Wave Function Renormalization Constant Research Articles

All-order renormalization of electric charge in the Standard Model and beyond

Electric charge, as defined in the Thomson limit of the electron–photon interaction vertex, is renormalized to all orders both in the Standard Model and in any spontaneously broken gauge theory with gauge group G\timesG×U(1) with a group factor U(1) that mixes with electromagnetic gauge symmetry. In the framework of the background-field method the charge renormalization constant Z_eZe is directly obtained from the photon wave-function renormalization constant, similar to the situation in QED, which proves charge universality as a byproduct. Exploiting charge universality in arbitrary R_\xiRξ gauge by formulating the charge renormalization condition for a ``fake fermion’’ that couples only via an infinitesimal electric charge, Z_eZe can be expressed in terms of renormalization constants that are obtained solely from gauge-boson self-energies.

Exact results for {Z}_m^{mathrm{OS}} and {Z}_2^{mathrm{OS}} with two mass scales and up to three loops

We consider the on-shell mass and wave function renormalization constants {Z}_m^{mathrm{OS}} and {Z}_2^{mathrm{OS}} up to three-loop order allowing for a second non-zero quark mass. We obtain analytic results in terms of harmonic polylogarithms and iterated integrals with the additional letters sqrt{1-{tau}^2} and sqrt{1-{tau}^2}/tau which extends the findings from ref. [1] where only numerical expressions are presented. Furthermore, we provide terms of order mathcal{O} (ϵ2) and mathcal{O} (ϵ) at two- and three-loop order which are crucial ingredients for a future four-loop calculation. Compact results for the expansions around the zero-mass, equal-mass and large-mass cases allow for a fast high-precision numerical evaluation.

High-precision four-loop mass and wave function renormalization in QED

The 4-loop QED mass and wave function renormalization constants Z2 and Zm have been evaluated in the on-shell subtraction scheme with 1100 digits of precision. We also worked out the coefficients of the five color structures of the QCD renormalization constants Z2OS and ZmOS which can be obtained from QED-like diagrams. The results agree with lower precision results available in the literature. Analytical fits were also obtained for all these quantities.

BRST in the exact renormalization group

AbstractWe show, explicitly within perturbation theory, that the quantum master equation and the Wilsonian renormalization group flow equation can be combined such that for the continuum effective action, quantum BRST invariance is not broken by the presence of an effective ultraviolet cutoff $\Lambda$, despite the fact that the structure demands quantum corrections that naïvely break the gauge invariance, such as a mass term for a non-Abelian gauge field. Exploiting the derivative expansion, BRST cohomological methods fix the solution up to choice of renormalization conditions, without inputting the form of the classical, or bare, interactions. Legendre transformation results in an equivalent description in terms of solving the modified Slavnov–Taylor identities and the flow of the Legendre effective action under an infrared cutoff $\Lambda$ (i.e. effective average action). The flow generates a canonical transformation that automatically solves the Slavnov–Taylor identities for the wavefunction renormalization constants. We confirm this structure in detail at tree level and one loop. Under flow of $\Lambda$, the standard results are obtained for the beta function, anomalous dimension, and physical amplitudes, up to the choice of the renormalization scheme.

Renormalization of the bilocal sine-Gordon model

The functional renormalization group treatment is presented for the two-dimensional sine-Gordon model including a bilocal term in the potential, which contributes to the flow at the tree level. It is shown that the flow of the bilocal term can substitute the evolution of the wave function renormalization constant, since it can recover the Kosterlitz–Thouless type phase transition. The flows can also reveal the connection between the sine-Gordon and the noninteracting Thirring models at a special value of the wave number parameter.

Addendum to “Critical behavior of ( 2+1 )-dimensional QED: 1/Nf -corrections in an arbitrary nonlocal gauge”

Dynamical chiral symmetry breaking (D$\chi$SB) is studied within ($2+1$)-dimensional QED with $N$ four-component fermions. The leading and next-to-leading orders of the $N$ expansion were computed exactly in Refs.~[\onlinecite{Gusynin:2016som,Kotikov:2016prf}] in an arbitrary non-local gauge. In this addendum to [\onlinecite{Kotikov:2016prf}], we show that the resummation of the wave-function renormalization constant at the level of the gap equation yields a {\it complete} cancellation of the gauge dependence of the critical fermion flavour number resulting in: $N_c= 2.8469$, which is such that D$\chi$SB takes place for $N<N_c$. The result is in full agreement with one of Ref.~[\onlinecite{Gusynin:2016som}].

Critical behavior of (2+1)-dimensional QED:1/Nfcorrections in an arbitrary nonlocal gauge

Dynamical chiral symmetry breaking (D$\chi$SB) is studied within ($2+1$)-dimensional QED with $N$ four-component fermions. The leading and next-to-leading orders of the $1/N$ expansion are computed exactly. The analysis is carried out in an arbitrary non-local gauge. Resumming the wave-function renormalization constant at the level of the gap equation yields a strong suppression of the gauge dependence of the critical fermion flavour number, $N_c(\xi)$ where $\xi$ is the gauge fixing parameter, which is such that D$\chi$SB takes place for $N<N_c(\xi)$. Neglecting the weak gauge-dependent terms yields $N_c = 2.8469$ while, in the general case, it is found that: $N_c(1) = 3.0084$ in the Feynman gauge, $N_c(0) = 3.0844$ in the Landau gauge and $N_c(2/3)= 3.0377$ in the $\xi=2/3$ gauge where the leading order fermion wave function is finite. These results suggest that D$\chi$SB should take place for integer values $N \leq 3$.

Electron contribution to the muon anomalous magnetic moment at four loops

We present results for the QED contributions to the anomalous magnetic moment of the muon containing closed electron loops. The main focus is on perturbative corrections at four-loop order where the external photon couples to the external muon. Furthermore, all four-loop contributions involving simultaneously a closed electron and tau loop are computed. In combination with our recent results on the light-by-light-type corrections (see Ref. \cite{Kurz:2015bia}) the complete four-loop electron-loop contribution to the anomalous magnetic moment of the muon has been obtained with an independent calculation. Our calculation is based on an asymptotic expansion in the ratio of the electron and the muon mass and shows the importance of higher order terms in this ratio. We perform a detailed comparison with results available in the literature and find good numerical agreement. As a by-product we present analytic results for the on-shell muon mass and wave function renormalization constants at three-loop order including massive closed electron and tau loops, which we also calculated using the method of asymptotic expansion.

On the meson mass spectrum in the covariant confined quark model

We provide a new insight into the problem of generating the hadron mass spectrum in the framework of the covariant confined quark model. One of the underlying principles of this model is the compositeness condition, which means that the wave function renormalization constant of the elementary hadron is equal to zero. In particular, this equation allows us to express the Yukawa coupling of the meson fields to the constituent quarks as a function of other model parameters. In addition to the compositeness condition, we also employ a further equation relating the meson mass function to the Fermi coupling. Both equations guarantee that the Yukawa-type theory is equivalent to the Fermi-type theory, thereby providing an interpretation of the meson field as the bound state of its constituent fermions (quarks). We evaluate the Fermi-coupling as a function of meson (pseudoscalar and vector) masses and vary the values of the masses in such a way to obtain a smooth behavior for the resulting curve. The mass spectrum obtained in this manner is found to be in good agreement with the experimental data.

Anatomy of relativistic pion loop corrections to the electromagnetic nucleon coupling

We present a relativistic formulation of pion loop corrections to the coupling of photons with nucleons on the light front. Vertex and wave function renormalization constants are computed to lowest order in the pion field, including their nonanalytic behavior in the chiral limit, and studied numerically as a function of the ultraviolet cutoff. Particular care is taken to explicitly verify gauge invariance and Ward-Takahashi identity constraints to all orders in the ${m}_{\ensuremath{\pi}}$ expansion. The results are used to compute the chiral corrections to matrix elements of local operators, related to moments of deep-inelastic structure functions. Finally, comparison of results for pseudovector and pseudoscalar coupling allows the resolution of a longstanding puzzle in the computation of pion cloud corrections to structure function moments.

Four-loop corrections with two closed fermion loops to fermion self energies and the lepton anomalous magnetic moment

We compute the eighth-order fermionic corrections involving two and three closed massless fermion loops to the anomalous magnetic moment of the muon. The required four-loop on-shell integrals are classified and explicit analytical results for the master integrals are presented. As further applications we compute the corresponding four-loop QCD corrections to the mass and wave function renormalization constants for a massive quark in the on-shell scheme.

Incorporating fermions in the Gaussian effective potential: The Higgs-top sector

The Higgs-Top model is studied by a non-perturbative variational extension of the Gaussian Effective Potential that incorporates fermions. In the limit of a very strong Yukawa coupling the one-loop result is shown to follow a single-parameter scaling while the gaussian fluctuations give rise to important deviations from scaling and to a reduction of the vev and of the top mass. A good general agreement is found with lattice data when a comparison can be made. The vacuum is shown to be stable for any choice of the Yukawa coupling, at variance with renormalized perturbation theory. Analytical results are provided for few observables like the renormalized mass of the Higgs boson and its wave function renormalization constant. Extensions to gauge theories like QCD are briefly discussed.

Effect of an imaginary part of the Schwinger-Dyson equation at finite temperature and density

We examined the effect of an imaginary part of the ladder approximation Schwinger-Dyson equation. We show the imaginary part enhances the effect of the first order transition, and affects a tricritical point. In particular, a chemical potential at a tricritical point is moved about 200(MeV). Thus, one should not ignore the imaginary part. On the other hand, since an imaginary part is small away from a tricritical point, one should be able to ignore an imaginary part. In addition, we also examined the contribution of the wave function renormalization constant.

Second order thermal corrections to electron wavefunction

Second order perturbative corrections to electron wavefunction are calculated here, for the first time, at generalized temperature. Calculations of electron self energy are important for the renormalizability of electron mass and wavefunction in QED through order by order cancellation of singularities up to order α 2 . Cancellation of temperature dependent singularities is demonstrated by incorporating the results of both orders of integration between cold and hot loops. For finite terms, we have rewritten second order thermal corrections as well, in a concise form, to calculate wavefunction renormalization constant. Our results are in a form that includes intermediate temperatures T ∼ m (where m is electron mass) while limits of high temperature T ≫ m and low temperature T ≪ m are also retrievable from them. A comparison with the existing results is included as well. The renormalized mass and wavefunction are used to calculate particle processes in extremely hot systems such as stellar cores and primordial nucleosynthesis during very early universe. An application to the latter case is also discussed.

Two loop low temperature corrections to electron self energy

We recalculate the two loop corrections in the background heat bath using real time formalism. The procedure of the integrations of loop momenta with dependence on finite temperature before the momenta without it has been followed. We determine the mass and wavefunction renormalization constants in the low temperature limit of QED, for the first time with this preferred order of integrations. The correction to electron mass and spinors in this limit is important in the early universe at the time of primordial nucleosynthesis as well as in astrophysics.

Renormalization in general gauge mediation

We revisit General Gauge Mediation (GGM) in light of the supersymmetric (linear) sigma model by utilizing the current superfield. The current superfield in the GGM is identified with supersymmetric extension of the vector symmetry current of the sigma model while spontaneous breakdown of supersymmetry in the GGM corresponds to soft breakdown of the axial vector symmetry of the sigma model. We first derive the current superfield from the supersymmetric linear sigma model and then compute 2-point functions of the current superfield using the (anti-)commutation relations of the messenger component fields. After the global symmetry are weakly gauged, the 2-point functions of the current superfield are identified with a part of the 2-point functions of the associated vector superfield. We renormalize them by dimensional regularization and show that physical gaugino and sfermion masses of the MSSM are expressed in terms of the wavefunction renormalization constants of the component fields of the vector superfield.

Numerical Calculation of Schwinger-Dyson Equation with Momentum-Dependent Gauge Parameter at Finite Temperature

Chiral symmetry at finite temperature is studied using the Schwinger-Dyson equation. We calculate numerically the critical temperature using the Schwinger-Dyson equation with the gauge parameter that depends on an external momentum. The critical temperature obtained by this method is similar to that with the Landau gauge and wave function renormalization constant 1. Moreover, the gauge invariance in the ladder approximation is examined using our method.

The SUSY-QCD β function to three loops

A number of $\overline {\mbox {\textsc{D}R}}$ renormalization constants in softly broken SUSY- QCD are evaluated to three-loop level: the wave function renormalization constants for quarks, squarks, gluons, gluinos, ghosts, and ε-scalars, and the renormalization constants for the quark and gluino mass as well as for all cubic vertices. The latter allow us to derive the corresponding β functions through three loops, all of which we find to be identical to the expression for the gauge β function obtained by Jack et al. (Phys. Lett. B 386:138, 1996, hep-ph/9606323 ) (see also Pickering et al. in Phys. Lett. B 510, 347, 2001, hep-ph/0104247 ). This explicitly demonstrates the consistency of DRED with SUSY and gauge invariance, an important pre-requisite for precision calculations in supersymmetric theories.

Functional Renormalization Group Approach to the Sine-Gordon Model

The renormalization group flow is presented for the two-dimensional sine-Gordon model within the framework of the functional renormalization group method by including the wave-function renormalization constant. The Kosterlitz-Thouless-Berezinski type phase structure is recovered as the interpolating scaling law between two competing IR attractive area of the global renormalization group flow.

Light quark mass effects in the on-shell renormalization constants

We compute the three-loop relation between the pole and the minimally subtracted quark mass allowing for virtual effects from a second massive quark. We also consider the analogue effects for the on-shell wave function renormalization constant.