Renormalization Scheme Research Articles

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.

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.

Gene prioritization based on random walks with restarts and absorbing states, to define gene sets regulating drug pharmacodynamics from single-cell analyses.

Prioritizing genes for their role in drug sensitivity, is an important step in understanding drugs mechanisms of action and discovering new molecular targets for co-treatment. To formalize this problem, we consider two sets of genes X and P respectively composing the gene signature of cell sensitivity at the drug IC50 and the genes involved in its mechanism of action, as well as a protein interaction network (PPIN) containing the products of X and P as nodes. We introduce Genetrank, a method to prioritize the genes in X for their likelihood to regulate the genes in P. Genetrank uses asymmetric random walks with restarts, absorbing states, and a suitable renormalization scheme. Using novel so-called saturation indices, we show that the conjunction of absorbing states and renormalization yields an exploration of the PPIN which is much more progressive than that afforded by random walks with restarts only. Using MINT as underlying network, we apply Genetrank to a predictive gene signature of cancer cells sensitivity to tumor-necrosis-factor-related apoptosis-inducing ligand (TRAIL), performed in single-cells. Our ranking provides biological insights on drug sensitivity and a gene set considerably enriched in genes regulating TRAIL pharmacodynamics when compared to the most significant differentially expressed genes obtained from a statistical analysis framework alone. We also introduce gene expression radars, a visualization tool embedded in MA plots to assess all pairwise interactions at a glance on graphical representations of transcriptomics data. Genetrank is made available in the Structural Bioinformatics Library (https://sbl.inria.fr/doc/Genetrank-user-manual.html). It should prove useful for mining gene sets in conjunction with a signaling pathway, whenever other approaches yield relatively large sets of genes.

Hydrodynamic theory of two-dimensional incompressible polar active fluids with quenched and annealed disorder.

We study the moving phase of two-dimensional (2D) incompressible polar active fluids in the presence of both quenched and annealed disorder. We show that long-range polar order persists even in this defect-ridden two-dimensional system. We obtain the large-distance, long-time scaling laws of the velocity fluctuations using three distinct dynamic renormalization group schemes. These are an uncontrolled one-loop calculation in exactly two dimensions, and two d=(d_{c}-ε) expansions to O(ε), obtained by two different analytic continuations of our 2D model to higher spatial dimensions: a "hard" continuation which has d_{c}=7/3, and a "soft" continuation with d_{c}=5/2. Surprisingly, the quenched and annealed parts of the velocity correlation function have the same anisotropy exponent and the relaxational and propagating parts of the dispersion relation have the same dynamic exponent in the nonlinear theory even though they are distinct in the linearized theory. This is due to anomalous hydrodynamics. Furthermore, all three renormalization schemes yield very similar values for the universal exponents, and therefore we expect the numerical values that we predict for them to be highly accurate.

Improving methods and predictions at high-energy e+e− colliders within collinear factorisation

We illustrate how electron Parton Distribution Functions (PDFs) with next-to-leading collinear logarithmic accuracy must be employed in the context of perturbative predictions for high-energy e+e−-collision processes. In particular, we discuss how the renormalisation group equation evolution of such PDFs is affected by the presence of multiple fermion families and their respective mass thresholds, and by the dependences on the choices of the factorisation and renormalisation schemes. We study the impact of the uncertainties associated with the PDFs on physical cross sections, in order to arrive at realistic precision estimates for observables computed with collinear-factorisation formulae. We do so by presenting results for the production of a heavy neutral object as well as for toverline{t} and W+W− pairs, including next-to-leading-order effects of electroweak origin.

One-loop Fierz transformations

Fierz transformations for four-fermion operators are generalized to the one-loop level. A general renormalization scheme is used to compute QCD and QED corrections to the tree-level relations, which result from Fierz-evanescent operators. The results can be used to perform general one-loop basis transformations involving four-fermi and evanescent operators. We illustrate the usefulness of our results by discussing two examples from a matching calculation and a one-loop basis change.

Operator mixing in massless QCD-like theories and Poincarè–Dulac theorem

Recently, a differential-geometric approach to operator mixing in massless QCD-like theories – that involves canonical forms, obtained by means of gauge transformations, based on the Poincarè–Dulac theorem for the system of linear differential equations that defines the renormalized mixing matrix in the coordinate representation Z(x,mu ) – has been proposed in [1]. Specifically, it has been determined under which conditions a renormalization scheme exists where Z(x,mu ) may be set in a diagonal canonical form that is one-loop exact to all perturbative orders – the nonresonant diagonalizable frac{gamma _{0}}{beta _{0}} case (I) –. Moreover, the remaining cases, (II), (III) and (IV), of operator mixing, where such diagonalization is not possible, have also been classified in [1]. Accordingly, if frac{gamma _{0}}{beta _{0}}, with gamma (g)=gamma _{0} g^{2}+cdots the matrix of the anomalous dimensions and beta (g)=-beta _{0} g^{3} + cdots the beta function, either is diagonalizable but a resonant condition for its eigenvalues and the system holds (II) or is nondiagonalizable and nonresonant (III), or is nondiagonalizable and resonant (IV), Z(x,mu ) is nondiagonalizable. In the cases (II), (III) and (IV), we demonstrate that its canonical form may be factorized into the exponential of a linear combination of upper triangular nilpotent constant matrices with coefficients that asymptotically in the UV are logs of the running coupling, i.e., asymptotically loglogs of the coordinates, and a diagonal matrix as in the nonresonant diagonalizable case (I). Hence, its ultraviolet asymptotics differs intrinsically from the case (I) and, for asymptotically free theories, this is the closest analog of logCFTs. We also work out actual physics realizations of the cases (I) and (II), while we argue that the cases (III) and (IV) are ruled out by a unitarity argument in the gauge-invariant sector.

Finite temperature contributions to cosmological constant

We reexamine the cosmological constant (CC) problem in a finite temperature setup and propose an intriguing possibility of carrying out perturbative analysis by employing a renormalization scheme in which the renormalized Higgs mass (or resummed mass, to be more precise) is taken to be on the order of the cosmic microwave background (CMB) temperature. Our proposal hinges on the fact that although the physical value of the CC does not depend on one’s renormalization scheme, whether or not a fine tuning is involved does. The CC problem is avoided in the sense that the renormalization process no longer requires finetuning. This is achieved essentially by renormalization scheme-independence of a physical quantity, which in turn is assured by bare perturbation theory. The proposal shifts the CC problem to a peculiarity of the consequent perturbation series for the Higgs mass (and other massive sectors of the Standard Model); the peculiarity is interpreted as an indicator of new physics after the expected mathematical structure of the series is scrutinized. Finite-temperature-induced complexification of the effective potential is observed and its interpretation is given. A consistency check in the cosmology context is suggested.

New analyses of event shape observables in electron-positron annihilation and the determination of αs running behavior in perturbative domain

In this paper, we give comprehensive analyses for event shape observables in electron-positron annihilation by using the Principle of Maximum Conformality (PMC) which is a rigorous scale-setting method to eliminate the renormalization scheme and scale ambiguities in perturbative QCD predictions. Conventionally the renormalization scale and theoretical uncertainties in event shape observables are often evaluated by setting the scale to the center-of-mass energy sqrt{s} . The event shape distributions using this conventional scale setting are plagued by the large renormalization scale uncertainty and underestimate the experimental data. Moreover, since the renormalization scale is simply fixed to the center-of-mass energy sqrt{s} , only one value of the coupling αs at the single scale sqrt{s} can be extracted. In contrast, the PMC renormalization scales are determined by absorbing the nonconformal β contributions that govern the behavior of the running coupling via the Renormalization Group Equation (RGE). The resulting PMC scales change with event shape kinematics, reflecting the virtuality of the underlying quark and gluon subprocess. The PMC scales thus yield the correct physical behavior of the scale and the PMC predictions agree with precise event shape distributions measured at the LEP experiment. More importantly, we can precisely determine the running of the QCD coupling constant αs(Q2) over a wide range of Q2 in perturbative domain from event shape distributions measured at a single center-of-mass energy sqrt{s} .

NNLO study of top-quark mass renormalization scheme uncertainties in Higgs boson production

The ambiguity in the choice of a renormalization scheme and scale for the top-quark mass leads to an additional source of theoretical uncertainty in the calculation of the Higgs boson production cross section via gluon fusion. These uncertainties were found to be dominant in the case of off-shell Higgs production at next-to-leading order in QCD for large values of the Higgs virtuality {m}_H^{ast } . In this work, we study the uncertainties related to the top-quark mass definition up to next-to-next-to-leading order (NNLO) in QCD. We include the full top-quark mass dependence up to three loops in the virtual corrections, and evaluate the real contributions in the soft limit, therefore obtaining the so-called soft-virtual (SV) approximation. We construct NNLOSV predictions for off-shell Higgs boson production renormalizing the top-quark mass within both the on-shell (OS) and the overline{mathrm{MS}} schemes, and study in detail the differences between them. While the differences between the two schemes are sizeable, we find that the predictions are always compatible within scale uncertainties. We also observe that the difference between renormalization schemes is largely reduced when increasing the order of the perturbative expansion. We analyze the quality of the convergence of the perturbative series in both schemes, and find that at large invariant masses the overline{mathrm{MS}} results present much larger corrections than their OS counterparts. We also comment on the more complicated case of Higgs boson pair production.

Renormalization of Transverse-Momentum-Dependent Parton Distribution on the Lattice.

To calculate the transverse-momentum-dependent parton distribution functions (TMDPDFs) from lattice QCD, an important goal yet to be realized, it is crucial to establish a viable nonperturbative renormalization approach for linear divergences in the corresponding Euclidean quasi-TMDPDF correlators in large-momentum effective theory. We perform a first systematic study of the renormalization property of the quasi-TMDPDFs by calculating the relevant matrix elements in a pion state at five lattice spacings ranging from 0.03fm to 0.12fm. We demonstrate that the square root of the Wilson loop combined with the short distance hadron matrix element provides a successful method to remove all ultraviolet divergences of the quasi-TMD operator, and thus provides the necessary justification to perform a continuum limit calculation of TMDPDFs. In contrast, the popular regularization independent momentum subtraction renormalization (RI/MOM) scheme fails to eliminate all linear divergences.

An iDTMC-based Monte Carlo depletion of a 3D SMR with intra-pin flux renormalization

Improved deterministic truncation of the Monte Carlo (iDTMC) solution method has been applied to neutronic depletion analyses of a nuclear reactor in this work. The theoretical background on the iDTMC method and material depletion calculation is briefly discussed, and the concept of an intra-pin renormalization scheme is presented to determine the detailed power and flux profiles with the pin-wise homogenized iDTMC solutions. The iDTMC method is applied and evaluated in a 100 MWth 3-dimensional small modular reactor (SMR) problem. Burnup-dependent multiplication factors, pin power distributions, and material densities are estimated by the iDTMC method and compared with the reference and standard Monte Carlo solutions. Numerical performance is investigated in terms of real standard deviation, root-mean-square and relative errors, computing time, and figure-of-merit. The numerical results demonstrate that the iDTMC method is superior to the standard MC method in estimating the burnup-dependent reactor eigenvalue and 3D pin-wise power distribution.

Gauge Symmetries and Renormalization

We study the perturbative renormalization of quantum gauge theories in the Hopf algebra setup of Connes and Kreimer. It was shown by van Suijlekom (Commun Math Phys 276:773–798, 2007) that the quantum counterparts of gauge symmetries—the so-called Ward–Takahashi and Slavnov–Taylor identities—correspond to Hopf ideals in the respective renormalization Hopf algebra. We generalize this correspondence to super- and non-renormalizable Quantum Field Theories, extend it to theories with multiple coupling constants and add a discussion on transversality. In particular, this allows us to apply our results to (effective) Quantum General Relativity, possibly coupled to matter from the Standard Model, as was suggested by Kreimer (Ann Phys 323:49–60, 2008). To this end, we introduce different gradings on the renormalization Hopf algebra and study combinatorial properties of the superficial degree of divergence. Then we generalize known coproduct and antipode identities to the super- and non-renormalizable cases and to theories with multiple vertex residues. Building upon our main result, we provide criteria for the compatibility of these Hopf ideals with the corresponding renormalized Feynman rules. A direct consequence of our findings is the well-definedness of the Corolla polynomial for Quantum Yang–Mills theory without reference to a particular renormalization scheme.

Renormalization scheme dependence of beta -functions in Lorentz-violating quantum field theory

Effective quantum field theories that allow for the possibility of Lorentz symmetry violation can sometimes also include redundancies of description in their Lagrangians. Explicit calculations in a Lorentz-violating generalization of Yukawa theory show that when this kind of redundancy exists, different renormalization schemes may lead to different expressions for the renormalization group beta -functions, even at only one-loop order. However, the renormalization group scaling of physically observable quantities appears not to share this kind of scheme dependence.

Loop-corrected Higgs masses in the NMSSM with inverse seesaw mechanism

In this study, we work in the framework of the Next-to-Minimal extension of the Standard Model (NMSSM) extended by six singlet leptonic superfields. Through the mixing with the three doublet leptonic superfields, the non-zero tiny neutrino masses can be generated through the inverse seesaw mechanism. While R-parity is conserved in this model lepton number is explicitly violated. We quantify the impact of the extended neutrino sector on the NMSSM Higgs sector by computing the complete one-loop corrections with full momentum dependence to the Higgs boson masses in a mixed on-shell-overline{text{ DR }} renormalization scheme, with and without the inclusion of CP violation. The results are consistently combined with the dominant two-loop corrections at mathcal{O}(alpha _t(alpha _s+alpha _t)) to improve the predictions for the Higgs mixing and the loop-corrected masses. In our numerical study we include the constraints from the Higgs data, the neutrino oscillation data, the charged lepton flavor-violating decays l_i rightarrow l_j + gamma , and the new physics constraints from the oblique parameters S, T, U. We present in this context the one-loop decay width for l_i rightarrow l_j + gamma . The loop-corrected Higgs boson masses are included in the Fortran code NMSSMCALC-nuSS.

On the NLO QCD corrections to gluon-initiated ZH production

We compute the QCD corrections at next-to-leading order for the process gg → ZH, including both the virtual two-loop terms and real-emission contributions. The two-loop box diagrams in the virtual corrections are approximated analytically over the complete phase space, combining the results of an expansion in the limit of small transverse momentum and an expansion in the regime of high energy. We obtain both inclusive and differential results for the cross section. We find that the NLO QCD corrections are of the same size as the LO contribution up to ZH invariant masses close to 1 TeV, but they increase significantly when higher energies are considered, due to a class of real-emission diagrams in which the Z boson is radiated from an open quark line. Finally, we estimate the uncertainty due to the renormalization scheme used for the top-quark mass both on the total and differential cross section.

NLO QCD renormalization group evolution for nonleptonic ΔF=2 transitions in the SMEFT

We present for the first time Next-to-Leading (NLO) QCD renormalization group (RG) evolution matrices for nonleptonic $\mathrm{\ensuremath{\Delta}}F=2$ transitions in the Standard Model effective field theory (SMEFT). To this end we transform first the known two-loop QCD anomalous dimension matrices (ADMs) of the BSM (Beyond the SM) operators in the so-called Buras Misiak Urban basis into the ones in the common weak effective theory (WET) basis (the so-called Jenkins Manohar Stoffer basis) for which tree-level and one-loop matching to the SMEFT are already known. This subsequently allows us to find the two-loop QCD ADMs for the SMEFT nonleptonic $\mathrm{\ensuremath{\Delta}}F=2$ operators in the Warsaw basis. Having all these ingredients we investigate the impact of these NLO QCD effects on the QCD RG evolution of SMEFT Wilson coefficients for nonleptonic $\mathrm{\ensuremath{\Delta}}F=2$ transitions from the new physics scale $\mathrm{\ensuremath{\Lambda}}$ down to the electroweak scale ${\ensuremath{\mu}}_{\mathrm{ew}}$. The main benefit of these new contributions is that they allow one to remove renormalization scheme dependences present in the one-loop matchings both between the WET and SMEFT and also between SMEFT and a chosen UV completion. But the Next-to-Leading (NLO) QCD effects, calculated here in the Naive dimensional regularisation minimal subtraction scheme, turn out to be small, in the ballpark of a few percent but larger than one-loop Yukawa top effects when only the $\mathrm{\ensuremath{\Delta}}F=2$ operators are considered. The more complicated class of nonleptonic $\mathrm{\ensuremath{\Delta}}F=1$ decays will be presented soon in another publication.

One-loop hybrid renormalization matching kernels for quasiparton distributions

Large-momentum effective theory allows extraction of hadron parton distribution functions in lattice QCD by matching them to quark bilinear matrix elements of hadrons with large momenta. We calculate the matching kernels for the unpolarized, helicity, and transversity-isovector parton distribution functions and skewless generalized parton distributions of all hadrons in the hybrid-regularization-invariant momentum subtraction (RI/MOM) scheme. This renormalization scheme uses RI/MOM when the Wilson line length is less than ${z}_{s}$, otherwise a mass subtraction scheme is used. By design, the nonhybrid scheme is recovered as ${z}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$. In the opposite limit, ${z}_{s}\ensuremath{\rightarrow}0$, the self-renormalization scheme is obtained. When the parameters ${p}_{z}^{R}=0$ and ${\ensuremath{\mu}}^{R}{z}_{s}\ensuremath{\ll}1$, the hybrid-RI/MOM scheme coincides with the hybrid-ratio scheme times the charge of the PDF. We also discuss the subtlety related to the commutativity of Fourier transform and $\ensuremath{\epsilon}$ expansion in the $\overline{\mathrm{MS}}$ scheme.

Singularity-free treatment of delta-function point scatterers in two dimensions and its conceptual implications

In two dimensions, the standard treatment of the scattering problem for a delta-function potential, , leads to a logarithmic singularity which is subsequently removed by a renormalization of the coupling constant . Recently, we have developed a dynamical formulation of stationary scattering (DFSS) which offers a singularity-free treatment of this potential. We elucidate the basic mechanism responsible for the implicit regularization property of DFSS that makes it avoid the logarithmic singularity one encounters in the standard approach to this problem. We provide an alternative interpretation of this singularity showing that it arises, because the standard treatment of the problem takes into account contributions to the scattered wave whose momentum is parallel to the detectors’ screen. The renormalization schemes used for removing this singularity has the effect of subtracting these unphysical contributions, while DFSS has a built-in mechanics that achieves this goal.

Germ-typicality of the coexistence of infinitely many sinks

In the spirit of Kolmogorov typicality, we introduce the notion of germ-typicality: in a space of dynamics, it encompasses all these phenomena that occur for a dense and open subset of parameters of any generic parametrized family contained in an open set of systems.For any 2≤r<∞, we prove that the Newhouse phenomenon (the coexistence of infinitely many sinks) is locally Cr-germ-typical, nearby a dissipative bicycle: a dissipative homoclinic tangency linked to a special heterodimensional cycle.During the proof we show a result of independent interest: the stabilization of some heterodimensional cycles for any regularity class r∈{1,…,∞}∪{ω} by introducing a new renormalization scheme. We also continue the study of the paradynamics done in [6,7,1] and prove that parablenders appear by unfolding some heterodimensional cycles.