Dimensional Regularization Technique Research Articles

Vlasov equation, waves and dispersion relations in fractal dimensions: Landau damping and the toroidal ion temperature gradient instability problem

In this study, we have used the concept of ‘product-like fractal measure’ introduced by Li and Ostoja-Starzewski in their formulation of anisotropic and elastic media to introduce Vlasov equation in fractal dimension. We have been motivated by this new concept since it is safely applied to several systems with length scales bounded by lower and upper cutoffs via the dimensional regularization technique. We have considered through this study collisionless plasma in fractal dimension and we have analyzed the impacts of the Vlasov equation in some points in plasma physics starting from the Hamiltonian dynamics. Several results have been obtained and discussed accordingly. The main outcomes of the present study concern first the facts that waves in cold plasma are associated with low fractal dimensions and hence by low complexity and that the effective damping is affected by the sign of the fractal dimension and differs from the conventional Landau damping in magnitude and in sign for negative dimension. Besides, for a very low fractal dimension, the Landau damping is negligible or removed. This result was observed in kinetic turbulent plasmas and may solve the paradox problem emerging in the application of critical balance to a kinetic turbulence cascade.

The exact renormalization group and dimensional regularization

The exact renormalization group (ERG) is formulated implementing the decimation of degrees of freedom by means of a particular momentum integration measure. The definition of this measure involves a distribution that links this decimation process with the dimensional regularization technique employed in field theory calculations. Taking the dimension [Formula: see text], the one-loop solutions to the ERG equations for the scalar field theory in this scheme are shown to coincide with the dimensionally regularized perturbative field theory calculation in the limit [Formula: see text], if a particular relation between the scale parameter [Formula: see text] and [Formula: see text] is employed. In general, it is shown that in this scheme the solutions to the ERG equations for the proper functions coincide when [Formula: see text] with the complete diagrammatic contributions appearing in field theory for these functions and this theory, provided that exact relations between [Formula: see text] and [Formula: see text] hold. In addition, a nonperturbative approximation is considered. This approximation consists in a truncation of the ERG equations, which by means of a low-momentum expansion leads to reasonable results.

Interesting QFT Problems Tackled in New Fashion

The Dimensional Regularization technique of Bollini and Giambiagi (BG) [Phys. Lett. B 40, 566 (1972); Il Nuovo Cim. B 12, 20 (1972); Phys. Rev. D 53, 5761 (1996)] cannot be employed for all Schwartz Tempered Distributions Explicitly Lorentz Invariant (STDELI) S′L. We lifted such limitation in [J. Phys. Comm. 2 115029 (2018)], which opens new QFT possibilities, centering in the use of STDELI that allows one to obtain a product in a ring with zero divisors. This in turn, overcomes all problems regrading QFT infinities. We provide here three examples of the application of our STDELI-extension to quantum field theory (A) the exact evaluation of an electron’s self energy to one loop, (B) the exact evaluation of QED’s vacuum polarization, and C) the theory for six dimensions, that is non-renormalizable.

A Review of the Classical Canonical Ensemble Treatment of Newton's Gravitation.

It is common lore that the canonical gravitational partition function associated with the classical Boltzmann-Gibbs (BG) exponential distribution cannot be built up because of mathematical pitfalls. The integral needed for writing up diverges. We review here how to avoid this pitfall and obtain a (classical) statistical mechanics of Newton’s gravitation. This is done using (1) the analytical extension treatment obtained of Gradshteyn and Rizhik and (2) the well known dimensional regularization technique.

Zero-Temperature Equation of State of a Two-Dimensional Bosonic Quantum Fluid with Finite-Range Interaction

We derive the two-dimensional equation of state for a bosonic system of ultracold atoms interacting with a finite-range effective interaction. Within a functional integration approach, we employ a hydrodynamic parameterization of the bosonic field to calculate the superfluid equations of motion and the zero-temperature pressure. The ultraviolet divergences, naturally arising from the finite-range interaction, are regularized with an improved dimensional regularization technique.

Emergence of non-Fermi-liquid type Weyl metals driven by doped magnetic impurities in spin-orbit coupled semiconductors

Constructing an effective field theory in terms of doped magnetic impurities (described by an O(3) vector model with a random mass term), itinerant electrons of spin-orbit coupled semiconductors (given by a Dirac theory with a relatively large mass term), and effective interactions between doped magnetic ions and itinerant electrons (assumed by an effective Zeeman coupling term), we perform the perturbative renormalization group analysis in the one-loop level based on the dimensional regularization technique. As a result, we find that the mass renormalization in dynamics of itinerant electrons acquires negative feedback effects due to quantum fluctuations involved with the Zeeman coupling term, in contrast with that of the conventional problem of quantum electrodynamics, where such interaction effects enhance the fermion mass more rapidly. Recalling that the applied magnetic field decreases the band gap in the presence of spin-orbit coupling, this renormalization group analysis shows that the external magnetic field overcomes the renormalized band gap, allowed by doped magnetic impurities even without ferromagnetic ordering. In other words, the Weyl metal physics can be controlled by doping magnetic impurities into spin-orbit coupled semiconductors, even if the external magnetic field alone cannot realize the Weyl metal phase due to relatively large band gaps of semiconductors. Furthermore, we emphasize that quasiparticles do not exist in this emergent disordered Weyl metal phase due to correlations with strong magnetic fluctuations. This non-Fermi liquid type Weyl metal state may be regarded to be a novel metallic phase in the respect that a topologically nontrivial band structure appears in the vicinity of quantum criticality.

Spin-liquid Mott quantum criticality in two dimensions: Destabilization of a spinon Fermi surface and emergence of one-dimensional spin dynamics

Resorting to a recently developed theoretical device called dimensional regularization for quantum criticality with a Fermi surface, we examine a metal-insulator quantum phase transition from a Landau's Fermi-liquid state to a U(1) spin-liquid phase with a spinon Fermi surface in two dimensions. Unfortunately, we fail to approach the spin-liquid Mott quantum critical point from the U(1) spin-liquid state within the dimensional regularization technique. Self-interactions between charge fluctuations called holons are not screened, which shows a run-away renormalization group flow, interpreted as holons remain gapped. This leads us to consider another fixed point, where the spinon Fermi surface can be destabilized across the Mott transition. Based on this conjecture, we reveal the nature of the spin-liquid Mott quantum critical point: Dimensional reduction to one dimension occurs for spin dynamics described by spinons. As a result, Landau damping for both spin and charge dynamics disappear in the vicinity of the Mott quantum critical point. When the flavor number of holons is over its critical value, an interacting fixed point appears to be identified with an inverted XY universality class, controlled within the dimensional regularization technique. On the other hand, a fluctuation-driven first order metal-insulator transition results when it is below the critical number. We propose that the destabilization of a spinon Fermi surface and the emergence of one-dimensional spin dynamics near the spin-liquid Mott quantum critical point can be checked out by spin susceptibility with a $2 k_{F}$ transfer momentum, where $k_{F}$ is a Fermi momentum in the U(1) spin-liquid state: The absence of Landau damping in U(1) gauge fluctuations gives rise to a divergent behavior at zero temperature while it vanishes in the presence of a spinon Fermi surface.

Vacuum polarization and classical self-action near higher-dimensional defects

We analyze the gravity-induced effects associated with a massless scalar field in a higher-dimensional spacetime being the tensor product of (d-n)-dimensional Minkowski space and n-dimensional spherically/cylindrically symmetric space with a solid/planar angle deficit. These spacetimes are considered as simple models for a multidimensional global monopole (if ngeqslant 3) or cosmic string (if n=2) with (d-n-1) flat extra dimensions. Thus, we refer to them as conical backgrounds. In terms of the angular-deficit value, we derive the perturbative expression for the scalar Green function, valid for any dgeqslant 3 and 2leqslant nleqslant d-1, and compute it to the leading order. With the use of this Green function we compute the renormalized vacuum expectation value of the field square {langle phi ^{{}2}(x)rangle }_{mathrm{ren}} and the renormalized vacuum averaged of the scalar-field energy-momentum tensor {langle T_{M N}(x)rangle }_{mathrm{ren}} for arbitrary d and n from the interval mentioned above and arbitrary coupling constant to the curvature xi . In particular, we revisit the computation of the vacuum polarization effects for a non-minimally coupled massless scalar field in the spacetime of a straight cosmic string. The same Green function enables to consider the old purely classical problem of the gravity-induced self-action of a classical point-like scalar or electric charge, placed at rest at some fixed point of the space under consideration. To deal with divergences, which appear in consideration of the two problems, we apply the dimensional-regularization technique, widely used in quantum field theory. The explicit dependence of the results upon the dimensionalities of both the bulk and conical submanifold is discussed.

Superconducting instability in non-Fermi liquids

We use renormalization group (RG) analysis and dimensional regularization techniques to study potential superconductivity-inducing four-fermion interactions in systems with critical Fermi surfaces of general dimensions ($m$) and co-dimensions ($d-m$), arising as a result of quasiparticle interaction with a gapless Ising-nematic order parameter. These are examples of non-Fermi liquid states in $d$ spatial dimensions. Our formalism allows us to treat the corresponding zero-temperature low-energy effective theory in a controlled approximation close to the upper critical dimension $d=d_c(m)$. The fixed points are identified from the RG flow equations, as functions of $d$ and $m$. We find that the flow towards the non-Fermi liquid fixed point is preempted by Cooper pair formation for both the physical cases of $(d=3, m=2)$ and $(d=2, m=1)$. In fact, there is a strong enhancement of superconductivity by the order parameter fluctuations at the quantum critical point.

Dimensional Regularization of the Spatial wave function for a singular contact interaction in the Relativistic Schrodinger Equation

Based on our previous work in PRD 89, 125023 (2014), we stress here (for the first time) the regularization of the spatial wave function for the δ-contact interaction within the relativistic Schrodinger equation. The D-dimensional inverse Fourier transform has been utilized to map the momentum-space wave function to the spatial one. To regularize the logarithmic blow up of the wave function as x → 0, we employed the dimensional regularization technique. We assert that although the technique has been used here for the bound state only, the form of the scattering states in momentum space assures the reliability of the same technique to regularize the spatial scattering wave functions.

Adler–Bardeen theorem and manifest anomaly cancellation to all orders in gauge theories

We reconsider the Adler-Bardeen theorem for the cancellation of gauge anomalies to all orders, when they vanish at one loop. Using the Batalin-Vilkovisky formalism and combining the dimensional-regularization technique with the higher-derivative gauge invariant regularization, we prove the theorem in the most general perturbatively unitary renormalizable gauge theories coupled to matter in four dimensions, and identify the subtraction scheme where anomaly cancellation to all orders is manifest, namely no subtractions of finite local counterterms are required from two loops onwards. Our approach is based on an order-by-order analysis of renormalization, and, differently from most derivations existing in the literature, does not make use of arguments based on the properties of the renormalization group. As a consequence, the proof we give also applies to conformal field theories and finite theories.

Weighted power counting and chiral dimensional regularization

We define a modified dimensional-regularization technique that overcomes several difficulties of the ordinary technique, and is specially designed to work efficiently in chiral and parity violating quantum field theories, in arbitrary dimensions greater than 2. When the dimension of spacetime is continued to complex values, spinors, vectors and tensors keep the components they have in the physical dimension, therefore the $\gamma $ matrices are the standard ones. Propagators are regularized with the help of evanescent higher-derivative kinetic terms, which are of the Majorana type in the case of chiral fermions. If the new terms are organized in a clever way, weighted power counting provides an efficient control on the renormalization of the theory, and allows us to show that the resulting chiral dimensional regularization is consistent to all orders. The new technique considerably simplifies the proofs of properties that hold to all orders, and makes them suitable to be generalized to wider classes of models. Typical examples are the renormalizability of chiral gauge theories and the Adler-Bardeen theorem. The difficulty of explicit computations, on the other hand, may increase.

Riemann zeta zeros and zero-point energy

The sequence of nontrivial zeros of the Riemann zeta function is zeta regularizable. Therefore, systems with countably infinite number of degrees of freedom described by self-adjoint operators whose spectra is given by this sequence admit a functional integral formulation. We discuss the consequences of the existence of such self-adjoint operators in field theory framework. We assume that they act on a massive scalar field coupled to a background field in a (d+1)-dimensional flat space–time where the scalar field is confined to the interval [0, a] in one of its dimensions and there are no restrictions in the other dimensions. The renormalized zero-point energy of this system is presented using techniques of dimensional and analytic regularization. In even-dimensional space–time, the series that defines the regularized vacuum energy is finite. For the odd-dimensional case, to obtain a finite vacuum energy per unit area, we are forced to introduce mass counterterms. A Riemann mass appears, which is the correction to the mass of the field generated by the nontrivial zeros of the Riemann zeta function.

Three-loop field renormalization for scalar field theory with Lorentz violation

Applying the counterterm method in minimal subtraction scheme we calculate the three-loop quantum correction to field anomalous dimension in a Lorentz-violating O(N) self-interacting scalar field theory. We compute the Feynman diagrams using dimensional regularization and ϵ-expansion techniques. As this approximation corresponds to a three-loop term, to our knowledge this is the first time in literature in which such a loop level is attained for a LV theory.

Renormalization group equation, the naturalness problem, and the understanding of the Higgs mass term

The naturalness problem might be studied on the complex two dimensional plane with the technique of dimensional regularization(DREG). The Renormalization group equation(RGE) of the Higgs mass on the plane suggests the Higgs mass approaches zero at ultraviolate (UV) scale, the scale can be Planck scale when the top quark pole mass $M_{t}=168$ GeV. The real issue of the naturalness problem in the sense of Wilsonian renormalization group method is not about quadratic divergences but the rescaling effect. The Higgs mass can be considered to be one composed mass. All terms in the lagrangian in this scenario are marginal terms and no relevant terms are left, thus no rescaling effect to cause the naturalness problem anymore. RGE of the vacuum expectation value (VEV) in the Landau gauge up to two-loop order is studied. Scale-dependent behavior of the composed Higgs mass shows that we can have one tiny Higgs mass at high energy scale, even around the Planck scale, when $M_{t}\leq170.7$ GeV.

Regularization, renormalization, and dimensional analysis: Dimensional regularization meets freshman E&M

We illustrate the dimensional regularization technique using a simple example from electrostatics. This example illustrates the virtues of dimensional regularization without the complications of a full quantum field theory calculation. We contrast the dimensional regularization approach with the cutoff regularization approach, and demonstrate that dimensional regularization preserves translational symmetry. We then introduce minimal subtraction and modified minimal subtraction schemes to renormalize the result. Finally, we consider dimensional transmutation as encountered in the case of compact “extra dimensions.”

Brane world corrections to the scalar vacuum force in the Randall-Sundrum II-pscenario

Vacuum force is an interesting low energy test for brane worlds due to its dependence on field's modes and its role in submillimeter gravity experiments. In this work, we generalize a previous model example: the scalar field vacuum force between two parallel plates lying in the brane of a Randall-Sundrum scenario extended by $p$ compact dimensions (RSII-$p$). Upon use of Green's function technique, for the massless scalar field, the 4D force is obtained from a zero mode while leading order corrections due to the noncompact dimension turn out attractive and depend on the separation between plates as ${l}^{\ensuremath{-}(6+p)}$. For the massive scalar field, a quasilocalized mode yields the 4D force with attractive corrections behaving like ${l}^{\ensuremath{-}(10+p)}$. Corrections are negligible with respect to 4D force for anti-de Sitter (${\mathrm{AdS}}_{(5+p)}$) radius much less than $\ensuremath{\sim}{10}^{\ensuremath{-}6}\text{ }\text{ }\mathrm{m}$. In the massless case we also determined, numerically, the corrections due to compact dimensions. To avoid conflict with experimental data we get $R\ensuremath{\le}0.4\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$, $0.3\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$ for the cases $p=1$, 2, respectively. Although the $p=0$ case is not physically viable due to the different behavior in regard to localization for the massless scalar and electromagnetic fields it yields a useful comparison between the dimensional regularization and Green's function techniques as we describe in the discussion.

Dimensionally continued infinite reduction of couplings

The infinite reduction of couplings is a tool to consistently renormalize a wide class of non-renormalizable theories with a reduced, eventually finite, set of independent couplings, and classify the non-renormalizable interactions. Several properties of the reduction of couplings, both in renormalizable and non-renormalizable theories, can be better appreciated working at the regularized level, using the dimensional-regularization technique. We show that, when suitable invertibility conditions are fulfilled, the reduction follows uniquely from the requirement that both the bare and renormalized reduction relations be analytic in epsilon=D-d, where D and d are the physical and continued spacetime dimensions, respectively. In practice, physically independent interactions are distinguished by relatively non-integer powers of epsilon. We discuss the main physical and mathematical properties of this criterion for the reduction and compare it with other equivalent criteria. The leading-log approximation is solved explicitly and contains sufficient information for the existence and uniqueness of the reduction to all orders.

DEFORMED DIMENSIONAL REGULARIZATION FOR ODD (AND EVEN) DIMENSIONAL THEORIES

A deformation of the dimensional-regularization technique that is useful for theories where the common dimensional regularization does not apply is formulated. The Dirac algebra is not dimensionally continued, to avoid inconsistencies with the trace of an odd product of gamma matrices in odd dimensions. The regularization is completed with an evanescent higher-derivative deformation, which proves to be efficient in practical computations. This technique is particularly convenient in three dimensions for Chern–Simons gauge fields, two-component fermions and four-fermion models in the large N limit, eventually coupled with quantum gravity. Differently from even dimensions, in odd dimensions it is not always possible to have propagators with fully Lorentz invariant denominators. The main features of the deformed technique are illustrated in a set of sample calculations. The regularization is universal, local, manifestly gauge-invariant and Lorentz invariant in the physical sector of space–time. In flat space power-like divergences are set to zero by default. Infinitely many evanescent operators are automatically dropped.

Cutoff schemes in chiral perturbation theory and the quark mass expansion of the nucleon mass

We discuss the use of cutoff methods in chiral perturbation theory. We develop a cutoff scheme based on the operator structure of the effective field theory that allows to suppress high momentum contributions in Goldstone boson loop integrals and by construction is free of the problems traditional cutoff schemes have with gauge invariance or chiral symmetries. As an example, we discuss the chiral expansion of the nucleon mass. We show that the mass of a nucleon in heavy baryon chiral perturbation theory has a well behaved chiral expansion up to effective Goldstone boson masses of 400 MeV when one utilizes standard dimensional regularization techniques. With the help of the here developed cutoff scheme we can demonstrate a well-behaved chiral expansion for the nucleon mass up to 600 MeV of effective Goldstone boson masses. We also discuss in detail the price, in numbers of additional short distance operators involved, that has to be paid for this extended range of applicability of chiral perturbation theory with cutoff regularization. We also compare the fourth-order result for the chiral expansion of the nucleon mass with lattice results and draw some conclusions about chiral extrapolations based on such type of representation.