrm{\ensuremath{\Phi}}}^{4}$ theory and obtain the renormalized two-point function of the dual operator to two loops, and the renormalized four-point function to one-loop order, for operators of any dimension $\mathrm{\ensuremath{\Delta}}$ and bulk spacetime dimension up to $d+1=7$.

The thermodynamic limit of an ideal Bose gas by asymptotic expansions and spectral ζ-functions

We analyze the thermodynamic limit—modeled as the open-trap limit of an isotropic harmonic potential—of an ideal, non-relativistic Bose gas with a special emphasis on the phenomenon of Bose–Einstein condensation. This is accomplished by the use of an asymptotic expansion of the grand potential, which is derived by ζ-regularization techniques. Herewith, we can show that the singularity structure of this expansion is directly interwoven with the phase structure of the system: In the non-condensation phase, the expansion has a form that resembles usual heat kernel expansions. By this, thermodynamic observables are directly calculable. In contrast, the expansion exhibits a singularity of infinite order above a critical density, and a renormalization of the chemical potential is needed to ensure well-defined thermodynamic observables. Furthermore, the renormalization procedure forces the system to exhibit condensation. In addition, we show that characteristic features of the thermodynamic limit, such as the critical density or the internal energy, are entirely encoded in the coefficients of the asymptotic expansion.

A probabilistic approach of ultraviolet renormalization in the boundary Sine-Gordon model

The Sine-Gordon model is obtained by tilting the law of a log-correlated Gaussian field X defined on a subset of \({\mathbb {R}}^d\) by the exponential of its cosine, namely \(\exp (\alpha \smallint \cos (\beta X))\). It has gathered significant attention due to its importance in quantum field theory and to its connection with the study of log-gases in statistical mechanics. In spite of its relatively simple definition, the model has a very rich phenomenology. While the integral \(\smallint \cos (\beta X)\) can be defined properly when \(\beta ^2<d\) using the standard Wick normalization of \(\cos (\beta X)\), a more involved renormalization procedure is needed when \(\beta ^2\in [d,2d)\). In particular it exhibits a countable sequence of phase transitions accumulating to the left of \(\beta =\sqrt{2d}\), each transition corresponding to the addition of an extra term in the renormalization scheme. The final threshold \(\beta =\sqrt{2d}\) corresponds to the Kosterlitz–Thouless (KT) phase transition of the \(\log \)-gas. In this paper, we present a novel probabilistic approach to renormalization of the two-dimensional boundary (or 1-dimensional) Sine-Gordon model up to the KT threshold \(\beta =\sqrt{2d}\). The purpose of this approach is to propose a simple and flexible method to treat this problem which, unlike the existing renormalization group techniques, does not rely on translation invariance for the covariance kernel of X or the reference measure along which \(\cos (\beta X)\) is integrated. To this purpose we establish by induction a general formula for the cumulants of a random variable defined on a filtered probability space expressed in terms of brackets of a family of martingales; to the best of our knowledge, the recursion formula is new and might have other applications. We apply this formula to study the cumulants of (approximations of) \(\smallint \cos (\beta X)\). To control all terms produced by the induction procedure, we prove a refinement of classical electrostatic inequalities, which allows us to bound the energy of configurations in terms of the Wasserstein distance between \(+\) and − charges.

One-loop corrections for WW to HH in Higgs EFT with the electroweak chiral Lagrangian

In this work we present the computation of the one-loop electroweak radiative corrections to the scattering process $WW\to HH$ within the context of the Higgs Effective Field Theory (HEFT). We assume that the fermionic interactions are like in the Standard Model, whereas the Beyond Standard Model interactions in the bosonic sector are given by the Electroweak Chiral Lagrangian. The computation of the one-loop amplitude and the renormalization program is performed in terms of the involved one-particle-irreducible functions (1PI) and using $R_\xi$ covariant gauges. The renormalization of 1PI functions at arbitrary external momenta is a more ambitious program than just renormalizing the amplitude with on-shell external legs and it has the advantage that they can be used in several scattering amplitudes. In fact, we use here some of the 1PI functions already computed in our previous work (devoted to $WZ \to WZ$). We will complement them here with the computation of the new 1PI functions required for $WW \to HH$. From this renormalization procedure we will also derive the full set of renormalized coefficients of the EChL that are relevant for this scattering process. In the last part, we will present the numerical results for the EChL predictions of the one-loop level cross section, $\sigma(WW \to HH)|_{\rm 1-loop}$, as a function of the center of mass energy, showing the relative size of the one-loop radiative corrections respect to the tree level prediction in terms of the EChL coefficients. The results of the one-loop corrections to $WW \to HH$ for the SM case will be also presented, for comparison with the EChL case, following the same computational method, i.e., by means of the renormalization of 1PI functions.

Local unitarity: cutting raised propagators and localising renormalisation

The Local Unitarity (LU) representation of differential cross-sections locally realises the cancellations of infrared singularities predicted by the Kinoshita-Lee-Nauenberg theorem. In this work we solve the two remaining challenges to enable practical higher-loop computations within the LU formalism. The first concerns the generalisation of the LU representation to graphs with raised propagators. The solution to this problem results in a generalisation of distributional Cutkosky rules. The second concerns the regularisation of ultraviolet and spurious soft singularities, solved using a fully automated and local renormalisation procedure based on Bogoliubov’s R-operation. We detail an all-order construction for the hybrid overline{textrm{MS}} and On-Shell scheme whose only analytic input is single-scale vacuum diagrams. We validate this novel technology by providing (semi-)inclusive results for two multi-leg processes at NLO, study limits of individual supergraphs up to N3LO and present the first physical NNLO cross-sections computed fully numerically in momentum-space, namely for the processes γ∗→ jj and {gamma}^{ast}to toverline{t} .

The Reissner-Nordström-Tangherlini solution from scattering amplitudes of charged scalars

The metric and the electromagnetic potential generated by a static, spherically symmetric charged massive object in any dimension are given by the Reissner-Nordström-Tangherlini solution. We derive the expansion of this solution up to third post-Minkowskian order by computing the classical contribution of scattering amplitudes describing the emission of either a graviton or a photon from a massive charged scalar field up to two loops. In four and five dimensions these amplitudes develop ultraviolet divergences that are cancelled by higher-derivative counterterms in a way that generalises what was recently shown to happen in the chargeless case. This renormalisation procedure produces logarithmic terms that match exactly those produced in the post-Minkowskian expansion of the classical solution in de Donder gauge.

AN INTERNET REVIEW TOPIC HIERARCHY MINING METHOD BASED ON MODIFIED CONTINUOUS RENORMALIZATION PROCEDURE

Mining the hierarchical structure of Internet review topics and realizing a fine classification of review texts can help alleviate users’ information overload. However, existing hierarchical topic classification methods primarily rely on external corpora and human intervention. This study proposes a Modified Continuous Renormalization (MCR) procedure that acts on the keyword co-occurrence network with fractal characteristics to achieve the topic hierarchy mining. First, the fractal characteristics in the keyword co-occurrence network of Internet review text are identified using a box-covering algorithm for the first time. Then, the MCR algorithm established on the edge adjacency entropy and the box distance is proposed to obtain the topic hierarchy in the keyword co-occurrence network. Verification data from the Dangdang.com book reviews shows that the MCR constructs topic hierarchies with greater coherence and independence than the HLDA and the Louvain algorithms. Finally, reliable review text classification is achieved using the MCR extended bottom-level topic categories. The accuracy rate ([Formula: see text], recall rate ([Formula: see text] and [Formula: see text]1 value of Internet review text classification obtained from the MCR-based topic hierarchy are significantly improved compared to four target text classification algorithms.

A general framework for gravitational charges and holographic renormalization

We develop a general framework for constructing charges associated with diffeomorphisms in gravitational theories using covariant phase space techniques. This framework encompasses both localized charges associated with space–time subregions, as well as global conserved charges of the full space–time. Expressions for the charges include contributions from the boundary and corner terms in the subregion action, and are rendered unambiguous by appealing to the variational principle for the subregion, which selects a preferred form of the symplectic flux through the boundaries. The Poisson brackets of the charges on the subregion phase space are shown to reproduce the bracket of Barnich and Troessaert for open subsystems, thereby giving a novel derivation of this bracket from first principles. In the context of asymptotic boundaries, we show that the procedure of holographic renormalization can be always applied to obtain finite charges and fluxes once suitable counterterms have been found to ensure a finite action. This enables the study of larger asymptotic symmetry groups by loosening the boundary conditions imposed at infinity. We further present an algorithm for explicitly computing the counterterms that renormalize the action and symplectic potential, and, as an application of our framework, demonstrate that it reproduces known expressions for the charges of the generalized Bondi–Metzner–Sachs algebra.

Renormalizing the vacuum energy in cosmological spacetime: implications for the cosmological constant problem

The renormalization of the vacuum energy in quantum field theory (QFT) is usually plagued with theoretical conundrums related not only with the renormalization procedure itself, but also with the fact that the final result leads usually to very large (finite) contributions incompatible with the measured value of Lambda in cosmology. As a consequence, one is bound to extreme fine-tuning of the parameters and so to sheer unnaturalness of the result and of the entire approach. We may however get over this adversity using a different perspective. Herein, we compute the zero-point energy (ZPE) for a nonminimally coupled (massive) scalar field in FLRW spacetime using the off-shell adiabatic renormalization technique employed in previous work. The on-shell renormalized result first appears at sixth adiabatic order, so the calculation is rather cumbersome. The general off-shell result yields a smooth function rho _{mathrm{vac}}(H) made out of powers of the Hubble rate and/or of its time derivatives involving different (even) adiabatic orders sim H^N (N=0, 2,4,6,ldots ), i.e. it leads, remarkably enough, to the running vacuum model (RVM) structure. We have verified the same result from the effective action formalism and used it to find the beta -function of the running quantum vacuum. No undesired contributions sim m^4 from particle masses appear and hence no fine-tuning of the parameters is needed in rho _{mathrm{vac}}(H). Furthermore, we find that the higher power sim H^6 could naturally drive RVM-inflation in the early universe. Our calculation also elucidates in detail the equation of state of the quantum vacuum: it proves to be not exactly -1 and is moderately dynamical. The form of rho _{mathrm{vac}}(H) at low energies is also characteristic of the RVM and consists of an additive term (the so-called ‘cosmological constant’) together with a small dynamical component sim nu H^2 (|nu |ll 1). Finally, we predict a slow (sim ln H) running of Newton’s gravitational coupling G(H). The physical outcome of our semiclassical QFT calculation is revealing: today’s cosmic vacuum and the gravitational strength should be both mildly dynamical.

Spontaneous symmetry breakings in the singlet scalar Yukawa model within the auxiliary field method

The aim of this work is to investigate the occurrence of two different spontaneous symmetry breakings at two levels of the description of fermion-scalar field model, by means of a set of gap equations and with a background field effective action. For that, we consider the Yukawa model, as a toy model for interactions between nonmassive fermions intermediated by a self-interacting real scalar field. This model has at stake two symmetries at the classical level that, as we know, might be spontaneously or dynamically broken with mass generation for the particles. The auxiliary field method is considered and it produces coupled renormalized gap equations. The effective action is then written with quantum contributions by the external background field method. We brought to light how the renormalization procedure affects the physical gaps, investigate its properties, and discuss the connection between the auxiliary fields not only to define composite states but also to compute the effective action.