Vacuum Polarization Diagrams Research Articles

The muon pair production in the process $$e^{+}e^{-}\rightarrow e^{+}e^{-}\mu ^{+}\mu ^{-}$$ through the two-photon mechanism at high energies: transverse momentum, rapidity and angular distributions

In the present paper, we investigate the muon pair production in the interaction between two quasi-real photons in $$e^+e^-$$ collision. We calculate the total and differential cross section of the process $$\gamma \gamma \rightarrow \mu ^+\mu ^-$$ at a beam energy of photons from 3 to 40 GeV in the center-of-mass and for different values of the muon transverse momentum, the muon rapidity and the muon angle. In the $$\gamma \gamma \rightarrow \mu ^+\mu ^-$$ process we also take the contribution of the Box+Vertex correction+ Soft photon emission+Vacuum polarization diagrams. We also study the cross section of the $$e^+ +e^- \rightarrow e^+ + e^- +\mu ^+ + \mu ^-$$ process as a function of the $$e^+ e^-$$ center-of-mass energy $$\sqrt{s}$$ in the region 5 GeV $$\le \sqrt{s} \le $$ 209 GeV using the two-photon mechanism. The obtained results are in satisfactory agreement with experimental data.

Quaternionic analysis, representation theory and physics II

We develop further quaternionic analysis introducing left and right doubly regular functions. We derive Cauchy-Fueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula for the second order pole, in addition to the one studied in the first paper with the same title. We also realize the doubly regular functions as a subspace of the quaternionic-valued functions satisfying a Euclidean version of Maxwell's equations for the electromagnetic field. Then we return to the study of the original quaternionic analogue of Cauchy's second order pole formula and its relation to the polarization of vacuum. We find the decomposition of the space of quaternionic-valued functions into irreducible components that include the spaces of doubly left and right regular functions. Using this decomposition, we show that a regularization of the vacuum polarization diagram is achieved by subtracting the component corresponding to the one-dimensional subrepresentation of the conformal group. After the regularization, the vacuum polarization diagram is identified with a certain second order differential operator which yields a quaternionic version of Maxwell equations. Next, we introduce two types of quaternionic algebras consisting of spaces of scalar-valued and quaternionic-valued functions. We emphasize that these algebra structures are invariant under the action of the conformal Lie algebra. This uses techniques from our study of the vacuum polarization diagram. These algebras are not associative, but we can define an infinite family of n-multiplications, and we conjecture that they have structures of weak cyclic A-infinity algebras. We also conjecture the relation between the multiplication operations of the scalar and non-scalar quaternionic algebras with the n-photon Feynman diagrams in the scalar and ordinary conformal QED.

Hadronic vacuum polarization contribution to aμ from full lattice QCD

We determine the contribution to the anomalous magnetic moment of the muon from the $\alpha^2_{\mathrm{QED}}$ hadronic vacuum polarization diagram using full lattice QCD and including $u/d$ quarks with physical masses for the first time. We use gluon field configurations that include $u$, $d$, $s$ and $c$ quarks in the sea at multiple values of the lattice spacing, multiple $u/d$ masses and multiple volumes that allow us to include an analysis of finite-volume effects. We obtain a result for $a_{\mu}^{\mathrm{HVP,LO}}$ of $667(6)(12)$, where the first error is from the lattice calculation and the second includes systematic errors from missing QED and isospin-breaking effects and from quark-line disconnected diagrams. Our result implies a discrepancy between the experimental determination of $a_{\mu}$ and the Standard Model of 3$\sigma$.

Anti De Sitter Deformation of Quaternionic Analysis and the Second-Order Pole

This is a continuation of a series of papers [FL1, FL2, FL3], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper we continue to study the quaternionic analogues of Cauchy’s formula for the second order pole. These quaternionic analogues are closely related to regularization of infinities of vacuum polarization diagrams in four-dimensional quantum field theory. In order to add some flexibility, especially when dealing with Cauchy’s formula for the second order pole, we introduce a one-parameter deformation of quaternionic analysis. This deformation of quaternions preserves conformal invariance and has a geometric realization as anti de Sitter space sitting inside the five-dimensional Euclidean space. We show that many results of quaternionic analysis – including the Cauchy-Fueter formula – admit a simple and canonical deformation. We conclude this paper with a deformation of the quaternionic analogues of Cauchy’s formula for the second order pole.

Self-healing of unitarity in effective field theories and the onset of new physics

In effective field theories it is common to identify the onset of new physics with the violation of tree-level unitarity. However, we show that this is parametrically incorrect in the case of chiral perturbation theory, and is probably theoretically incorrect in general. In the chiral theory, we explore perturbative unitarity violation as a function of the number of colors and the number of flavors, holding the scale of the "new physics" (i.e. QCD) fixed. This demonstrates that the onset of new physics is parametrically uncorrelated with tree-unitarity violation. When the latter scale is lower than that of new physics, the effective theory must heal its unitarity violation itself, which is expected because the field theory satisfies the requirements of unitarity. In the chiral theory, the self-healing results in a resonant structure with scalar quantum numbers. In the electroweak variant of this argument, the structure must have the properties of the Higgs and must couple proportional to the mass in both gauge boson and fermion scattering. A similar example can be seen in the case of general relativity coupled to multiple matter fields, where iteration of the vacuum polarization diagram restores unitarity. We present arguments that suggest the correct identification should be connected to the onset of inelasticity rather than unitarity violation. We describe how the onset of inelasticity can occur in the effective theory, although it does not appear possible to predict the onset reliably.

Explicit derivation of the QED trace anomaly in symmetry-preserving loop regularization at one-loop level

The QED trace anomaly is calculated at one-loop level based on the loop regularization method which is realized in 4-dimensional space-time and preserves gauge symmetry and Poincar\'e symmetry in spite of the introduction of two mass scales, namely, the ultraviolet (UV) cutoff ${M}_{c}$ and infrared (IR) cutoff ${\ensuremath{\mu}}_{s}$. It is shown that the dilation Ward identity which relates the three-point diagrams with the vacuum polarization diagrams gets the standard form of trace anomaly through quantum corrections in taking the consistent limit ${M}_{c}\ensuremath{\rightarrow}\ensuremath{\infty}$ and ${\ensuremath{\mu}}_{s}=0$ which recovers the original integrals. This explicitly demonstrates that the loop regularization method is indeed a self-consistent regularization scheme which is applicable to the calculations not only for the chiral anomaly but also for the trace anomaly, at least at one-loop level. It is also seen that the consistency conditions which relate the tensor-type and scalar-type irreducible loop integrals (ILIs) are crucial for obtaining a consistent result. As a comparison, we also present the one-loop calculations by using the usual Pauli-Villars regularization and the dimensional regularization.

Tenth-order QED contribution to lepton anomalous magnetic moment: Fourth-order vertices containing sixth-order vacuum-polarization subdiagrams

This paper reports the tenth-order contributions to the $g\ensuremath{-}2$ of the electron ${a}_{e}$ and those of the muon ${a}_{\ensuremath{\mu}}$ from the gauge-invariant Set II(c), which consists of 36 Feynman diagrams, and Set II(d), which consists of 180 Feynman diagrams. Both sets are obtained by insertion of sixth-order vacuum-polarization diagrams in the fourth-order anomalous magnetic moment. The mass-independent contributions from Set II(c) and Set II(d) are $\ensuremath{-}0.116\text{ }489\text{ }(32)(\ensuremath{\alpha}/\ensuremath{\pi}{)}^{5}$ and $\ensuremath{-}0.243\text{ }00\text{ }(29)(\ensuremath{\alpha}/\ensuremath{\pi}{)}^{5}$, respectively. The leading contributions to ${a}_{\ensuremath{\mu}}$, which involve electron loops only, are $\ensuremath{-}3.888\text{ }27\text{ }(90)(\ensuremath{\alpha}/\ensuremath{\pi}{)}^{5}$ and $0.4972\text{ }(65)(\ensuremath{\alpha}/\ensuremath{\pi}{)}^{5}$ for Set II(c) and Set II(d), respectively. The total contributions of the electron, muon, and tau-lepton loops to ${a}_{e}$ are $\ensuremath{-}0.116\text{ }874\text{ }(32)(\ensuremath{\alpha}/\ensuremath{\pi}{)}^{5}$ for the Set II(c), and $\ensuremath{-}0.243\text{ }10\text{ }(29)(\ensuremath{\alpha}/\ensuremath{\pi}{)}^{5}$ for the Set II(d), respectively. The contributions of the electron, muon, and tau-lepton loop to ${a}_{\ensuremath{\mu}}$ are $\ensuremath{-}5.5594\text{ }(11)(\ensuremath{\alpha}/\ensuremath{\pi}{)}^{5}$ for the Set II(c) and $0.2465\text{ }(65)(\ensuremath{\alpha}/\ensuremath{\pi}{)}^{5}$ for the Set II(d), respectively.

Evaluation of the screened QED corrections to thegfactor and the hyperfine splitting of lithiumlike ions

The screened QED corrections of first order in $\ensuremath{\alpha}$ and $1/Z$ to the $g$ factor and the hyperfine splitting of lithiumlike ions are evaluated within ab initio quantum-electrodynamical approach. The complete gauge-invariant set of the two-electron self-energy diagrams in the presence of the magnetic field and a dominant part of the two-electron vacuum-polarization diagrams are calculated. The most accurate values of the $g$ factor of Li-like lead and uranium are presented. The theoretical prediction for the specific difference of the hyperfine splittings of H- and Li-like bismuth is improved.

Relativistic recoil, electron-correlation, and QED effects on the2pj-2stransition energies in Li-like ions

The relativistic nuclear recoil, higher-order interelectronic-interaction, and screened QED corrections to the transition energies in Li-like ions are evaluated. The calculation of the relativistic recoil effect is performed to all orders in $1/Z$. The interelectronic-interaction correction to the transition energies beyond the two-photon-exchange level is evaluated to all orders in $1/Z$ within the Breit approximation. The evaluation is carried out employing the large-scale configuration-interaction Dirac-Fock-Sturm method. The rigorous calculation of the complete gauge invariant sets of the screened self-energy and vacuum-polarization diagrams is performed utilizing a local screening potential as the zeroth-order approximation. The theoretical predictions for the $2{p}_{j}$-$2s$ transition energies are compiled and compared with available experimental data in the range of the nuclear charge number $Z=10$--60.

Higher derivative operators in the noncommutative Schwinger model

A study of the noncommutative Schwinger model is presented. It is shown that the Schwinger mass is not modified by the noncommutativity of spacetime till the first nontrivial order in the noncommutative parameter. Instead, a higher derivative kinetic term is dynamically generated by the lowest-order vacuum polarization diagrams. We argue that in the framework of the Seiberg–Witten map the feature of non-unitarity for a field theory with spacetime noncommutativity is characterized by the presence of higher derivative kinetic terms. The θ-expanded version of a unitary theory will not generate the lowest-order higher derivative quadratic terms.

Quantum stabilization of Z-strings, a status report on D = 3 + 1 dimensions

We investigate an extension to the phase shift formalism for calculating one-loop determinants. This extension is motivated by requirements of the computation of Z-string quantum energies in D = 3 + 1 dimensions. A subtlety that seems to imply that the vacuum polarization diagram in this formalism is (erroneously) finite is thoroughly investigated.

The light gluino mass window revisited

The precise measurements of the “electroweak observables” performed at LEP and SLC are well consistent with the standard model predictions. Deviations from the standard model arising from vacuum polarization diagrams (also called “weak loop corrections”) have been constrained in a model-independent manner with the ε formalism. Within the same formalism, additional deviations from new physics production processes can also be constrained, still in a model-independent way. For instance, a 95% C.L. limit of ΔΓhad<3.9 MeV is set on the partial width of any purely hadronic exotic contribution to Z decays. When applied to the e+e−→qq̄g̃g̃ process, it allows an absolute lower limit to be set on the gluino mass, mg̃>6.3 GeV/c2 at 95% C.L., which definitely closes the so-called light gluino mass window.

Renormalization of the two-photon vacuum polarization and the self energy vacuum polarization for a tightly bound electron

The renormalization method of Bogoljubov-Parasiuk-Hepp-Zimmermann (BPHZ) is used in order to derive the renormalized energy shift due to the gauge invariant Kallen-Sabry diagram of the two-photon vacuum polarization (VPVP) as well as the self energy vacuum polarization S(VP)E beyond the Uehling approximation. It is outlined, that no outer renormalization is required for the two-photon vacuum polarization and that only the inner renormalization has to be accomplished. It is shown that the so-called nongauge invariant spurious term is absent for a wide class of vacuum polarization (VP) diagrams if one applies the widely used spherical expansion of bound and free-electron propagator. This simplifies significantly calculations in bound state quantum electrodynamics. As one result of our paper the use of the BPHZ-approach in bound state QED is established.

Renormalization of the two-photon vacuum polarization and the self energy vacuum polarization for a tightly bound electron

The renormalization method of Bogoljubov-Parasiuk-Hepp-Zimmermann (BPHZ) is used in order to derive the renormalized energy shift due to the gauge invariant K\"all\'en-Sabry diagram of the two-photon vacuum polarization (VPVP) as well as the self energy vacuum polarization S(VP)E beyond the Uehling approximation. It is outlined, that no outer renormalization is required for the two-photon vacuum polarization and that only the inner renormalization has to b e accomplished. It is shown that the so-called nongauge invariant spurious term is absent for a wide class of vacuum polarization (VP) diagrams if one applies the widely used spherical expansion of bound and free-electron propagat or. This simplifies significantly calculations in bound state quantum electrodynamis . As one result of our paper the use of the BPHZ-approach in bound state QED is established.

Regularization of covariant calculations of meson decay amplitudes at one-loop order: Properties of thea0(980)resonance

The unitary quark model of T\"ornqvist and Roos provides an extremely interesting description of the properties of the scalar meson nonet. The model is quite phenomenological and has six parameters. In this work we are interested in seeing whether a theoretical foundation for the unitary quark model can be created using our generalized Nambu--Jona-Lasinio (NJL) model, which includes a covariant model of confinement. To make contact with T\"ornqvist's work, it is necessary to go beyond the leading vacuum polarization diagram of the NJL model, ${J(P}^{2}),$ which is of order ${n}_{c},$ and calculate the (complex) function ${K(P}^{2}),$ which is of order 1, and which describes the decay of $q\overline{q}$ states to the continuum of two-meson states. The central issue is the creation of a model for the regularization of the one-loop amplitudes with three vertices. We choose a regulator that has been used in our recent study of ${f}_{0}$ mesons. The same regularization is used for the \ensuremath{\pi}\ensuremath{\eta}, $K\overline{K},$ and $\ensuremath{\pi}{\ensuremath{\eta}}^{\ensuremath{'}}$ channels, when studying the properties of the ${a}_{0}(980)$ resonance. The coupling to the decay channels described by ${K(P}^{2})$ is an important feature of the model, since the mass of the ${a}_{0}$ mass would be 1090 MeV in the absence of such coupling. The peak width obtained for the ${a}_{0}(980)$ resonance is 23 MeV, which is smaller than the values of 50--100 MeV usually quoted. (Larger values may be obtained in our calculation with another form of the regulator.) While our model is not parameter free, we believe we have made significant progress toward putting the unitary quark model on a firmer foundation.

Accuracy of calculations involvingα3vacuum-polarization diagrams: Muonic hydrogen Lamb shift and muong−2

The contribution of the ${\ensuremath{\alpha}}^{3}$ single electron-loop vacuum-polarization diagrams to the Lamb shift of muonic hydrogen has been evaluated recently by two independent methods. One uses the exact parametric representation of the vacuum-polarization function while the other relies on the Pad\'e approximation method. The high precision of these values offers an opportunity to examine the reliability of the Monte Carlo integration as well as that of the Pad\'e method. Our examination covers both the muonic hydrogen atom and muon $g\ensuremath{-}2.$ We test them further for the cases involving two-loop vacuum polarization, where an exact analytic result is known. Our analysis justifies the result for the Lamb shift of muonic hydrogen and also resolves the long-standing discrepancy between two previous evaluations of the muon $g\ensuremath{-}2$ value.

Covariant confinement model for the calculation of radial excitations of the pion

We describe the mixing of $q\overline{q}$ pseudoscalar states with longitudinal $q\overline{q}$ axial-vector states, making use of a relativistic quark model that includes a model of confinement. (In the absence of the confinement model, our model reduces to the Nambu--Jona-Lasinio model.) In addition to the pion, we find ${J}^{P}{=0}^{\ensuremath{-}}$ states at 1.18, 1.36, 1.47, 1.63, and 1.68 GeV. The first two of these states are in the region of the \ensuremath{\pi}(1300) that is assigned a mass of $1300\ifmmode\pm\else\textpm\fi{}100\mathrm{MeV}$ and a width of 200--600 MeV in the data tables. We provide values of the coupled-channel $q\overline{q}$ T matrix, as well as the mixing angle, which is energy-dependent in our analysis. In addition, we describe a model of confinement for longitudinal axial-vector $q\overline{q}$ states that is used in the calculation of vacuum polarization diagrams. (That analysis supplements our previous study of confinement in the case of pseudoscalar mesons.) We show that our confinement model may be made covariant. We use the covariant model to calculate the decay of the various states, ${\ensuremath{\pi}}^{\ensuremath{'}},$ to the $\ensuremath{\pi}+\ensuremath{\rho}$ and $\ensuremath{\pi}+\ensuremath{\sigma}$ channels at one-loop order. At one-loop order, it is found that only the nodeless state at 1.18 GeV and the state at 1.36 GeV have significant widths for ${\ensuremath{\pi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\sigma}.$ These states have somewhat larger widths for the decay ${\ensuremath{\pi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\rho},$ leading to ${\ensuremath{\Gamma}}_{\mathrm{tot}}=0.368\mathrm{GeV}$ for the state at 1.18 GeV and 0.150 GeV for the state at 1.36 GeV. We note that the state 1.18 GeV is a mixed pseudoscalar--axial-vector state, while the state at 1.36 GeV is the $\ensuremath{\pi}(2S)$ state to a good approximation, since it has a very small admixture of axial-vector components. There is information concerning the decay ${\ensuremath{\pi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\pi}+(\ensuremath{\pi}+\ensuremath{\pi}{)}_{L=0}$ that is extracted from experimental data for three-body final states. Our (nodeless) state at 1.18 GeV has the correct energy and width to fit that data. However, our widths for ${\ensuremath{\pi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\pi}+(\ensuremath{\pi}+\ensuremath{\pi}{)}_{L=1}$ are larger than those for ${\ensuremath{\pi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\pi}+(\ensuremath{\pi}+\ensuremath{\pi}{)}_{L=0}.$ That suggests that final-state interactions are probably quite important in understanding the branching ratios for ${\ensuremath{\pi}}^{\ensuremath{'}}$ decays to states of three pions. Our results also suggest that, if we were to study the \ensuremath{\pi}(1300), and include final-state interactions, it is necessary to include both the 1.18 GeV and the 1.36 GeV states in the analysis. (On the other hand, since the 1.36 GeV state is a $2S$ state, it may be only weakly excited in the reactions used to generate final states of three pions.)

Gauge invariance and confinement in a generalized Nambu-Jona-Lasinio model.

We discuss the quark-antiquark loop diagrams of an extended version of the Nambu{endash}Jona-Lasinio (NJL) model that includes a description of confinement. We show how the constraints of gauge invariance affect the calculations of such diagrams, if the {ital q{bar q}} system has the quantum numbers of the rho or omega mesons. In the NJL model without confinement, gauge invariance may be maintained by using a regularization scheme that respects that symmetry. Alternatively, gauge-invariance constraints may be imposed by introducing a substraction procedure when evaluating the {ital q{bar q}} loop integrals (vacuum polarization diagrams). We discuss the merits of such subtraction procedures in the presence of a confining interaction. We also provide a calculation of the vacuum polarization diagrams of the NJL model that includes the effects of confinement and also satisfies the constraints imposed by gauge invariance. Our analysis provides some justification for the subtraction procedure used in several studies of the NJL model. We also show, in a rather clear fashion, how confinement serves to change the analytic properties of the polarization tensors, so that the tensor for the theory with confinement has only a physical cut structure in the region of interest. {copyright} {ital 1996 The American Physicalmore » Society.}« less

Evaluating loop diagrams in a cavity (II). The vacuum polarization in scalar QED

In a previous paper (part (I)) we have introduced a new numerical technique to evaluate divergent loop diagrams in renormalizable cavity field theories. Subsequently, we have applied this method to the linearly divergent quark self-energy of QCD in a spherical cavity. In this paper (part (II)) it is shown that the cavity regularization technique also works for the quadratically divergent photon self-energy of scalar QED in a spherical cavity. The success of this calculation depends critically on the non-trivial cancellation of non-renormalizable divergences contained in the reflection parts of the vacuum polarization diagrams.

The analytic four-loop corrections to the QED β-function in the MS scheme and to the QED ψ-function. Total reevaluation

The results of the total reevaluation of the four-loop correction to the QED β-function in the minimal and momentum subtraction schemes (β MS and ψ-functions) and for the functions F 1( α) and F( α) determined by the photon vacuum polarization diagrams without fermion-loop contributions to the internal photon lines are presented. It is found that the ζ(3)-, ζ(4)- and ζ(5)-terms cancel in the ultimate result for the four-loop coefficient of the F 1-function. The results obtained are briefly discussed.