Showing papers on "Vertex function published in 2010"

A new look at loop quantum gravity

Abstract: I describe a possible perspective on the current state of loop quantum gravity, at the light of the developments of the last years. I point out that a theory is now available, having a well-defined background-independent kinematics and a dynamics allowing transition amplitudes to be computed explicitly in different regimes. I underline the fact that the dynamics can be given in terms of a simple vertex function, largely determined by locality, diffeomorphism invariance and local Lorentz invariance. I emphasize the importance of approximations. I list open problems.

Nonequilibrium functional renormalization group with frequency-dependent vertex function: A study of the single-impurity Anderson model

Abstract: We investigate nonequilibrium properties of the single-impurity Anderson model by means of the functional renormalization group fRG within Keldysh formalism. We present how the level broadening / 2 can be used as flow parameter for the fRG. This choice preserves important aspects of the Fermi-liquid behavior that the model exhibits in case of particle-hole symmetry. An approximation scheme for the Keldysh fRG is developed which accounts for the frequency dependence of the two-particle vertex in a way similar but not equivalent to a recently published approximation to the equilibrium Matsubara fRG. Our method turns out to be a flexible tool for the study of weak to intermediate on-site interactions U 3. In equilibrium we find excellent agreement with numerical RG results for the linear conductance at finite gate voltage, magnetic field, and temperature. In nonequilibrium, our results for the current agree well with time-dependent density-matrix RG. For the nonlinear conductance as function of the bias voltage, we propose reliable results at finite magnetic field and finite temperature. Furthermore, we demonstrate the exponentially small scale of the Kondo temperature to appear in the second-order derivative of the self-energy. We show that the approximation is, however, not able to reproduce the scaling of the effective mass at large interactions.

Conductivity of interacting massless Dirac particles in graphene: Collisionless regime

Abstract: We provide detailed calculation of the ac conductivity in the case of $1/r$ Coulomb interacting massless Dirac particles in graphene in the collisionless limit when $\ensuremath{\omega}⪢T$. The analysis of the electron self-energy, current vertex function, and polarization function, which enter into the calculation of physical quantities including the ac conductivity, is carried out by checking the Ward-Takahashi identities associated with the electrical charge conservation and making sure that they are satisfied at each step. We adopt a variant of the dimensional regularization of Veltman and 't Hooft by taking the spatial dimension $D=2\ensuremath{-}ϵ$ for $ϵg0$. The procedure adopted here yields a result for the conductivity correction which, while explicitly preserving charge conservation laws, is nevertheless different from the results reported previously in literature.

Fermi liquid near Pomeranchuk quantum criticality

Abstract: We analyze the behavior of an itinerant two-dimensional Fermi system near a charge nematic $(n=2)$ Pomeranchuk instability in terms of the Landau Fermi-liquid (FL) theory. A key object of our study is the fully renormalized vertex function ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$, related to the Landau interaction function. We derive ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$ for a model case of the long-range interaction in the nematic channel. Already within the random-phase approximation (RPA), the vertex is singular near the instability. The full vertex, obtained by resumming the ladder series composed of the RPA vertices, differs from the RPA result by a multiplicative renormalization factor ${Z}_{\ensuremath{\Gamma}}$, related to the single-particle residue $Z$ and effective-mass renormalization ${m}^{\ensuremath{\ast}}/m$. We employ the Pitaevski-Landau identities, which express the derivatives of the self-energy in terms of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$, to obtain and solve a set of coupled nonlinear equations for ${Z}_{\ensuremath{\Gamma}}$, $Z$, and ${m}^{\ensuremath{\ast}}/m$. We show that near the transition the system enters a critical FL regime, where ${Z}_{\ensuremath{\Gamma}}\ensuremath{\sim}Z\ensuremath{\propto}{(1+{g}_{c,2})}^{1/2}$ and ${m}^{\ensuremath{\ast}}/m\ensuremath{\approx}1/Z$, where ${g}_{c,2}$ is the $n=2$ charge Landau component which approaches $\ensuremath{-}1$ at the instability. We construct the Landau function of the critical FL and show that all but ${g}_{c,2}$ Landau components diverge at the critical point. We also show that in the critical regime the one-loop result for the self-energy $\ensuremath{\Sigma}(K)\ensuremath{\propto}\ensuremath{\int}dPG(P)D(K\ensuremath{-}P)$ is asymptotically exact if one identifies the effective interaction $D$ with the RPA form of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$.

Infrared analysis of Dyson-Schwinger equations taking into account the Gribov horizon in Landau gauge

Abstract: The low momentum behavior of the Landau gauge Gribov-Zwanziger action is investigated using the respective Dyson-Schwinger equations. Because of the mixing of the gluon and the auxiliary fields four scenarios can be distinguished for the infrared behavior. Two of them lead to inconsistencies and can be discarded. Another one corresponds to the case where the auxiliary fields behave exactly like the Faddeev-Popov ghosts and the same scaling relation as in standard Landau gauge, {kappa}{sub A}+2{kappa}{sub c}=0, is valid. Even the parameter {kappa} is found to be the same, 0.595. The mixed propagators, which appear, are suppressed in all loops, and their anomalous infrared exponent can also be determined. A fourth case provides an even stricter scaling relation that includes also the mixed propagators, but possesses the same qualitative feature, i.e. the propagators of the Faddeev-Popov ghost and the auxiliary fields are infrared enhanced and the mixed and the gluon propagators are infrared suppressed. In this case the system of equations to obtain the parameter {kappa} is nonlinear in all variables.

Analysis of the vertexes $ \Xi_{Q}^{*}$ ′QV , $ \Sigma_{Q}^{*}$ $ \Sigma_{Q}^{}$ V and radiative decays $ \Xi_{Q}^{*}$ $ \rightarrow$ ′Q $ \gamma$ , $ \Sigma_{Q}^{*}$ $ \rightarrow$ $ \Sigma_{Q}^{}$ $ \gamma$

Abstract: In this article, we study the vertexes $ \Xi_{Q}^{*}$ ′Q V and $ \Sigma_{Q}^{*}$ $ \Sigma_{Q}^{}$ V with the light-cone QCD sum rules, then assume the vector meson dominance of the intermediate $ \phi$ (1020) , $ \rho$ (770) and $ \omega$ (782) , and calculate the radiative decays $ \Xi_{Q}^{*}$ $ \rightarrow$ ′Q $ \gamma$ and $ \Sigma_{Q}^{*}$ $ \rightarrow$ $ \Sigma_{Q}^{}$ $ \gamma$ .

On the infrared scaling solution of SU(N) Yang–Mills theories in the maximally Abelian gauge

Abstract: An improved method for extracting infrared exponents from functional equations is presented. The generalizations introduced allow for an analysis of quite complicated systems such as Yang–Mills theory in the maximally Abelian gauge. Assuming the absence of cancellations in the appropriately renormalized integrals the only consistent scaling solution yields an infrared enhanced diagonal gluon propagator in support of the Abelian dominance hypothesis. This is explicitly shown for SU(2) and subsequently verified for SU(N), where additional interactions exist. We also derive the most infrared divergent scaling solution possible for vertex functions in terms of the propagators’ infrared exponents. We provide general conditions for the existence of a scaling solution for a given system and comment on the cases of linear covariant gauges and ghost–anti-ghost symmetric gauges.

Renormalization versus strong form factors for one-boson-exchange potentials

Abstract: We analyze the one-boson-exchange potential from the point of view of renormalization theory. We show that the nucleon-meson Lagrangian, while predicting the $\mathit{NN}$ force, does not predict the $\mathit{NN}$ scattering matrix nor the deuteron properties unambiguously due to the appearance of short distance singularities. While the problem has traditionally been circumvented by introducing vertex functions via phenomenological strong form factors, we propose to impose physical renormalization conditions on the scattering amplitude at low energies. Working in the large ${N}_{c}$ approximation with $\ensuremath{\pi}$, $\ensuremath{\sigma}$, $\ensuremath{\rho}$, and $\ensuremath{\omega}$ mesons we show that, once these conditions are applied, results for low-energy phases of proton-neutron scattering as well as deuteron properties become largely insensitive to the form factors and to the vector mesons yielding reasonable agreement with the data and for realistic values of the coupling constants.

Holographic QCD integrated back to hidden local symmetry

Abstract: We develop a previously proposed gauge-invariant method to integrate out an infinite tower of Kaluza-Klein (KK) modes of vector and axial-vector mesons in a class of models of holographic QCD (HQCD). The HQCD is reduced by our method to chiral perturbation theory with hidden local symmetry (HLS), having only the lowest KK mode identified as the HLS gauge boson. We take the Sakai-Sugimoto model as a concrete HQCD, and completely determine the $\mathcal{O}({p}^{4})$ terms as well as the $\mathcal{O}({p}^{2})$ terms from the Dirac-Born-Infeld part and the anomaly-related (intrinsic-parity odd) gauge-invariant terms from the Chern-Simons part. Effects of higher KK modes are fully included in these terms. To demonstrate the power of our method, we compute momentum dependences of several form factors, such as the pion electromagnetic form factors, and the ${\ensuremath{\pi}}^{0}\mathrm{\text{\ensuremath{-}}}\ensuremath{\gamma}$ and $\ensuremath{\omega}\mathrm{\text{\ensuremath{-}}}{\ensuremath{\pi}}^{0}$ transition form factors, compared with experiment, which was not achieved before due to the complication of handling infinite sums. We also study other anomaly-related quantities like ${\ensuremath{\gamma}}^{*}\mathrm{\text{\ensuremath{-}}}{\ensuremath{\pi}}^{0}\mathrm{\text{\ensuremath{-}}}{\ensuremath{\pi}}^{+}\mathrm{\text{\ensuremath{-}}}{\ensuremath{\pi}}^{\ensuremath{-}}$ and $\ensuremath{\omega}\mathrm{\text{\ensuremath{-}}}{\ensuremath{\pi}}^{0}\mathrm{\text{\ensuremath{-}}}{\ensuremath{\pi}}^{+}\mathrm{\text{\ensuremath{-}}}{\ensuremath{\pi}}^{\ensuremath{-}}$ vertex functions.

Electron interactions in bilayer graphene: Marginal Fermi liquid and zero-bias anomaly

Abstract: We analyze the many-body properties of bilayer graphene (BLG) at charge neutrality, governed by long range interactions between electrons Perturbation theory in a large number of flavors is used in which the interactions are described within a random phase approximation, taking account of dynamical screening effect Crucially, the dynamically screened interaction retains some long range character, resulting in $\log^2$ renormalization of key quantities We carry out the perturbative renormalization group calculations to one loop order, and find that BLG behaves to leading order as a marginal Fermi liquid Interactions produce a log squared renormalization of the quasiparticle residue and the interaction vertex function, while all other quantities renormalize only logarithmically We solve the RG flow equation for the Green function with logarithmic accuracy, and find that the quasiparticle residue flows to zero under RG At the same time, the gauge invariant quantities, such as the compressibility, remain finite to $\log^2$ order, with subleading logarithmic corrections The key experimental signature of this marginal Fermi liquid behavior is a strong suppression of the tunneling density of states, which manifests itself as a zero bias anomaly in tunneling experiments in a regime where the compressibility is essentially unchanged from the non-interacting value

CP violation due to compactification

Abstract: We address the challenging issue of how $CP$ violation is realized in higher dimensional gauge theories without higher dimensional elementary scalar fields. In such theories interactions are basically governed by a gauge principle and therefore to get $CP$ violating phases is a nontrivial task. It is demonstrated that $CP$ violation is achieved as the result of compactification of extra dimensions, which is incompatible with the 4-dimensional $CP$ transformation. As a simple example we adopt a 6-dimensional U(1) model compactified on a 2-dimensional orbifold ${T}^{2}/{Z}_{4}$. We argue that the 4-dimensional $CP$ transformation is related to the complex structure of the extra space and show how the ${Z}_{4}$ orbifolding leads to $CP$ violation. We confirm by explicit calculation of the interaction vertices that $CP$ violating phases remain even after the rephasing of relevant fields. For completeness, we derive a rephasing invariant $CP$ violating quantity, following a similar argument in the Kobayashi-Maskawa model which led to the Jarlskog parameter. As an example of a $CP$ violating observable we briefly comment on the electric dipole moment of the electron.

Analysis of the vertices {Omega}{sub Q}*{Omega}{sub Q{phi}} and radiative decays {Omega}{sub Q}*{yields}{Omega}{sub Q{gamma}}

Abstract: In this article, we study the vertices {Omega}{sub Q}*{Omega}{sub Q{phi}} with the light-cone QCD sum rules, then assume the vector meson dominance of the intermediate {phi}(1020), and calculate the radiative decays {Omega}{sub Q}*{yields}{Omega}{sub Q{gamma}}.

Relativistic effects in the double S- and P-wave charmonium production in e+ e- annihilation

Abstract: On the basis of perturbative QCD and the relativistic quark model we calculate relativistic and bound state corrections in the pair production of S-wave and P-wave charmonium states. Relativistic factors in the production amplitude connected with the relative motion of heavy quarks and the transformation law of the bound state wave function to the reference frame of the moving S- and P-wave mesons are taken into account. For the gluon and quark propagators entering the production vertex function we use a truncated expansion in the ratio of the relative quark momenta to the center-of-mass energy {radical}(s) up to the second order. The relativistic treatment of the wave functions makes all such second order terms convergent, thus allowing the reliable calculation of their contributions to the production cross section. Relativistic corrections to the quark bound state wave functions in the rest frame are considered by means of the QCD generalization of the standard Breit potential. It turns out that the examined effects change essentially the nonrelativistic results of the cross section for the reaction e{sup +}+e{sup -{yields}}J/{Psi}({eta}{sub c})+{chi}{sub cJ}(h{sub c}) at the center-of-mass energy {radical}(s)=10.6 GeV.

Towards next-to-leading order transport coefficients from the four-particle irreducible effective action

Abstract: Transport coefficients can be obtained from two-point correlators using the Kubo formulas. It has been shown that the full leading order result for electrical conductivity and (QCD) shear viscosity is contained in the resummed two-point function that is obtained from the three-loop three-particle irreducible resummed effective action. The theory produces all leading order contributions without the necessity for power counting, and in this sense it provides a natural framework for the calculation. In this article we study the four-loop four-particle irreducible effective action for a scalar theory with cubic and quartic interactions, with a nonvanishing field expectation value. We obtain a set of integral equations that determine the resummed two-point vertex function. A next-to-leading order contribution to the viscosity could be obtained from this set of coupled equations.

Decay b - sγ in the presence of a constant antisymmetric tensor field

Abstract: A constant antisymmetric 2-tensor can arise in general relativity with spontaneous symmetry breaking or in field theories formulated in a noncommutative space-time. In this work, the one-loop contribution of a nonstandard WW{gamma} vertex on the flavor violating quark transition q{sub i}{yields}q{sub j}{gamma} is studied in the context of the electroweak Yang-Mills sector extended with a Lorentz-violating constant 2-tensor. An exact analytical expression for the on-shell case is presented. It is found that the loop amplitude is gauge independent, electromagnetic gauge invariant, and free of ultraviolet divergences. The dipolar contribution to the b{yields}s{gamma} transition together with the experimental data on the B{yields}X{sub s{gamma}} decay is used to derive the constraint {Lambda}{sub LV}>1.96 TeV on the Lorentz-violating scale.

Finite width induced modification to the electromagnetic form factors of spin-1 particles

Abstract: The inclusion of the unstable features of a spin-1 particle, without breaking the electromagnetic gauge invariance, can be properly accomplished by including higher order contributions as done in the so-called fermion loop scheme (for the $W$ gauge boson) and the boson loop scheme (for vector mesons). This induces a nontrivial modification to the electromagnetic vertex of the particle, which must be considered in addition to any other contribution computed as stable particles. Considering the modified electromagnetic vertex, we obtain general expressions for the corresponding corrections to the multipoles as a function of the mass of the particles in the loop. For the $W$ gauge boson no substantial deviations from the stable case are observed. For the $\ensuremath{\rho}$ and ${K}^{*}$ mesons the mass of the particles in the loop has a significant effect, and can be comparable with corrections of a different nature.

The algebra of physical observables in non-linearly realized gauge theories

Abstract: We classify the physical observables in spontaneously broken non-linearly realized gauge theories in the recently proposed loopwise expansion governed by the Weak Power-Counting (WPC) and the Local Functional Equation. The latter controls the non-trivial quantum deformation of the classical non-linearly realized gauge symmetry, to all orders in the loop expansion. The Batalin–Vilkovisky (BV) formalism is used. We show that the dependence of the vertex functional on the Goldstone fields is obtained via a canonical transformation w.r.t. the BV bracket associated with the BRST symmetry of the model. We also compare the WPC with strict power-counting renormalizability in linearly realized gauge theories. In the case of the electroweak group we find that the tree-level Weinberg relation still holds if power-counting renormalizability is weakened to the WPC condition.

Bounding the $ B_{s} \rightarrow \gamma\gamma$ decay from Higgs mediated FCNC transitions

Abstract: The Higgs-mediated flavor violating bottom-strange quarks transitions induced at the one-loop level by a nondiagonal Hbs coupling are studied within the context of an effective Yukawa sector that comprises SU{sub L}(2)xU{sub Y}(1)-invariant operators of up to dimension six. The most recent experimental result on B{yields}X{sub s{gamma}} with hard photons is employed to constrain the Hbs vertex, which is used to estimate the branching ratio for the B{sub s{yields}{gamma}{gamma}} decay. It is found that the B{sub s{yields}{gamma}{gamma}} decay can reach a branching ratio of the order of 4x10{sup -8}, which is 2 orders of magnitude smaller than the current experimental limit.

Static and dynamic properties of the pion from continuum model ling of strong QCD

Abstract: We present nonperturbative numerical solutions for the quark propagator Schwinger-Dyson equation (SDE) and pseudoscalar meson Bethe-Salpeter equation (BSE) at and beyond the rainbow-ladder truncation level of this system of equations. We solve this coupled system of integral equations using a phenomenological model for the dressed gluon propagator in Landau gauge as input. In the rainbow-ladder truncation scheme, we systematically calculate static properties of the pion and kaon. After combining the rainbow-ladder truncation for the SDE-BSE system with the impulse approximation for the pion-photon vertex, we present numerical results for the pion form factor using the Ball-Chiu and bare vertices for the nonperturbative quark-photon vertex. We find that the Ball-Chiu vertex satisfies electromagnetic current conservation automatically, however, this vertex gives a charge pion radius that is less than its experimental value, leaving room for further improvement. We go beyond the rainbow-ladder truncation by including pion cloud effects into the quark propagation, and then all the way up into the pion form factor. Here we find significant changes for the mass and decay constant of the pion. For the pion form factor, on the other hand, we find no qualitative changes in the $Q^{2}$ region studied for both vertices. Nevertheless, more work remains to be done at and beyond the rainbow-ladder truncation in order to connect the pion form factor to the model-independent perturbative result.

Discrete phase space - III: The divergence-free S-matrix elements

Abstract: In the arena of the discrete phase space and continuous time, the theory of S-matrix is formulated. In the special case of quantum electrodynamics (QED), the Feynman rules are precisely developed. These rules in the four-momentum turn out to be identical to the usual QED, except for the vertex function. The new vertex function is given by an infinite series that can only be treated in an asymptotic approximation at the present time. Preliminary approximations prove that the second-order self-energies of a fermion and a photon in the discrete model have convergent improper integrals. In the final section, a sharper asymptotic analysis is employed. It is proved that in case where the number of external photon or fermion lines is at least one, then the S-matrix elements converge in all orders. Moreover, there are no infrared divergences in this formulation.

Local interactions of higher-spin potentials that are gauge invariant in linear approximation

Abstract: We study connected Wightman functions of N conserved currents, each of which is formed from a scalar field and has even spin l i . The UV divergence of this vertex function is regularized by the analytic continuation in the space dimension D → D − e. We evaluate the residue of e −1 only, which is a local interaction Lagrangian density and gauge invariant in linear approximation. The text was submitted by the author in English.

Baryons in QCD(AS) at Large N(c): A Roundabout Approach

Abstract: QCD{sub AS}, a variant of large N{sub c} QCD in which quarks transform under the color two-index antisymmetric representation, reduces to standard QCD at N{sub c}=3 and provides an alternative to the usual large N{sub c} extrapolation that uses fundamental representation quarks. Previous strong plausibility arguments assert that the QCD{sub AS} baryon mass scales as N{sub c}{sup 2}; however, the complicated combinatoric problem associated with quarks carrying two color indices impeded a complete demonstration. We develop a diagrammatic technique to solve this problem. The key ingredient is the introduction of an effective multigluon vertex: a ''traffic circle'' or roundabout diagram. We show that arbitrarily complicated diagrams can be reduced to simple ones with the same leading N{sub c} scaling using this device, and that the leading contribution to baryon mass does, in fact, scale as N{sub c}{sup 2}.

A simple strategy for renormalization: QED at one-loop level

Abstract: We demonstrate our simple strategy for renormalization with QED at one-loop level, basing on an elaboration of the effective field theory philosophy. No artificial regularization or deformation of the original theory is introduced here and hence no manipulation of infinities, ambiguities arise instead of infinities. Ward identities first come to reduce the number of ambiguities, the residual ones could in principle be removed by imposing physical boundary conditions. Renormalization group equations arise as "decoupling theorems" in the underlying theory perspective. In addition, a technical theorem concerning routing of external momenta is also presented and illustrated with the self-energy and vertex function as examples.

Improvement on the GW$\Gamma$ Scheme for the Electron Self-Energy and Relevance of the $G_0W_0$ Approximation from this Perspective

Abstract: Based on an exact functional form derived for the three-point vertex function $\Gamma$, we propose a self-consistent calculation scheme for the electron self-energy with $\Gamma$ always satisfying the Ward identity. This scheme is basically equivalent to the one proposed in 2001, but it is improved in the aspects of computational costs and its applicability range; it can treat a low-density electron system with a dielectric catastrophe. If it is applied to semiconductors and insulators, we find that the obtained quasiparticle dispersion is virtually the same as that in the one-shot $GW$ approximation (or $G_0W_0$A), indicating that the $G_0W_0$A actually takes proper account of both vertex and high-order self-energy corrections in a mutually cancelling manner.

A simple strategy for renormalization: qed at one-loop level

Abstract: We demonstrate our simple strategy for renormalization with QED at one-loop level, basing on an elaboration of the effective field theory philosophy. No artificial regularization or deformation of the original theory is introduced here and hence no manipulation of infinities, ambiguities arise instead of infinities. Ward identities first come to reduce the number of ambiguities, the residual ones could in principle be removed by imposing physical boundary conditions. Renormalization group equations arise as "decoupling theorems" in the underlying theory perspective. In addition, a technical theorem concerning routing of external moment a is also presented and illustrated with the self-energy and vertex function as example.

Inclusion of vertex corrections for superconductivity in gauge-invariant self-consistent approximations

Abstract: Based on the Ward–Takahashi identities representing both charge and velocity local conservation laws, we provide a scheme for constructing the vertex function Γ ˆ in the self-consistent iteration loop to determine the self-energy Σ ˆ for superconductivity. For a singlet superconductor without spin-dependent interaction, the relationship between Γ ˆ and Σ ˆ in this iteration loop is mainly controlled by the electronic compressibility of the system, which is directly related to the velocity of the Nambu-Goldstone mode.