Vacuum Polarization Tensor Research Articles

Smoothed asymptotics: From number theory to QFT

Inspired by the method of smoothed asymptotics developed by Terence Tao, we introduce a new ultraviolet regularization scheme for loop integrals in quantum field theory which we call η regularization. This reveals a connection between the elimination of divergences in divergent series of powers and the preservation of gauge invariance in the regularization of loop integrals in quantum field theory. In particular, we note that a method for regularizing the series of natural numbers so that it converges to minus one-twelfth inspires a regularization scheme for non-Abelian gauge theories coupled to Dirac fermions that preserves the Ward identity for the vacuum polarization tensor and other higher point functions. We also comment on a possible connection to Schwinger proper time integrals. Published by the American Physical Society 2024

Lorentz invariance violation and the CPT -odd electromagnetic response of a tilted anisotropic Weyl semimetal

We derive the electromagnetic response of a particular fermionic sector in the minimal QED contribution to the Standard Model Extension (SME), which can be physically realized in terms of a model describing a tilted and anisotropic Weyl semimetal (WSM). The contact is made through the identification of the Dirac-like Hamiltonian resulting from the SME with that corresponding to the WSM in the linearized tight-binding approximation. We first calculate the effective action by computing the nonperturbative vacuum polarization tensor using thermal field theory techniques, focusing upon the corrections at finite chemical potential and zero temperature. Next, we confirm our results by a direct calculation of the anomalous Hall current within a chiral kinetic theory approach. In an ideal Dirac cone picture of the WSM (isotropic and nontilted) such response is known to be governed by axion electrodynamics, with the space-time dependent axion angle Θ(r,t)=2(b·r−b0t), being 2b and 2b0 the separation of the Weyl nodes in momentum and energy, respectively. In this paper we demonstrate that the node tilting and the anisotropies induce novel corrections at a finite density which however preserve the structure of the axionic field theory. We apply our results to the ideal Weyl semimetal EuCd2As2 and to the highly anisotropic and tilted monopnictide TaAs. Published by the American Physical Society 2024

Erratum — The full Lorentz-violating vacuum polarization tensor: Low- and high-energy limits

Erratum — The full Lorentz-violating vacuum polarization tensor: Low- and high-energy limits

The full Lorentz-violating vacuum polarization tensor: Low- and high-energy limits

In this paper, we compute the full vacuum polarization tensor in the fermion sector of Lorentz-violating quantum electrodynamics (QED). It turns out to be that even if we assume momentum routing invariance of the Feynman diagrams, it is not possible to fix all surface terms and find an ambiguity-free vacuum polarization tensor. The high- and low-energy limits of this tensor are obtained explicitly. In the high-energy limit, only [Formula: see text] coefficients contribute to the result. In the low-energy limit, we find that Lorentz-violating-induced terms depend on [Formula: see text], [Formula: see text] and [Formula: see text] coefficients and vanish at [Formula: see text]. At small [Formula: see text], we succeeded to obtain implications for condensed matter systems, explicitly, for the Hall effect in Weyl semi-metals.

Coupling sum rules and oblique corrections in gauge-Higgs unification

Abstract In grand-unified-theory-inspired SO(5) × U(1)X × SU(3)C gauge-Higgs unification (GHU) in the Randall–Sundrum warped spacetime, the W- and Z-couplings of all 4D fermion modes become nontrivial. The W- and Z-couplings of zero-mode quarks and leptons slightly deviate from those in the SM, and the couplings take the matrix form in the space of Kaluza–Klein (KK) states. In particular, the 4D couplings and mass spectra in the KK states depend on the Aharonov–Bohm phase θH in the fifth dimension. Nevertheless there emerge three astonishing sum rules among those coupling matrices, which guarantees the finiteness of certain combinations of corrections to vacuum polarization tensors. We confirm by numerical evaluation that the equality in the sum rules holds with 5- to 7-digit accuracy. Based on the sum rules we propose improved oblique parameters in GHU. Oblique corrections due to fermion 1-loop diagrams are found to be small.

QED as a many-body theory of worldlines. II. All-order S -matrix formalism

In arXiv:2206.04188, we developed a first-quantized worldline formalism for all-order computations of amplitudes in QED. In particular, we demonstrated in this framework an all-order proof of the infrared safety of the Faddeev-Kulish (FK) S-matrix for virtual exchanges in the scattering of charged fermions. In this work, we extend the worldline formalism for both the Dyson and FK S-matrix to consider further the emission and absorption of arbitrary numbers of photons. We show how Low's theorem follows in this framework and derive Weinberg's theorem for the exponentiation of IR divergences. In particular, we extend our all-order proof of the IR safety of the FK S-matrix to both virtual exchanges and real photon emissions. We argue that the worldline approach leads to a modern Wilsonian interpretation of the IR safety of the FK S-matrix and provides a novel template for the treatment of IR divergences in real-time problems. Using Grassmannian integration methods, we derive a simple and powerful result for N-th rank vacuum polarization tensors. Applications of these methods will be discussed in follow-up work.

Effective electromagnetic actions for Lorentz violating theories exhibiting the axial anomaly

The CPT odd contribution to the effective electromagnetic action deriving from the vacuum polarization tensor in a large class of fermionic systems exhibiting Lorentz invariance violation (LIV) is calculated using thermal field theory methods, focusing upon corrections depending on the chemical potential. The systems considered exhibit the axial anomaly and their effective actions are described by axion electrodynamics whereby all the LIV parameters enter in the coupling Θ(x) to the unmodified Pontryagin density. A preliminary application to type-I tilted Weyl semimetals is briefly presented.

Chiral anomaly, induced current, and vacuum polarization tensor for a Dirac field in the presence of a defect

We evaluate the vacuum polarization tensor (VPT) for a massless Dirac field in 1+1 and 3+1 dimensions, in the presence of a particular kind of defect, which in a special limit imposes bag boundary conditions.We also show that the chiral anomaly in the presence of such a defect is the same as when no defects are present, both in 1+1 and 3+1 dimensions. This implies that the induced vacuum current in 1+1 dimensions due to the lowest order VPT is exact.

Derivative corrections to the Heisenberg-Euler effective action

We show that the leading derivative corrections to the Heisenberg-Euler effective action can be determined efficiently from the vacuum polarization tensor evaluated in a homogeneous constant background field. After deriving the explicit parameter-integral representation for the leading derivative corrections in generic electromagnetic fields at one loop, we specialize to the cases of magnetic- and electric-like field configurations characterized by the vanishing of one of the secular invariants of the electromagnetic field. In these cases, closed-form results and the associated all-orders weak- and strong-field expansions can be worked out. One immediate application is the leading derivative correction to the renowned Schwinger-formula describing the decay of the quantum vacuum via electron-positron pair production in slowly-varying electric fields.

On some short-distance properties of the fourth-rank hadronic vacuum polarization tensor and the anomalous magnetic moment of the muon

Some short-distance properties of the fourth-rank hadronic vacuum polarization tensor are re-examined. Their consequences are critically discussed in the context of the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon.

Noncommutative Momentum and Torsional Regularization

We show that in the presence of the torsion tensor $S^k_{\phantom{k}ij}$, the quantum commutation relation for the four-momentum, traced over spinor indices, is given by $[p_i,p_j]=2i\hbar S^k_{\phantom{k}ij}p_k$. In the Einstein--Cartan theory of gravity, in which torsion is coupled to spin of fermions, this relation in a coordinate frame reduces to a commutation relation of noncommutative momentum space, $[p_i,p_j]=i\epsilon_{ijk}Up^3 p_k$, where $U$ is a constant on the order of the squared inverse of the Planck mass. We propose that this relation replaces the integration in the momentum space in Feynman diagrams with the summation over the discrete momentum eigenvalues. We derive a prescription for this summation that agrees with convergent integrals: \[ \int\frac{d^4p}{(p^2+\Delta)^s}\rightarrow 4\pi U^{s-2}\sum_{l=1}^\infty \int_0^{\pi/2} d\phi \frac{\sin^4\phi\,n^{s-3}}{[\sin\phi+U\Delta n]^s}, \] where $n=\sqrt{l(l+1)}$ and $\Delta$ does not depend on $p$. We show that this prescription regularizes ultraviolet-divergent integrals in loop diagrams. We extend this prescription to tensor integrals. We derive a finite, gauge-invariant vacuum polarization tensor and a finite running coupling. Including loops from all charged fermions, we find a finite value for the bare electric charge of an electron: $\approx -1.22\,e$. This torsional regularization may therefore provide a realistic, physical mechanism for eliminating infinities in quantum field theory and making renormalization finite.

Planar phenomena in CPT -even chiral QED with Lorentz violation

Quantum gravity models as string theory, loop quantum gravity, and noncommutative field theories allow for the violation of Lorentz symmetry in the Planck scale. In order to examine the implications of such violation in planar models, we present here a $CPT$-even effective model, in the Standard Model extension framework, for Maxwell-Chern-Simons quantum electrodynamics $({\mathrm{QED}}_{3})$, with a Lorentz violation tensor parameter coupled to the photon field. The magnetoelectric effect induced by the space-time anisotropy in the model is discussed. We then derive the exact classical photon propagator. The electrostatic interaction is investigated and the long-range classical potential, corrected to second order in the Lorentz-violating tensor parameter, is obtained as well as its radiative correction in the isotropic violation scenario, derived at one-loop approximation from the vacuum polarization tensor. Also, the contribution of the space-time anisotropy to the magnetic moment of the fermion trough the vertex diagram is evaluated on shell and in the forward scattering approximation, in the absence of the topological Chern-Simons term.

Scalar quantum electrodynamics via Duffin–Kemmer–Petiau gauge theory in the Heisenberg picture: Vacuum polarization

Scalar Quantum Electrodynamics is investigated in the Heisenberg picture via the Duffin-Kemmer-Petiau gauge theory. On this framework, a perturbative method is used to compute the vacuum polarization tensor and its corresponding induced current for the case of a charged scalar field in the presence of an external electromagnetic field. Charge renormalization is brought into discussion for the interpretation of the results for the vacuum polarization.

Dilepton production rate in a hot and magnetized quark–gluon plasma

The differential multiplicity of dileptons in a hot and magnetized quark–gluon plasma, ΔB≡dNB/d4xd4q, is derived from first principles. The constant magnetic field B is assumed to be aligned in a fixed spatial direction. It is shown that the anisotropy induced by the B field is mainly reflected in the general structure of photon spectral density function. This is related to the imaginary part of the vacuum polarization tensor, Im[Πμν], which is derived in a first order perturbative approximation. As expected, the final analytical expression for ΔB includes a trace over the product of a photonic part, Im[Πμν], and a leptonic part, Lμν. It is shown that ΔB consists of two parts, ΔB∥ and ΔB⊥, arising from the components (μ,ν)=(∥,∥) and (μ,ν)=(⊥,⊥) of Im[Πμν] and Lμν. Here, the transverse and longitudinal directions are defined with respect to the direction of the B field. Combining ΔB∥ and ΔB⊥, a novel anisotropy factor νB is introduced. Using the final analytical expression of ΔB, the possible interplay between the temperature T and the magnetic field strength eB on the ratio ΔB/Δ0 and νB is numerically studied. Here, Δ0 is the Born approximated dilepton multiplicity in the absence of external magnetic fields. It is, in particular, shown that for each fixed T and B, in the vicinity of certain threshold energies of virtual photons, ΔB≫Δ0 and ΔB⊥≫ΔB∥. The latter anisotropy may be interpreted as one of the microscopic sources of the macroscopic anisotropies, reflecting themselves, e.g., in the elliptic asymmetry factor v2 of dileptons.

Radiatively induced CPT-odd Chern-Simons term in massless QED

In this work, we study the radiative generation of the CPT-odd Lorentz-violating Chern-Simons term, arising from massless fermions. For this, we calculate the vacuum polarization tensor using the ’t Hooft-Veltman regularization scheme, in which the result obtained for the coefficient of the Chern-Simons term is . This result leads us precisely to the same conductivity found in Weyl semimetals, i.e., the ’t Hooft-Veltman regularization scheme is the correct one to be used in this context. We also discuss the temperature dependence of , in which at high temperature, and . In the context of Weyl semimetals, these results are in accordance with the fact that the chiral magnetic current vanishes at high temperature, whereas the anomalous Hall current remains unaffected by the finite temperature.

Interaction Induced Quantum Valley Hall Effect in Graphene

We use pseudo-quantum electrodynamics in order to describe the full electromagnetic interaction of the p electrons in graphene in a consistent 2D formulation. We first consider the effect of this interaction in the vacuum polarization tensor or, equivalently, in the current correlator. This allows us to obtain the T→0 conductivity after a smooth zero-frequency limit is taken in Kubo’s formula. Thereby, we obtain the usual expression for the minimal conductivity plus corrections due to the interaction that bring it closer to the experimental value. We then predict the onset of an interaction-driven spontaneous quantum valley Hall effect below an activation temperature of the order of 2 K. The transverse (Hall) valley conductivity is evaluated exactly and shown to coincide with the one in the usual quantum Hall effect. Finally, by considering the effects of pseudo-quantum electrodynamics, we show that the electron self-energy is such that a set of P- and T-symmetric gapped electron energy eigenstates are dynamically generated, in association with the quantum valley Hall effect.Received 8 October 2013DOI:https://doi.org/10.1103/PhysRevX.5.011040This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical Society

The hadronic vacuum polarization and automatic O a $$ \mathcal{O}(a) $$ improvement for twisted mass fermions

The vacuum polarization tensor and the corresponding vacuum polarization function are the basis for calculations of numerous observables in lattice QCD. Examples are the hadronic contributions to lepton anomalous magnetic moments, the running of the electroweak and strong couplings and quark masses. Quantities which are derived from the vacuum polarization tensor often involve a summation of current correlators over all distances in position space leading thus to the appearance of short-distance terms. The mechanism of O(a) improvement in the presence of such short-distance terms is not directly covered by the usual arguments of on-shell improvement of the action and the operators for a given quantity. If such short-distance contributions appear, the property of O(a) improvement needs to be reconsidered. We discuss the effects of these short-distance terms on the vacuum polarization function for twisted mass lattice QCD and find that even in the presence of such terms automatic O(a) improvement is retained if the theory is tuned to maximal twist.

Graphene transparency in weak magnetic fields

We carry out an explicit calculation of the vacuum polarization tensor for an effective low-energy model of monolayer graphene in the presence of a weak magnetic field of intensity B perpendicularly aligned to the membrane. By expanding the quasiparticle propagator in the Schwinger proper time representation up to order (eB)2, where e is the unit charge, we find an explicitly transverse tensor, consistent with gauge invariance. Furthermore, assuming that graphene is irradiated with monochromatic light of frequency ω along the external field direction, from the modified Maxwell equations we derive the intensity of transmitted light. Corrections to this quantity, both calculated and measured, are of the order of (eB)2/ω4. Our findings generalize and complement previously known results reported in the literature regarding the light absorption problem in graphene from the experimental and theoretical points of view, with and without external magnetic fields.

The Epstein–Glaser causal approach to the light-front QED4. II: Vacuum polarization tensor

In this work we show how to construct the one-loop vacuum polarization for light-front QED4 in the framework of the perturbative causal theory. Usually, in the canonical approach, it is considered for the fermionic propagator the so-called instantaneous term, but it is known in the literature that this term is controversial because it can be omitted by computational reasons; for instance, by compensation or vanishing by dimensional regularization. In this work we propose a solution to this paradox. First, in the Epstein–Glaser causal theory, it is shown that the fermionic propagator does not have instantaneous term, and with this propagator we calculate the one-loop vacuum polarization, from this calculation it follows the same result as those obtained by the standard approach, but without reclaiming any extra assumptions. Moreover, since the perturbative causal theory is defined in the distributional framework, we can also show the reason behind our obtaining the same result whether we consider or not the instantaneous fermionic propagator term.

Consistency and universality in odd and even dimensional space–time QFT perturbative calculations

The questions related to the consistent interpretation of QFT perturbative amplitudes are considered in light of a novel procedure, alternative to the traditional ones based on regularization prescriptions. A detailed discussion about the aspects associated to the space–time dimension is performed. For this purpose, it is considered a simple model having a fermionic vector current, coupled to a vector field, as well as a fermionic scalar current, coupled to a scalar field, both of them composed by different species of massive fermions. The referred currents are related in a precise way, which is reflected in the Ward identities for the perturbative physical amplitudes. The double vector two-point fermionic function, related to the vacuum polarization tensor of QED, as well as the amplitudes related to such quantity through relations among Green functions are explicit evaluated in space–time dimensions d = 2, 3, 4, 5 and 6. In the adopted procedure the perturbative amplitudes are not modified in intermediary steps of the calculations, as occurs in regularization procedures. Divergent Feynman integrals are not really solved. They appear only in standard objects, conveniently defined, where no physical parameter is present. Only very general properties for such quantities are assumed. For the finite parts, a set of functions is introduced which allows universal forms for the results. We show that scale independent, ambiguity free amplitudes are automatically obtained in a regularization independent way. As a consequence, interesting and, in certain way, surprising aspects are revealed in a clear and transparent way when the Ward identities and low-energy limits are verified for the simple amplitudes considered in the presently reported investigation. The obtained results suggest that the procedure can be considered as an advantageous tool to handle with the problem of divergences in perturbative solutions of QFT's, relative to the traditional regularization techniques, since the obtained results are so consistent as desirable and there are no limitations of applicability. In particular, the method can be applied in odd and even space–time dimensions having extra dimensions, which is not possible within the context of traditional regularization.