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Fermionic condensate is investigated in $(D+1)$-dimensional de Sitter spacetime by using the cutoff function regularization. In order to fix the renormalization ambiguity for massive fields an additional condition is imposed, requiring the condensate to vanish in the infinite mass limit. For large values of the field mass the condensate decays exponentially in odd dimensional spacetimes and follows a power law decay in even dimensional spacetimes. For a massless field the fermionic condensate vanishes for odd values of the spatial dimension $D$ and is nonzero for even $D$. Depending on the spatial dimension the fermionic condensate can be either positive or negative. The change in the sign of the condensate may lead to instabilities in interacting field theories., 12 pages, 2 figures, to appear in Astrophysics

Loop diagrams with near-Planck energies create a strong external gravitational field, which slows down local processes for distant observers up to their freezing. Since Planck length is the gravitational radius of the system of quanta, the events of this and smaller scale cannot occur in finite world time t and do not contribute to the S-matrix. Consequently, gravitational time dilation, leading to a strong redshift of local frequencies, provides gravitational self-regularization of the loop diagrams. The loop corrections without gravity effects, cut off at Planck energy, give upper bounds for the corrections with gravity effects and this fact leads to simple rules of gravitational regularization. The corrections with quanta of gauge fields and gravitons are small, and the perturbation theory series converge. At pre-Planck energies, one-loop graviton contributions are sufficient, since the multi-loop ones are damped by high degrees of the relation “energy/Planck energy”. Scalar field with power-law growing corrections should be effective field. Non-linearity of fields enhances gravity and get faster freezing, which suppresses the high energy terms. Nonrenormalizable models are finite, but become consistent only when their loop corrections remain small on Planck scale and this occurs in quantum gravity. Gravitationally regularized Extended Standard Model (ESM), including gravitons and Standard Model with effective scalars, is renormalizable and finite, which simplifies its further generalization.

We present two-loop renormalization of $\phi^3$-model effective action by using the background field method and cutoff momentum regularization. In this paper, we also study a derivation of the quantum equation of motion and its application to the renormalization., Comment: LaTeX, 15 pages, 3 figures; The work has been published three years ago. In this version we have made some corrections and added calculations for the counterterm

We consider a diffuse interface model describing flow and phase separation of a binary isothermal mixture of (partially) immiscible viscous incompressible Newtonian fluids having different densities. The model is the nonlocal version of the one derived by Abels, Garcke and Grun and consists in a inhomogeneous Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. This model was already analyzed in a paper by the same author, for the case of singular potential and non-degenerate mobility. Here, we address the physically more relevant situation of degenerate mobility and we prove existence of global weak solutions satisfying an energy inequality. The proof relies on a regularization technique based on a careful approximation of the singular potential. Existence and regularity of the pressure field is also discussed. Moreover, in two dimensions and for slightly more regular solutions, we establish the validity of the energy identity. We point out that in none of the existing contributions dealing with the original (local) Abels, Garcke Grun model, an energy identity in two dimensions is derived (only existence of weak solutions has been proven so far).

Recently a $D$-dimensional regularization approach leading to the non-trivial $(3+1)$-dimensional Einstein-Gauss-Bonnet (EGB) effective description of gravity was formulated which was claimed to bypass the Lovelock's theorem and avoid Ostrogradsky instability. Later it was shown that the regularization is possible only for some broad, but limited, class of metrics and Aoki, Gorji and Mukohyama [arXiv:2005.03859] formulated a well-defined four-dimensional EGB theory, which breaks the Lorentz invariance in a theoretically consistent and observationally viable way. The black-hole solution of the first naive approach proved out to be also the exact solution of the well-defined theory. Here we calculate quasinormal modes of scalar, electromagnetic and gravitational perturbations and find the radius of shadow for spherically symmetric and asymptotically flat black holes with Gauss-Bonnet corrections. We show that the black hole is gravitationally stable when ($-16 M^2, Comment: RevTex, 12 pages, 8 figures, the version published in the journal, the title is slightly changed

Recently, a novel four-dimensional Einstein-Gauss-Bonnet (EGB) theory was presented to bypass the Lovelock's theorem and to give nontrivial effects on the four-dimensional local gravity. The main mechanism is to introduce a redefinition $\alpha\rightarrow\alpha/(D-4)$ and to take the limit $D\rightarrow4$. However, this theory does not have standard four-dimensional field equations. Some regularization procedures are then proposed to address this problem [arXiv:2003.11552, arXiv:2003.12771, arXiv:2004.08362, arXiv:2004.09472, arXiv:2004.10716]. The resultant regularized four-dimensional EGB theory has the same on-shell action as the original theory. Thus it is expected that the novel four-dimensional EGB theory is equivalent to its regularized version. However, the equivalence of these two theories is symmetry-dependent. In this paper, we test the equivalence in a cylindrically symmetric spacetime. The well-defined field equations of the two theories are obtained, with which our follow-up analysis shows that they are equivalent in such spacetime. Cylindrical cosmic strings are then considered as specific examples of the metric. Three sets of solutions are obtained and the corresponding string mass densities are evaluated. The results reveal how the Gauss-Bonnet term in four dimensions contributes to the string geometry in the new theory., Comment: 20 pages, 9 figures, published version

In this article, we establish in a rigorous mathematical way that Koba-Nielsen amplitudes defined on any local field of characteristic zero are bona fide integrals that admit meromorphic continuations in the kinematic parameters. Our approach allows us to study in a uniform way open and closed Koba-Nielsen amplitudes over arbitrary local fields of characteristic zero. In the regularization process we use techniques of local zeta functions and embedded resolution of singularities. As an application we present the regularization of p-adic open string amplitudes with Chan-Paton factors and constant B-field. Finally, all the local zeta functions studied here are partition functions of certain 1D log-Coulomb gases, which shows an interesting connection between Koba-Nielsen amplitudes and statistical mechanics., Some additional comments were added

A model describing the dynamics of a set of quantum states generated by a nonlinear Schrodinger equation is studied. The relationship between the blow-up of a solution with self-focusing and the transition from pure to mixed states of a quantum system was investigated in [1]. In this context, a natural question is concerned with the dynamics generated by the nonlinear Schrodinger equation in the set of mixed quantum states. The dynamics of mixed quantum states is described by the Liouville–von Neumann equation corresponding to the nonlinear Schrodinger equation. For the former equation, conditions for the global existence of a unique solution of the Cauchy problem and blow-up conditions are obtained.

We apply the harmonic superspace approach for calculating the divergent part of the one-loop effective action of $6D$, ${\cal N}=(1,0)$ supersymmetric higher-derivative gauge theory with a dimensionless coupling constant. Our consideration uses the background superfield method allowing to carry out the analysis of the effective action in a manifestly gauge covariant and ${\cal N}=(1,0)$ supersymmetric way. We exploit the regularization by dimensional reduction in which the divergences are absorbed into a renormalization of the coupling constant. Having the expression for the one-loop divergences, we calculate the relevant $\beta$-function. Its sign is specified by the overall sign of the classical action which in higher-derivative theories is not fixed {\it a priori}. The result agrees with the earlier calculations in the component approach. The superfield calculation is simpler and provides possibilities for various generalizations., Comment: 18 pages, 1 figure, final version accepted for publication in JHEP

A functional version of the Kato one-parametric regularisation for the construction of a dynamical semigroup generator of a relative bound one perturbation is introduced. It does not require that the minus generator of the unperturbed semigroup is a positivity preserving operator. The regularisation is illustrated by an example of a boson-number cut-off regularisation.

The problem of rigorous substantiation of the so-called microscopic solutions to the Boltzmann–Enskog kinetic equations discovered by N.N. Bogolyubov is considered. The solutions have the form of sums of delta functions and correspond to the reversible dynamics of a finite number of particles. However, the direct substitution of delta functions into the equation is formally incorrect. A regularization of the collision integral is proposed in this paper at which the substitution of delta functions becomes possible.

We studied the Schwarzschild solution using Colombeau distributional geometry [M. Kunzinger and R. Steinbauer, Trans. Am. Math. Soc. 354, 4179 (2002)], thus without leaving Schwarzschild coordinates.

The two-loop anomalous dimension of the chiral matter superfields is calculated for a general ${\cal N}=1$ supersymmetric gauge theory regularized by higher covariant derivatives. We obtain both the anomalous dimension defined in terms of the bare couplings, and the one defined in terms of the renormalized couplings for an arbitrary renormalization prescription. For the one-loop finite theories we find a simple relation between the higher derivative regulators under which the anomalous dimension defined in terms of the bare couplings vanishes in the considered approximation. In this case the one-loop finite theory is also two-loop finite in the HD+MSL scheme. Using the assumption that with the higher covariant derivative regularization the NSVZ equation is satisfied for RGFs defined in terms of the bare couplings, we construct the expression for the three-loop $\beta$-function. Again, the result is written both for the $\beta$-function defined in terms of the bare couplings and for the one defined in terms of the renormalized couplings for an arbitrary renormalization prescription., Comment: 26 pages, 3 figures, two minor misprints corrected, the final version to appear in JHEP

In this article, we study the two-flavor Nambu and Jona-Lasinio (NJL) phase diagrams on the T–μ plane through three regularization methods. In one of these, we introduce an infrared three-momentum cutoff in addition to the usual ultraviolet regularization to the quark loop integrals and compare the obtained phase diagrams with those obtained from the NJL model with proper time regularization and Pauli–Villars regularization. We have found that the crossover appears as a band with a well-defined width in the T–μ plane. To determine the extension of the crossover zone, we propose a novel criterion, comparing it to another criterion that is commonly reported in the literature; we then obtain the phase diagrams for each criterion. We study the behavior of the phase diagrams under all these schemes, focusing on the influence of the regularization procedure on the crossover zone and the presence or absence of critical end points.

We formulate a class of stochastic partial differential equations based on Kelvin’s circulation theorem for ideal fluids. In these models, the velocity field is randomly transported by white-noise vector fields, as well as by its own average over realizations of this noise. We call these systems the Lagrangian averaged stochastic advection by Lie transport (LA SALT) equations. These equations are nonlinear and non-local, in both physical and probability space. Before taking this average, the equations recover the Stochastic Advection by Lie Transport (SALT) fluid equations introduced by Holm (Proc R Soc A 471(2176):20140963, 2015). Remarkably, the introduction of the non-locality in probability space in the form of momentum transported by its own mean velocity gives rise to a closed equation for the expectation field which comprises Navier–Stokes equations with Lie–Laplacian ‘dissipation’. As such, this form of non-locality provides a regularization mechanism. The formalism we develop is closely connected to the stochastic Weber velocity framework of Constantin and Iyer (Commun Pure Appl Math 61(3):330–345, 2008) in the case when the noise correlates are taken to be the constant basis vectors in $$\mathbb {R}^3$$ R 3 and, thus, the Lie–Laplacian reduces to the usual Laplacian. We extend this class of equations to allow for advected quantities to be present and affect the flow through exchange of kinetic and potential energies. The statistics of the solutions for the LA SALT fluid equations are found to be changing dynamically due to an array of intricate correlations among the physical variables. The statistical properties of the LA SALT physical variables propagate as local evolutionary equations which when spatially integrated become dynamical equations for the variances of the fluctuations. Essentially, the LA SALT theory is a non-equilibrium stochastic linear response theory for fluctuations in SALT fluids with advected quantities.

The contributions of the matter superfields and of the Faddeev--Popov ghosts to the $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories defined in terms of the bare couplings are calculated in all orders in the case of using the higher covariant derivative regularization. For this purpose we use the recently proved statement that the $\beta$-function in these theories is given by integrals of double total derivatives with respect to the loop momenta. These integrals do not vanish due to singularities of the integrands. This implies that the $\beta$-function beyond the one-loop approximation is given by the sum of the singular contributions, which is calculated in all orders for singularities produced by the matter superfields and by the Faddeev--Popov ghosts. The result is expressed in terms of the anomalous dimensions of these superfields. It coincides with the corresponding part of the new form of the NSVZ equation, which can be reduced to the original one with the help of the non-renormalization theorem for the triple gauge-ghost vertices., Comment: 38 pages, 4 figures

We are grateful to Rafael Ortega for bringing to our attention the position-energy system that has been crucial for the regularization of collisions. We also thank the anonymous referees for their valuable comments and suggestions that contributed to improve the initial version of the paper. This work has been realized thanks to the Agencia Estatal de Investigaci´on, Spain and Ministerio de Ciencia, Innovaci´on y Universidades, Spain grants MTM2017-82348-C2-1-P and MTM2017-86795-C3-1-P., We prove the existence and multiplicity of periodic solutions of bouncing type for a second-order differential equation with a weak repulsive singularity. Such solutions can be cataloged according to the minimal period and the number of elastic collisions with the singularity in each period. The proof relies on the Poincaré–Birkhoff Theorem., Agencia Estatal de Investigación, Spain and Ministerio de Ciencia, Innovación y Universidades, Spain grants MTM2017-82348-C2-1-P and MTM2017-86795-C3-1-P

We analyze the lattice spacing dependence for the pion unpolarized matrix element of a quark bilinear operator with a Wilson link (parton quasidistribution functions operator, quasi-PDFs) in the rest frame, using 13 lattice spacings ranging from 0.032 to 0.121 fm. We compare results for three different fermion actions with or without good chiral symmetry on dynamical gauge ensembles from three collaborations. This investigation is motivated by the fact that the gauge link generates a 1/a divergence, the cancellation of which in many ratios can be numerically tricky. Indeed, our results show that this cancellation deteriorates with decreasing lattice spacing, and that the RI/MOM method leaves a linearly divergent residue for quasi-PDFs. We also show that in the Landau gauge the interaction between the Wilson link and the external state results in a linear divergence which depends on the discretized fermion action.

In applying large-momentum effective theory, renormalization of the Euclidean correlators in lattice regularization is a challenge due to linear divergences in the self-energy of Wilson lines. Based on lattice QCD matrix elements of the quasi-PDF operator at lattice spacing $a$= 0.03 fm $\sim$ 0.12 fm with clover and overlap valence quarks on staggered and domain-wall sea, we design a strategy to disentangle the divergent renormalization factors from finite physics matrix elements, which can be matched to a continuum scheme at short distance such as dimensional regularization and minimal subtraction. Our results indicate that the renormalization factors are universal in the hadron state matrix elements. Moreover, the physical matrix elements appear independent of the valence fermion formulations. These conclusions remain valid even with HYP smearing which reduces the statistical errors albeit reducing control of the renormalization procedure. Moreover, we find a large non-perturbative effect in the popular RI/MOM and ratio renormalization scheme, suggesting favor of the hybrid renormalization procedure proposed recently., Comment: 29 pages, 30 figures

We describe the effect of the marginal deformation of the $\cal N = (4, 4)$ superconformal $(T^4)^N /S_N$ orbifold theory on a doublet of R-neutral twisted Ramond fields, in the large-$N$ approximation. Our analysis of their dynamics explores the explicit analytic form of the genus-zero four-point function involving two R-neutral Ramond fields and two deformation operators. We compute this correlation function with two different approaches: the Lunin-Mathur path-integral technique and the stress-tensor method. From its short distance limits, we extract the OPE structure constants and the scaling dimensions of non-BPS fields appearing in the fusion. In the deformed CFT, at second order in the deformation parameter, the two-point function of the $n$-twisted Ramond fields is UV-divergent. We perform an appropriate regularization, together with a renormalization of the undeformed fields, obtaining finite, well-defined corrections to their two-point functions and their bare conformal weights, for $n < N$. The fields with maximal twist $n=N$ remain protected from renormalization, with vanishing anomalous dimensions., 27 pages, 1 figure; V2 - corrected and improved version; V3 - 37 pages, 1 figure; extended, improved version; new discussions and references added; to appear in JHEP

We construct a quantum kinetic theory of a weakly interacting critical boson gas using the expectation values of products of Heisenberg field operators in the grand canonical ensemble. Using a functional representation for the Wick theorem for time-ordered products, we construct a perturbation theory for the generating functional of these time-dependent Green’s functions at a finite temperature. We note some problems of the functional-integral representation and discuss unusual apparent divergences of the perturbation expansion. We propose a regularization of these divergences using attenuating propagators. Using a linear transformation to variables with well-defined scaling dimensions, we construct an infrared effective field theory. We show that the structure of the regularized model is restored by renormalization. We propose a multiplicatively renormalizable infrared effective model of the quantum dynamics of a boson gas.

We study the contour dependence of the out-of-time-ordered correlation function (OTOC) both in weakly coupled field theory and in the Sachdev-Ye-Kitaev (SYK) model. We show that its value, including its Lyapunov spectrum, depends sensitively on the shape of the complex time contour in generic weakly coupled field theories. For gapless theories with no thermal mass, such as SYK, the Lyapunov spectrum turns out to be an exception; their Lyapunov spectra do not exhibit contour dependence, though the full OTOCs do. Our result puts into question which of the Lyapunov exponents computed from the exponential growth of the OTOC reflects the actual physical dynamics of the system. We argue that, in a weakly coupled $\Phi^4$ theory, a kinetic theory argument indicates that the symmetric configuration of the time contour, namely the one for which the bound on chaos has been proven, has a proper interpretation in terms of dynamical chaos. Finally, we point out that a relation between these OTOCs and a quantity which may be measured experimentally --- the Loschmidt echo --- also suggests a symmetric contour configuration, with the subtlety that the inverse periodicity in Euclidean time is half the physical temperature. In this interpretation the chaos bound reads $\lambda \leq \frac{2\pi}{\beta}= \pi T_{\text{physical}}$., Comment: Comment on regularization dependence in 2d-CFTs added. Published version

We explore several new converse bounds for classical communication over quantum channels in both the one-shot and asymptotic regimes. First, we show that the Matthews-Wehner meta-converse bound for entanglement-assisted classical communication can be achieved by activated, no-signalling assisted codes, suitably generalizing a result for classical channels. Second, we derive a new efficiently computable meta-converse on the amount of classical information unassisted codes can transmit over a single use of a quantum channel. As applications, we provide a finite resource analysis of classical communication over quantum erasure channels, including the second-order and moderate deviation asymptotics. Third, we explore the asymptotic analogue of our new meta-converse, the $\Upsilon$-information of the channel. We show that its regularization is an upper bound on the classical capacity, which is generally tighter than the entanglement-assisted capacity and other known efficiently computable strong converse bounds. For covariant channels we show that the $\Upsilon$-information is a strong converse bound., Comment: v3: published version; v2: 18 pages, presentation and results improved

By imposing the weighted renormalization condition and the (super)symmetry requirements, we construct a Lifshitz-like extension of the three-dimensional Wess-Zumino model, with dynamical critical exponent z = 2 . In this context, the auxiliary field F plays a key role by introducing the appropriate Lifshitz operator in the bosonic sector of the theory, avoiding so undesirable time-space mixing derivatives and inconsistencies concerning the critical z exponent, as reported in the literature. The consistency of the proposed model is verified by building explicitly the susy algebra through the Noether method in the canonical formalism. This component-field Lifshitz-Wess-Zumino model is in addition rephrased in the Lifshitz superspace, a natural modification of the conventional one. The one-loop effective potential is computed to study the possibility of symmetry breaking. It is found that supersymmetry remains intact at one-loop order, while the U ( 1 ) phase symmetry suffers a spontaneous breakdown above the critical value of the renormalization point. By renormalizing the one-loop effective potential within the cutoff regularization scheme, it is observed an improvement of the UV behavior of the theory compared with the relativistic Wess-Zumino model.

We continue the study of the Wilson line representation of conformal blocks in two-dimensional conformal field theory; these have an alternative interpretation as gravitational Wilson lines in the context of the AdS$_3$/CFT$_2$ correspondence. The gravitational Wilson line involves a path-ordered exponential of the stress tensor, and its expectation value can be computed perturbatively in an expansion in inverse powers of the central charge $c$. The short-distance singularities which occur in the associated stress tensor correlators require systematic regularization and renormalization prescriptions, whose consistency with conformal Ward identities presents a subtle problem. The regularization used here combines dimensional regularization and analytic continuation. Representation theoretic arguments, based on SL(2,R) current algebra, predict an exact result for the Wilson line anomalous dimension and, by building on previous work, we verify that the perturbative calculations using our regularization and renormalization prescriptions reproduce the exact result to order $1/c^3$ included. We also discuss a related, but somewhat simpler, Wilson line in Wess-Zumino-Witten models that yields current algebra conformal blocks, and we emphasize the distinction between Wilson lines constructed out of non-holomorphic and purely holomorphic currents., Comment: 47 pages

We study a geometric variational problem for sets in the plane in which the perimeter and a regularized dipolar interaction compete under a mass constraint. In contrast to previously studied nonlocal isoperimetric problems, here the nonlocal term asymptotically localizes and contributes to the perimeter term to leading order. We establish existence of generalized minimizers for all values of the dipolar strength, mass and regularization cutoff and give conditions for existence of classical minimizers. For subcritical dipolar strengths we prove that the limiting functional is a renormalized perimeter and that for small cutoff lengths all mass-constrained minimizers are disks. For critical dipolar strength, we identify the next-order $\Gamma$-limit when sending the cutoff length to zero and prove that with a slight modification of the dipolar kernel there exist masses for which classical minimizers are not disks., Comment: 54 pages, 6 figures

We complete the proof of Weinberg's no-go theorem on the cosmological constant problem in classical gravity when the theory has a (global) scale symmetry. Stimulated with this proof, we explore a solution to the cosmological constant problem by the help of renormalization group equations. We find that the manifestly scale invariant regularization method provides a physically plausible solution to the cosmological constant problem, in particular, to the issue of radiative instability of the cosmological constant., Comment: 17 pages

We present an account of dual pairs and the Kummer shapes for $n:m$ resonances that provides an alternative to Holm and Vizman's work. The advantages of our point of view are that the associated Poisson structure on $\mathfrak{su}(2)^{*}$ is the standard $(+)$-Lie--Poisson bracket independent of the values of $(n,m)$ as well as that the Kummer shape is regularized to become a sphere without any pinches regardless of the values of $(n,m)$. A similar result holds for $n:-m$ resonance with a paraboloid and $\mathfrak{su}(1,1)^{*}$. The result also has a straightforward generalization to multidimensional resonances as well., 14 pages, 1 figure

We compute the two-loop $\beta$-function of scalar and spinorial quantum electrodynamics as well as pure Yang-Mills and quantum chromodynamics using the background field method in a fully quadridimensional setup using Implicit Regularization (IREG). Moreover, a thorough comparison with dimensional approaches such as conventional dimensional regularization (CDR) and dimensional reduction (DRED) is presented. Subtleties related to Lorentz algebra contractions/symmetric integrations inside divergent integrals as well as renormalisation schemes are carefully discussed within IREG where the renormalisation constants are fully defined as basic divergent integrals to arbitrary loop order. Moreover, we confirm the hypothesis that momentum routing invariance in the loops of Feynman diagrams implemented via setting well-defined surface terms to zero deliver non-abelian gauge invariant amplitudes within IREG just as it has been proven for abelian theories., Comment: Section 2 expanded, references revised, misprints corrected. Final results unchanged. Comply with published version

Alocally-finitequantity is one for which there is no region of divergence anywhere in the space of real loop momenta; it can therefore be computed (in principle) without regularization. In this work, we prove thatalltwo-loop ratio functions in planar, maximally supersymmetric Yang-Mills theory are locally-finite.

We investigate the lattice regularization of $\mathcal{N} = 4$ supersymmetric Yang-Mills theory, by stochastically computing the eigenvalue mode number of the fermion operator. This provides important insight into the non-perturbative renormalization group flow of the lattice theory, through the definition of a scale-dependent effective mass anomalous dimension. While this anomalous dimension is expected to vanish in the conformal continuum theory, the finite lattice volume and lattice spacing generically lead to non-zero values, which we use to study the approach to the continuum limit. Our numerical results, comparing multiple lattice volumes, 't Hooft couplings, and numbers of colors, confirm convergence towards the expected continuum result, while quantifying the increasing significance of lattice artifacts at larger couplings., 24 pages, 13 figures

We do not observe quantum effects on cosmological scales. Thus, if loop quantum cosmology (LQC) is to provide an accurate depiction of the real world, it must allow for quantum states of spacetime geometry which are semi-classical in two respects: they must be sharply peaked around a single, classical geometry, and they must have small quantum fluctuations. It is generally assumed that Gaussian states exhibit both of these properties. After all, they do in ordinary quantum mechanics. In this paper, we derive exact closed-form expressions for the fluctuations of Gaussian states in LQC and their lower bound given by the Robertson-Schr\"odinger inequality. We demonstrate that, contrary to ordinary quantum mechanics, fluctuations for Gaussian states in spatially flat, homogeneous and isotropic LQC diverge as the state variance increases (as well as in related cosmological models with the same kinematic Hilbert space and canonical observables). However, when the holonomy length is made to scale with a volume regularization parameter, these fluctuations may be arbitrarily suppressed by taking the fiducial volume to be large, providing analytic control over their divergence. Finally, we show that, despite this, Gaussian states in LQC generally do not minimize uncertainty. Moreover, it is conjectured that no such minimal-uncertainty states exist. Throughout this work, it becomes clear how important the often-assumed condition of holonomy length volume-scaling is; we show that when this condition is violated, the resulting theory exhibits operator closure pathologies and other exotic algebraic features., Comment: 11 pages, 5 figures; To be published in Physical Review D

We show the existence of ground states in the massless spin boson model without any infrared regularization. Our proof is non-perturbative and relies on a compactness argument. It works for arbitrary values of the coupling constant under the hypothesis that the second derivative of the ground state energy as a function of a constant external magnetic field is bounded., 14 pages

Singularity of the potential function makes quantum tunneling problem mathematically underdetermined. To circumvent the difficulties it introduced in physics, a potential singularity cutoff is often used, followed by a reverse limit transition, or is a suitable self-adjoint extension of the Hamiltonian along the entire coordinate axis made. However, both of them somehow affect the singular nature of the problem, and so I discuss here how quantum tunneling will behave if the original singular nature of the Schrodinger equation left untuched. To do this, I use the property of the probability density current that the singularities are mutually destroyed in it. It is found that the mildly singular potential has a finite, but unusual tunneling transparency, in particular, a non-zero value at zero energy of incident particle. The tunneling of one dimensional Coulomb potential exhibits infinitely fast and complete oscillation at the zero energy boundary and a suppresion to zero in the high-energy limit. In the more singular region, the tunneling becomes forbidden, theby repeating the well-known result of the regularized counterparts., The quantum tunneling of a potential is considered with the preservation of the singular character of the corresponding Scrodinger equation. It is found that that the weakly singular and Coulomb potentials are partially penetrable but reveals non-standard behavior, in particular, accelerating complete oscillations with the approach to the zero energy

Classical black holes shield us from the singularities that inevitably appear in general relativity. Being singularity regularization one of the main landmarks for a successful theory of quantum gravity, quantum black holes are not obliged to hide their inner core from the outside world. Notwithstanding the aforesaid, it is often implicitly assumed that quantum gravity effects must remain confined to black hole interiors. In this essay we argue in the opposite direction, discussing theoretical evidence for the existence of strong correlations between the physics inside and outside non-singular black holes. We conclude that astronomical tests of the surroundings of black holes can provide invaluable information about their so-far unexplored interiors., Comment: 7 pages, 2 figure,Awarded an Honorable Mention in the 2021 Gravity Research Foundation Essays Competition

Incompressible fluid equations are studied with UV cut-off and in periodic boundary conditions. Properties of the resulting ODEs holding uniformly in the cut-off are considered and, in particular, are conjectured to be equivalent to properties of other time reversible equations. Reversible equations with the same regularization and describing equivalently the fluid, and the fluctuations of large classes of observables, are examined in the context of the "Chaotic Hypothesis", "Axiom C" and the "Fluctuation Theorem"., Comment: 12 pages, in v.2 Eq. 4.2 (and references to it) corrected; in v.3 typo in 4.3 and 5.1 corrected

We investigate a particular regularization of big bang singularity, which remains within the domain of 4-dimensional general relativity but allowing for degenerate metrics. We study the geodesics and geodesic congruences in the modified Friedmann-Lema\^itre-Robertson-Walker universe. In particular, we calculate the expansion of timelike and null geodesic congruences. Based on these results, we also briefly discuss the cosmological singularity theorems., Comment: 16 pages, 1 figure, published version

The functional renormalization group is a particular implementation of the renormalization group concept which combines functional methods of quantum field theory with the renormalization group idea of Kenneth Wilson. It interpolates smoothly between the known microscopic laws and the complex macroscopic phenomena in physical systems. The renormalization group is formulated directly for a continuum field theory—no lattice regularization is required. The flow from microscopic to macroscopic scales is given by technically demanding flow equations. We derive the Polchinski equation for the scale-dependent Schwinger functional and the Wetterich equation for the scale-dependent effective action. We use the latter to calculate the effective potential, ground state energy and energy gap of quantum mechanical systems. Next we consider scalar fields and calculate the flow of the effective potential and several critical exponents in the local potential approximation. Then we present an exact solution for the scale-dependent effective potential of O(N) models and the critical exponents in the large-N limit. All numerical results were obtained with the matlab/octave programs listed at the end of the chapter.

The encoding of all possible sets of Bethe equations for a spin chain with SU(N|M) symmetry into a QQ-system calls for an expression of spin chain overlaps entirely in terms of Q-functions. We take a significant step towards deriving such a universal formula in the case of overlaps between Bethe eigenstates and integrable boundary states, of relevance for AdS/dCFT, by determining the transformation properties of the overlaps under fermionic as well as bosonic dualities which allows us to move between any two descriptions of the spin chain encoded in the QQ-system. An important part of our analysis involves introducing a suitable regularization for singular Bethe root configurations., Comment: 28 pages, 3 figures; v2: minor changes

We revisit a nearly ten-year old controversy on the diphoton decay of the Higgs particle. To a large extent, the controversy turned around the respective merits of the regularization techniques employed. The novel aspect of our approach is that no regularization techniques are brought to bear: we work within the Bogoliubov--Epstein--Glaser scheme of renormalization by extension of distributions. Solving the problem actually required an expansion of this method's toolkit, furnished in the paper., 45 pages, to appear in Eur. Phys. J. C

The quantum vacuum fluctuations of a neutral scalar field induced by background zero-range potentials concentrated on a flat hyperplane of co-dimension $1$ in $(d+1)$-dimensional Minkowski spacetime are investigated. Perfectly reflecting and semitransparent surfaces are both taken into account, making reference to the most general local, homogeneous and isotropic boundary conditions compatible with the unitarity of the quantum field theory. The renormalized vacuum polarization is computed for both zero and non-zero mass of the field, implementing a local version of the zeta regularization technique. The asymptotic behaviours of the vacuum polarization for small and large distances from the hyperplane are determined to leading order. It is shown that boundary divergences are soften in the specific case of a pure Dirac delta potential., Comment: 25 pages

The regularization of quantum electrodynamics in the space of functions $\psi_a(x)$, which depend on both the position $x$ and the scale $a$, is presented. The scale-dependent functions are defined in terms of the continuous wavelet transform in $\mathbb{R}^4$ Euclidean space, with the derivatives of Gaussian served as basic wavelets. The vacuum polarization and the dependence of the effective coupling constant on the scale parameters are calculated in one-loop approximation in the limit $p^2 \gg 4m^2$., Comment: 19 LaTeX pages, 6 eps figures; to appear in Phys. Rev. D

Let $\Phi$ be a locally convex space and let $\Phi'$ denote its strong dual. In this paper we introduce sufficient conditions for the existence of a continuous or a c\`{a}dl\`{a}g $\Phi'$-valued version to a cylindrical process defined on $\Phi$. Our result generalizes many other known results on the literature and their different connections will be discussed. As an application, we use our results to show the existence of a $\Phi'$-valued c\`{a}dl\`{a}g L\'{e}vy process version to a given cylindrical L\'{e}vy process in $\Phi'$., Comment: arXiv admin note: text overlap with arXiv:1701.06630, arXiv:1902.03981, arXiv:1806.10231, arXiv:1511.08443

We study the reduction and regularization processes of perturbed Keplerian systems from an astronomical point of view. Our approach connects axially symmetric perturbed 4-DOF oscillators with Keplerian systems, including the case of rectilinear solutions. This is done through a preliminary reduction recently studied by the authors. Then, the reduction program continues by removing the Keplerian energy. For each value of the semi-major axis, we explain the astronomical meaning of the sextuples defining the orbit space $$\mathbb {S}^2\times \mathbb {S}^2$$ and its connection with the orbital elements. More precisely, we present alternative sextuple coordinates for the set of bounded Keplerian orbits that ‘separate’ the node of the orbital plane from the argument of perigee giving the Laplace vector in that plane. Still, the reduction of the axial symmetry defined by the third component of the angular momentum is performed. For the thrice reduced space $$\varGamma _{0,L,H}$$ we propose the Cushman–Deprit coordinates, a variant to the set given by Cushman. The main feature of these variables is that they are all with the same dimensions, which is convenient for the normalization procedure. As an application of the proposed scheme, we study the spatial lunar problem.

We study one-loop corrections to the two-point correlation function of tensor perturbations in primordial cosmology induced by massless spectator matter fields. Using the Schwinger-Keldysh formalism in cosmological perturbation theory, we employ dimensional regularization and cutoff regularization to study the finite quantum corrections at one-loop arising from isocurvature fields of the massless scalar, fermion and abelian gauge field which are freely propagating on the FRW spacetime. For all cases, we find a logarithmic running of the form $\frac{C}{q^3} \, \frac{H^4}{M^4_p} \log \left( \frac{H}{\mu } \right)$ where $C$ is a negative constant related to the beta function, $H$ is the Hubble parameter at horizon exit and $\mu$ is the renormalization scale., Comment: 33 pages, 2 figures: improved discussion of the covariant cutoff regularization method, some reorganization of materials, references added

In this article, we successfully construct various kinds of exact traveling wave solutions like, hyperbolic (kink and singular kink-shaped soliton), trigonometric (periodic and singular periodic) as well as rational function solutions for the best known mixed hyperbolic-elliptic system of conservation laws, namely the van der Waals gas system in the viscosity-capillarity regularization version by means of ($\frac{G’}{G^2}$)-expansion and advanced $\mbox{e}^{-\phi(\tau)}$-expansion functions methods. These techniques are very useful and exceptionally helpful in a contrast with other analytical schemes, which show the effectiveness and the simplicity to discuss the exact solutions. 3D and contour figures are sketched in order to understand the physical movement of the gained results under the selections ofunknown parameters.

We show that, contrary to popular belief, lower order dispersive regularization of hyperbolic systems does not exclude the development of the localized shock-like transition fronts. To guide the numerical search of such solutions, we generalize Rankine-Hugoniot relations to cover the case of higher order dispersive discontinuities and study their properties in an idealized case of a transition between two periodic wave trains with different wave lengths. We present evidence that smoothed stationary fronts of this type are numerically stable in the case when regularization is temporal and one of the adjacent states is homogeneous. In the zero dispersion limit such shock-like transition fronts, that are not traveling waves and apparently require for their description more complex anzats, evolve into traveling wave type jump discontinuities.

The hydrogen ground-state problem is a touchstone for the theory of Stochastic Electrodynamics. Recently, we have shown numerically and theoretically that the H-atom self-ionizes after a characteristic time. In another approach, we reconsidered the harmonic oscillator and renormalized the stochastic force in order to suppress high-frequency tails so that all frequency integrals are dominated by the physical resonances. In the present work, we consider the regularization of the noise in the hydrogen ground-state problem. Several renormalization schemes are considered. Some are well-behaved, whereas in others the high frequency renormalization induces pathologies at low frequencies. In no situation did we find a way to escape from the previously signaled self-ionization.

We present an analysis of two-current correlations for the pion in the Nambu--Jona-Lasinio model, with Pauli--Villars regularization. We provide explicit expressions in momentum space for two-current correlations corresponding to the zeroth component of the vector Dirac bilinear in the quark vertices, which has been evaluated on the lattice. The numerical results show a remarkable qualitative agreement with recent lattice data. The factorization approximation into one-body currents is discussed., Comment: 19 pages, 9 figures. arXiv admin note: text overlap with arXiv:1909.09530

We propose a new regularization scheme to study the bound state of two-nucleon systems in lattice effective field theory. Inspired by a continuum effective field theory calculation, we study an exponential regulator acting on the leading-order and next-to-leading order interactions, consisting of local contact terms. By fitting the low-energy coefficients to deuteron binding energy and the asymptotic normalization coefficient on a lattice simulation, we extract the effective range expansion (ERE) parameters in the ${}^{3}{S}_{1}$ channel to order ${p}^{2}$. We explore the impact of different powers of the regulator on the extracted ERE parameters for the lattice spacing $a=1.97$ fm. Moreover, we investigate how the implementation of the regularization scheme improves the predicted ERE parameters on the lattice spacing in the range of $1.4\ensuremath{\le}a\ensuremath{\le}2.6\phantom{\rule{0.16em}{0ex}}\mathrm{fm}$. Our numerical analysis indicates that for lattice spacing greater than $2\phantom{\rule{0.16em}{0ex}}\mathrm{fm}$, the predicted observables are very close to the experimental data.