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In this paper, we investigate a class of non-invertible piecewise isometries on the upper half-plane known as Translated Cone Exchanges. These maps include a simple interval exchange on a boundary we call the baseline. We provide a geometric construction for the first return map to a neighbourhood of the vertex of the middle cone for a large class of parameters, then we show a recurrence in the first return map tied to Diophantine properties of the parameters, and subsequently prove the infinite renormalizability of the first return map for these parameters. [ABSTRACT FROM AUTHOR]

We show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g. piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as generalized interval exchange transformations (GIETs) of the interval and rely on the notion of combinatorial rotation number for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting. [ABSTRACT FROM AUTHOR]

In the presence of spacetime torsion, the momentum components do not commute; therefore, in quantum field theory, summation over the momentum eigenvalues will replace integration over the momentum. In the Einstein–Cartan theory of gravity, in which torsion is coupled to spin, the separation between the eigenvalues increases with the magnitude of the momentum. Consequently, this replacement regularizes divergent integrals in Feynman diagrams with loops by turning them into convergent sums. In this article, we apply torsional regularization to the self-energy of a charged lepton in quantum electrodynamics. We show that torsion eliminates the ultraviolet divergence of the standard self-energy. We also show that the infrared divergence is absent. In the end, we calculate the finite bare masses of the electron, muon, and tau lepton: 0.4329 MeV , 90.95 MeV , and 1543 MeV , respectively. These values constitute about 85% of the observed, re-normalized masses. [ABSTRACT FROM AUTHOR]

The explicit expression for the propagator of the Dirac delta potential in three dimensional hyperbolic spaces is derived using the integral transform of the Krein’s type of the resolvent formula, obtained after the renormalization procedure. [ABSTRACT FROM AUTHOR]

An approach to renormalization of scalar fields on the Doplicher–Fredenhagen–Roberts (DFR) quantum spacetime is presented. The effective nonlocal theory obtained through the use of states of optimal localization for the quantum spacetime is reformulated in the language of (perturbative) Algebraic Quantum Field Theory. The structure of the singularities associated to the nonlocal kernel that codifies the effects of non-commutativity is analyzed using the tools of microlocal analysis. [ABSTRACT FROM AUTHOR]

The normalization in the path integral approach to quantum field theory, in contrast with statistical field theory, can contain physical information. The main claim of this paper is that the inner product on the space of field configurations, one of the fundamental pieces of data required to be added to quantize a classical field theory, determines the normalization of the path integral. In fact, dimensional analysis shows that the introduction of this structure necessarily introduces a scale that is left unfixed by the classical theory. We study the dependence of the theory on this scale. This allows us to explore mechanisms that can be used to fix the normalization based on cutting and gluing different integrals. "Self-normalizing" path integrals, those independent of the scale, play an important role in this process. Furthermore, we show that the scale dependence encodes other important physical data: we use it to give a conceptually clear derivation of the chiral anomaly. Several explicit examples, including the scalar and compact bosons in different geometries, supplement our discussion. [ABSTRACT FROM AUTHOR]

Consider an expansive Lorenz map f defined on I = [ 0 , 1 ] and let c be its discontinuity. The survivor set under consideration is represented as S f + (a , b) : = { x ∈ I : f (b) ≤ f n (x) ≤ f (a) , ∀ n ≥ 0 } , where (a , b) ⊆ I satisfies a ≤ c ≤ b and a ≠ b . We mainly study the following two bifurcation sets related to survivor set and its topological entropy h top : E f (a) : = { b ≥ c : S f + (a , ϵ) ≠ S f + (a , b) , ∀ ϵ > b } , and B f (a) : = { b ≥ c : h top (f | S f + (a , ϵ) ) ≠ h top (f | S f + (a , b) ) , ∀ ϵ > b }. By utilizing combinatorial renormalization techniques, we give the kneading sequences of the endpoints of the platform P (b) : = { b ′ ≥ c : h top (f | S f + (a , b ′) ) = h top (f | S f + (a , b) ) }. Moreover, we obtain a sufficient and necessary condition for when E f (a) = B f (a) , extending the results of Baker and Kong (2020), Allaart and Kong (2023). [ABSTRACT FROM AUTHOR]

The magnetohydrodynamics system forced by space-time white noise has been studied by physicists for decades, and the rigorous proof of its solution theory was recently established by Yamazaki (2023, Electron. J. Probab., 28 , 1–66). When an equation is well-posed, and it is approximated by replacing the differentiation operator by reasonable discretization schemes, it is widely believed that a solution of the approximating equation should converge to the solution of the original equation as the discretization parameter approaches zero. We prove otherwise for the three-dimensional magnetohydrodynamics system forced by space-time white noise. Specifically, we prove that the limit of the solution to the approximating system with an additional 32 drift terms solves the original system. These 32 drift terms depend on the choice of approximations, can be calculated explicitly and essentially represent a spatial version of Itô-Stratonovich correction terms. [ABSTRACT FROM AUTHOR]

Significance: Hyperspectral imaging (HSI) of murine tumor models grown in dorsal skinfold window chambers (DSWCs) offers invaluable insight into the tumor microenvironment. However, light loss in a glass coverslip is often overlooked, and particular tissue characteristics are improperly modeled, leading to errors in tissue properties extracted from hyperspectral images. Aim: We highlight the significance of spectral renormalization in HSI of DSWC models and demonstrate the benefit of incorporating enhanced green fluorescent protein (EGFP) excitation and emission in the skin tissue model for tumors expressing genes to produce EGFP. Approach: We employed an HSI system for intravital imaging of mice with 4T1 mammary carcinoma in a DSWC over 14 days. We performed spectral renormalization of hyperspectral images based on the measured reflectance spectra of glass coverslips and utilized an inverse adding-doubling (IAD) algorithm with a two-layer murine skin model, to extract tissue parameters, such as total hemoglobin concentration and tissue oxygenation (StO2). The model was upgraded to consider EGFP fluorescence excitation and emission. Moreover, we conducted additional experiments involving tissue phantoms, human forearm skin imaging, and numerical simulations. Results: Hyperspectral image renormalization and the addition of EGFP fluorescence in the murine skin model reduced the mean absolute percentage errors (MAPEs) of fitted and measured spectra by up to 10% in tissue phantoms, 0.55% to 1.5% in the human forearm experiment and numerical simulations, and up to 0.7% in 4T1 tumors. Similarly, the MAPEs for tissue parameters extracted by IAD were reduced by up to 3% in human forearms and numerical simulations. For some parameters, statistically significant differences (p < 0.05) were observed in 4T1 tumors. Ultimately, we have shown that fluorescence emission could be helpful for 4T1 tumor segmentation. Conclusions: The results contribute to improving intravital monitoring of DWSC models using HSI and pave the way for more accurate and precise quantitative imaging. [ABSTRACT FROM AUTHOR]

We revisit the non-perturbative renormalization of a class of simple polaron models with resting fermions. The considered dispersion relations and form factors are allowed to be highly singular, such that infinite self-energies and wave function renormalizations may occur and the diagonalizing dressing transformations might not be implementable on Fock space. Instead of taking cutoffs, we rigorously interpret the self-energies and wave function renormalizations as elements of suitable vector spaces, as well as a field extension of the complex numbers. Moreover, we define two extended state vector spaces (ESSs) over this field extension, which contain a dense subspace of Fock space, but also incorporate non-square-integrable wave functions. Elements of these ESSs can be seen as states of virtual particles, described in a mathematically rigorous way. The Hamiltonian without cutoffs, formally infinite counterterms, and the dressing transformation can then be defined as linear operators between certain subspaces of the two Fock space extensions. This way, we obtain a renormalized Hamiltonian which can be realized as a densely defined self-adjoint operator on Fock space. [ABSTRACT FROM AUTHOR]

I suggest that the current situation in quantum field theory (QFT) provides some reason to question the universal validity of ontological reductionism. I argue that the renormalization group flow is reversible except at fixed points, which makes the relation between large and small distance scales quite symmetric in QFT, opening up at least the technical possibility of a non-reductionist approach to QFT. I suggest that some conceptual problems encountered within QFT may potentially be mitigated by moving to an alternative picture in which it is no longer the case that the large supervenes on the small. Finally, I explore some specific models in which a form of non-reductionism might be implemented, and consider the prospects for future development of these models. [ABSTRACT FROM AUTHOR]

In this paper, we study Feynman checkers, one of the most elementary models of electron motion. It is also known as a one-dimensional quantum walk or an Ising model at an imaginary temperature. We add the simplest non-trivial electromagnetic field and find the limits of the resulting model for small lattice step and large time, analogous to the results by Narlikar from 1972 and Grimmet–Jason–Scudo from the 2000s. It turns out that the limits in the model with the added field are obtained from the ones without field by mass renormalization. Also, we find an exact solution of the resulting model. [ABSTRACT FROM AUTHOR]

We present on shell-scheme for the 2PI formalism with a particular focus on the renormalized equations of motion. We first revisit the so-called Hartree approximation where we give the counterterms for both the broken and unbroken phases. Moreover, we give explicit formulas for the renormalized three- and four-point functions in the broken phase. We then turn to the sunset approximation, with only scalars and then including fermions. We give explicit formulas for the wave function and mass counterterms. Moreover, we show that, in particular, the two-point functions can be obtained numerically in a fast converging scheme even for large couplings of order one. [ABSTRACT FROM AUTHOR]

The one-dimensional quasi-geostrophic equation is the one-dimensional Fourier-space analogue of the famous Navier--Stokes equations. In [D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp. 1945--1950], Li and Sinai have proposed a renormalization approach to the problem of the existence of finite-time blow-up solutions of this equation. In this paper, we revisit the renormalization problem for the quasi-geostrophic blow-ups, prove the existence of a family of renormalization fixed points, and deduce the existence of real C∞ ([0, T), C∞ (R ) ∩ L² (R )) solutions to the quasi-geostrophic equation whose energy and enstrophy become unbounded in finite time, different from those found in [D. Li and Ya. G. Sinai, Phys. D, 237 (2008), pp. 1945--1950]. [ABSTRACT FROM AUTHOR]

The exponential decay of lattice Green functions is one of the main technical ingredients of the Bałaban's approach to renormalization. We give here a self-contained proof, whose various ingredients were scattered in the literature. The main sources of exponential decay are the Combes–Thomas method and the analyticity of the Fourier transforms. They are combined using a renormalization group equation and the method of images. [ABSTRACT FROM AUTHOR]

We present concrete evidence that Yang-Mills theory exhibits non-unitarity in non-integer spacetime dimensions. This violation of unitarity stems from evanescent operators that, while vanishing in four dimensions, are non-zero in general d dimensions. We demonstrate that these evanescent operators lead to the emergence of both negative-norm states and complex anomalous dimensions. [ABSTRACT FROM AUTHOR]

in this article we analyze the reflective accounts written by mentors/coaches at the end of their training, in order to retrace the transformation of their professional identity and professional action. The analysis focuses on a thematic and linguistic analysis of the texts as embodiments of a developing relationship with the professional world. We highlight that these texts bear traces of processes of renormalization, resulting from conceptual and formal learning as much as from experiential learning achieved during training. This analysis ultimately calls into question the training itself, along with its multiple expectations. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation with a large gradient of the initial function and a small dispersion parameter. The renormalization method is used to construct an asymptotic solution in the explicit form of integral convolution. An asymptotic analogue of the renormalization group property is established under scaling transformations determined by the dispersion parameter. In the case of a negative focusing coefficient, a clarifying expression is obtained for the asymptotic solution in terms of known elliptic special functions. [ABSTRACT FROM AUTHOR]

The paper studies the quantum action for the four-dimensional real ϕ4-theory in the case of a general formulation using the background field method. The three-loop renormalization is performed with the usage of a cutoff regularization in the coordinate representation. The absence of non-local singular contributions and the correctness of the renormalization R-operation on the example of separate three-loop diagrams are also discussed. The explicit forms of the first three coefficients for the renormalization constants and for the β-function are presented. Consistency with previously known results is shown.

We establish a version of the Pommerenke–Levin–Yoccoz inequality for the modulus of a polynomial-like (PL) restriction of a polynomial and give two applications. First we show that if the modulus of a PL restriction of a polynomial is bounded from below then this restricts the combinatorics of the polynomial. The second application concerns parameter slices of cubic polynomials given by the non-repelling multiplier of a fixed point. Namely, the intersection of the so-called Main Cubioid and the multiplier slice lies in the closure of the principal hyperbolic domain, with possible exception of queer components. [ABSTRACT FROM AUTHOR]

Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a periodic domain. There are two new ingredients as compared to the parabolic setting. (i) In constructing stochastic objects, we have to carefully exploit dispersion at a multilinear level. (ii) We introduce novel random operators and leverage their regularity to overcome the lack of smoothing of usual paradifferential commutators. [ABSTRACT FROM AUTHOR]

To simulate quantum systems on classical or quantum computers, the continuous observables (e.g., coordinate and momentum or energy and time) must be reduced to discrete ones. In this paper, we consider the continuous observables represented in the positional systems as power series in the radix multiplied over the summands ("digits"), which turn out to be Hermitian operators with discrete spectrum. We investigate the obtained quantum mechanical operators of digits, the commutation relations between them, and the effects of the choice of a numeral system on lattices and representations. Renormalizations of diverging sums naturally occur in constructing the digital representation. [ABSTRACT FROM AUTHOR]

Geometric algebras of dimension n<6$$ n<6 $$ are becoming increasingly popular for the modeling of 3D and 3 + 1D geometry. With this increased popularity comes the need for efficient algorithms for common operations such as normalization, square roots, and exponential and logarithmic maps. The current work presents a signature‐agnostic analysis of these common operations in all geometric algebras of dimension n<6$$ n<6 $$ and gives efficient numerical implementations in the most popular algebras ℝ4,ℝ3,1,ℝ3,0,1$$ {\mathbb{R}}_4,{\mathbb{R}}_{3,1},{\mathbb{R}}_{3,0,1} $$, and ℝ4,1$$ {\mathbb{R}}_{4,1} $$, in the hopes of lowering the threshold for adoption of geometric algebra solutions by code maintainers. [ABSTRACT FROM AUTHOR]

The non-decoupling effects of heavy scalars and vector fields play an important role in the indirect search for Beyond the Standard Model (BSM) physics at the LHC. By exploiting some new differential equations for the 1-PI amplitudes, we show that such non-decoupling effects are absent for quite a general class of effective field theories involving dimension six two-derivative and dimension eight four-derivative operators, once the resummation in certain BSM couplings is taken into account and some particular regimes of the relevant couplings are considered. [ABSTRACT FROM AUTHOR]

For 0 < k < 1, a finite-type k-surface in 3-dimensional hyperbolic space is a complete, immersed surface of finite area and of constant extrinsic curvature equal to k. In Smith (2021), we showed that such surfaces have finite genus and finitely many cusp-like ends. Each of these cusps is asymptotic to an immersed cylinder of exponentially decaying radius about a complete geodesic and terminates at an ideal point which we call the extremity of the cusp. We show that every cusp of any finite-type k-surface has a well-defined axis, which we will call the Steiner geodesic of the cusp. One of the end-points of this axis is the extremity, and we will call the other, which constitutes new geometric data, the Steiner point of the cusp. We prove a new identity involving extremities and Steiner points in terms of Möbius invariant vector fields over the Riemann sphere. We define two new functionals over the space of finite-type k-surfaces. The first, which will be called the generalized volume, is defined by the integral of a certain well-chosen form, and extends to the non-embedded case the concept of volume of the set bounded by the surface. The second, which will be called the renormalized energy, is related to the integral of the mean curvature of the surface, and is well-defined up to a choice of Busemann function. Upon describing natural parametrizations of the strata of the space of finite-type k-surfaces by open complex manifolds, we prove a new Schläfli-type formula relating the extremities and Steiner points to the first order variations of the generalized volume and the renormalized energy. In particular, Möbius invariance of this formula yields the aforementioned identity. We conclude by studying some applications of this identity and Schläfli-type formula. [ABSTRACT FROM AUTHOR]

Systematic iterative algorithms, that are solely dictated by the principles of supersymmetric quantum mechanics and do not rest on any input from the traditional methods, are developed for constructing the discrete eigen-spectra of a generic principal hypergeometric-like differential operator and also for iteratively building its associated hypergeometric-like differential operator as well as the eigen-spectra of the latter in one go. These pure algebraic algorithms reveal the simple supersymmetric quantum mechanical reason for why, for the same principal hypergeometric-like differential operator, there exist simultaneously a tower of principal as well as of associated special functions in their canonical forms. That is, two distinct types of quantum momentum operators as well as the initial superpotentials are rooted in this type of operator, i.e. the same hypergeometric-like operator admits two distinct supersymmetric factorizations, and each of these superpotentials/supersymmetric factorizations can proliferate into a hierarchy of descendant superpotentials/supersymmetric factorizations in the same shape-invariant fashion. These algebraic algorithms can be implemented in equal efficiency either directly in the non-standard x-coordinate representation of the underlying supersymmetric quantum mechanics of the hypergeometric-like differential operator or in its standard y/z-coordinate representation, and the two representations are isomorphically connected to each other through two types of active supersymmetrization transformations plus the accompanying momentum operator maps, which are readily constructed by supersymmetrizing the two types of trivial asymmetric factorizations of that principal hypergeometric-like operator in an elementary algebraic way. Due to their conceptual simplicity, relatively high efficiency and nature of doing analysis without analysis, these algorithms make the hypergeometric-like special functions and their associated cousins not so special at all as in their guises out of the popular traditional methods, and therefore could become the hopefuls for supplanting the latter.

In this paper, we study the dominant nonuniversal contributions to muon decay for the five scalar leptoquarks (LQs) allowed by the Standard Model gauge symmetries. By using a common computational framework, we isolate the New-Physics (NP) one-loop corrections over the observable Δ r and explore the connection with the W boson mass. We highlight the role of nonuniversal terms considering, for the SU (2) doublet and triplet LQs, a completely degenerate scenario, thus cancelling/moderating the effects of the oblique parameters S, T and U. We determine simple formulas that exhibit the leading nonuniversal behavior generated by the introduction of extra Yukawa terms and confront them against a present tension in M W and the projected FCC-ee accuracy. [ABSTRACT FROM AUTHOR]

We show that the dynamics of the scalar Higgs field in the Abelian Higgs–Kibble model supplemented with a dimension-6 derivative operator can be constrained at the quantum level by a certain stability equation. It holds in the Landau gauge and is derived within the recently proposed extended field formalism, where the physical scalar is described by a gauge-invariant field variable. Physical implications of the stability equation are discussed. [ABSTRACT FROM AUTHOR]

For QFT on a lattice of dimension , the vacuum energy (both bosonic and fermionic) is zero if the Hamiltonian is a function of the square of the momentum, and the calculation of the vacuum energy is performed in the ring of residue classes modulo . This fact is related to a problem from number theory about the number of ways to represent a number as a sum of squares in the ring of residue classes modulo . [ABSTRACT FROM AUTHOR]

Let f ∗ be a polynomial of degree d ( ≥ 3 ) with all critical points escaping to infinity except one periodic critical point with multiplicity d ∗ - 1 . We show that f ∗ can be tuned by an arbitrary monic centered polynomial Q of degree d ∗ with connected Julia set. This generalizes a celebrate result on the existence of the baby Mandelbrot sets by Douady–Hubbard. In particular, if d = 3 , we prove that E (f ∗) is homeomorphic to the Mandelbrot set, where E (f ∗) is the set of all the cubic polynomials that are externally conjugate to f ∗ . [ABSTRACT FROM AUTHOR]

In this Letter we introduce some field-theoretic approach for computing the critical properties of γKLS-generalized systems undergoing continuous phase transitions, namely γKLS-statistical field theory. From this new approach emerges the new generalized O(N)γKLS universality class, which is capable of encompassing nonconventional critical exponents for real imperfect systems known as manganites not described by standard statistical field theory. We compare the generalized results with those obtained from measurements in manganites. The agreement was satisfactory, where the relative errors are

We present a removal of the infrared divergences in the Beliaev-Dyson equation for spin-1 spinor Bose–Einstein-condensed systems. Unlike the Beliaev-Dyson equations for spin-0 Bose–Einstein-condensates, the Beliaev-Dyson equations for spin-1 spinor Bose–Einstein-condensates are given as 6 × 6 matrix elements. The divergent terms remaining in a ferromagnetic phase and a polar phase are contained identically in some elements of self-energy matrix. Based on experiences removing infrared divergences in spin-0 Bose–Einstein-condensates and identity of divergent terms, we remove the infrared divergences in Beliaev-Dyson equation for spinor Bose–Einstein-condensed systems. [ABSTRACT FROM AUTHOR]

We study singular limits of stochastic evolution equations in the interplay of disappearing strength of the noise and insufficient regularity, where the equation in the limit with noise would not be defined due to lack of regularity. We recover previously known results on vanishing small noise with increasing roughness, but our main focus is to study for fixed noise the singular limit where the leading order differential operator in the equation may vanish. Although the noise is disappearing in the limit, additional deterministic terms appear due to renormalization effects. We separate the analysis of the equation from the convergence of stochastic terms and give a general framework for the main error estimates. This first reduces the result to bounds on a residual and in a second step to various bounds on the stochastic convolution. Moreover, as examples we apply our result to the singularly regularized Allen–Cahn (AC) equation with a vanishing Bilaplacian, and the Cahn–Hilliard/AC homotopy with space-time white noise in two spatial dimensions. [ABSTRACT FROM AUTHOR]

We consider directed polymers in random environment in the critical dimension d = 2 , focusing on the intermediate disorder regime when the model undergoes a phase transition. We prove that, at criticality, the diffusively rescaled random field of partition functions has a unique scaling limit: a universal process of random measures on R 2 with logarithmic correlations, which we call the Critical 2d Stochastic Heat Flow. It is the natural candidate for the long sought solution of the critical 2d Stochastic Heat Equation with multiplicative space-time white noise. [ABSTRACT FROM AUTHOR]

Renormalized homotopy continuation on toric varieties is introduced as a tool for solving sparse systems of polynomial equations or sparse systems of exponential sums. The cost of continuation depends on a renormalized condition length, defined as a line integral of the condition number along all the lifted renormalized paths. The theory developed in this paper leads to a continuation algorithm tracking all the solutions between two generic systems with the same structure. The algorithm is randomized, in the sense that it follows a random path between the two systems. The probability of success is one. In order to produce an expected cost bound, several invariants depending solely on the supports of the equations are introduced. For instance, the mixed area is a quermassintegral that generalizes surface area in the same way that mixed volume generalizes ordinary volume. The facet gap measures for each 1-cone in the fan and for each support polytope, how close is the supporting hyperplane to the nearest vertex. Once the supports are fixed, the expected cost depends on the input coefficients solely through two invariants: the renormalized toric condition number and the imbalance of the absolute values of the coefficients. This leads to a nonuniform polynomial complexity bound for polynomial solving in terms of those two invariants. [ABSTRACT FROM AUTHOR]

The running of the top quark mass is experimentally investigated for the first time. The mass of the top quark in the modified minimal subtraction (MS‾) renormalization scheme is extracted from a comparison of the differential top quark-antiquark (tt¯) cross section as a function of the invariant mass of the tt¯ system to next-to-leading-order theoretical predictions. The differential cross section is determined at the parton level by means of a maximum-likelihood fit to distributions of final-state observables. The analysis is performed using tt¯ candidate events in the e± μ∓ channel in proton-proton collision data at a centre-of-mass energy of 13 TeV recorded by the CMS detector at the CERN LHC in 2016, corresponding to an integrated luminosity of 35.9fb−1. The extracted running is found to be compatible with the scale dependence predicted by the corresponding renormalization group equation. In this analysis, the running is probed up to a scale of the order of 1 TeV.

The running of the top quark mass is experimentally investigated for the first time. The mass of the top quark in the modified minimal subtraction (MS‾) renormalization scheme is extracted from a comparison of the differential top quark-antiquark (tt¯) cross section as a function of the invariant mass of the tt¯ system to next-to-leading-order theoretical predictions. The differential cross section is determined at the parton level by means of a maximum-likelihood fit to distributions of final-state observables. The analysis is performed using tt¯ candidate events in the e± μ∓ channel in proton-proton collision data at a centre-of-mass energy of 13 TeV recorded by the CMS detector at the CERN LHC in 2016, corresponding to an integrated luminosity of 35.9fb−1. The extracted running is found to be compatible with the scale dependence predicted by the corresponding renormalization group equation. In this analysis, the running is probed up to a scale of the order of 1 TeV.

We show renormalization in Quantum Brain Dynamics (QBD) in 3+1 dimensions, namely Quantum Electrodynamics with water rotational dipole fields. First, we introduce the Lagrangian density for QBD involving terms of water rotational dipole fields, photon fields and their interactions. Next, we show Feynman diagrams with 1-loop self-energy and vertex function in dipole coupling expansion in QBD. The counter-terms are derived from the coupling expansion of the water dipole moment. Our approach will be applied to numerical simulations of Kadanoff–Baym equations for water dipoles and photons to describe the breakdown of the rotational symmetry of dipoles, namely memory formation processes. It will also be extended to the renormalization group method for QBD with running parameters in multi-scales.