Showing papers on "Vertex function published in 2014"

Spin correlations in polarizations of P-wave charmonia $\chi_{cJ}$ and impact on $J/\psi$ polarization

Abstract: Based on a general form of the effective vertex functions for the decays of P-wave charmonia $\chicj$, angular distribution formulas for the subsequent decays $\chicj\rightarrow \jpsi \gamma$ and $\jpsi \to \mu^+\mu^-$ are derived. The formulas are the same as those obtained in a different approach in the literature. Our formulas are expressed in a more general form, including parity violation effects and the full angular dependence of $\jpsi$ and muon in the cascade decay $\chicj\to\jpsi\gamma\to\mu^+\mu^-\gamma$. The $\chicj$ polarization observables are expressed in terms of rational functions of the spin density matrix elements of $\chicj$ production. Generalized rotation-invariant relations for arbitrary integer-spin particles are also derived and their expressions in terms of observable angular distribution parameters are given in the $\chi_{c1}$ and $\chi_{c2}$. To complement our previous direct-$\jpsi$ polarization result, we also discuss the impact on the observable prompt-$\jpsi$ polarization. As an illustrative application of our angular distribution formulas, we present the angular distributions in terms of the tree-level spin density matrix elements of $\chi_{c1}$ and $\chi_{c2}$ production in several different frames at the Large Hadron Collider. Moreover, a reweighting method is also proposed to determine the entire set of the production spin density matrix elements of the $\chi_{c2}$, some of which disappear or are suppressed for vanishing higher-order multipole effects making the complete extraction difficult experimentally.

Transmon-based simulator of nonlocal electron-phonon coupling: A platform for observing sharp small-polaron transitions

Abstract: We propose an analog superconducting quantum simulator for a one-dimensional model featuring momentum-dependent (nonlocal) electron-phonon couplings of Su-Schrieffer-Heeger and ``breathing-mode'' types. Because its corresponding coupling vertex function depends on both the electron and phonon quasimomenta, this model does not belong to the realm of validity of the Gerlach-L\"owen theorem that rules out any nonanalyticities in single-particle properties. The superconducting circuit behind the proposed simulator entails an array of transmon qubits and microwave resonators. By applying microwave driving fields to the qubits, a small-polaron Bloch state with an arbitrary quasimomentum can be prepared in this system within times several orders of magnitude shorter than the typical qubit decoherence times. We demonstrate that---by varying the externally tunable parameters---one can readily reach the critical coupling strength required for observing the sharp transition from a nondegenerate (single-particle) ground state corresponding to zero quasimomentum (${K}_{\mathrm{gs}}=0$) to a twofold-degenerate small-polaron ground state at nonzero quasimomenta ${K}_{\mathrm{gs}}$ and $\ensuremath{-}{K}_{\mathrm{gs}}$. Through exact numerical diagonalization of our effective Hamiltonian, we show how this nonanalyticity is reflected in the relevant single-particle properties (ground-state energy, quasiparticle residue, average number of phonons). We also show that the proposed setup provides an ideal testbed for studying the nonequilibrium dynamics of small-polaron formation in the presence of strongly momentum-dependent electron-phonon interactions.

Pion electromagnetic form factor in the covariant spectator theory

Abstract: The pion electromagnetic form factor at spacelike momentum transfer is calculated in relativistic impulse approximation using the Covariant Spectator Theory. The same dressed quark mass function and the equation for the pion bound-state vertex function as discussed in the companion paper are used for the calculation, together with a dressed quark current that satisfies the WardTakahashi identity. The results obtained for the pion form factor are in agreement with experimental data, they exhibit the typical monopole behavior at high-momentum transfer, and they satisfy some remarkable scaling relations.

Gauge invariant electromagnetic properties of fermions induced by CPT-violation in the Standard Model Extension

Abstract: Low-energy Lorentz-invariant quantities could receive contributions from a fundamental theory producing small Lorentz-violating effects. Within the Lorentz-violating extension of quantum electrodynamics, we investigate, perturbatively, the contributions to the one-loop ffγ vertex from the CPT-violating axial coupling of a vector background field to fermions. We find that the resulting vertex function has a larger set of Lorentz structures than the one characterizing the usual, Lorentz-invariant, parametrization of the ffγ vertex. We prove gauge invariance of the resulting one-loop expression through a set of gauge invariant nonrenormalizable operators introducing new-physics effects at the first- and second-orders in Lorentz-violation, and which generate tree-level contributions to the ffγ vertex. Whereas loop contributions involving parameters that violate Lorentz-invariance at the first-order are CPT-odd, those arising at the second-order are CPT-even, so that contributions to low-energy physics are restricted to emerge for the first time at the second-order. In this context, we derive a contribution to anomalous magnetic moment (AMM) of fermions, which we use to set a bound on Lorentz-violation.

One-loop nonbirefringent effects on the electromagnetic vertex in the Standard Model Extension

Abstract: Lorentz violation emerged from a fundamental description of nature may impact, at low energies, the Maxwell sector, so that contributions from such new physics to the electromagnetic vertex would be induced. Particularly, nonbirefringent CPT-even effects from the electromagnetic sector modified by the Lorentz- and CPT-violating Standard Model Extension alter the structure of the free photon propagator. We calculate Lorentz-violating contributions to the electromagnetic vertex, at the one-loop level, by using a modified photon propagator carrying this sort of effects. We take the photon off shell, and find an expression that involves both isotropic and anisotropic effects of nonbirefringent violation of Lorentz invariance. Our analysis of the one-loop vertex function includes gauge invariance, transformation properties under C, P and T, and tree-level contributions from Lorentz-violating nonrenormalizable interactions. These elements add to previous studies of the one-loop contributions to the electromagnetic vertex in the context of Lorentz violation in the photon sector. Finally, we restrict our analysis to the isotropic case and derive a finite contribution from isotropic Lorentz violation to the anomalous magnetic moment of fermions that coincides with the result already reported in the literature.

Quasiparticle interaction function in a two-dimensional Fermi liquid near an antiferromagnetic critical point

Abstract: We present the expression for the quasiparticle vertex function ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ (proportional to the Landau interaction function) in a 2D Fermi liquid (FL) near an instability towards antiferromagnetism. This function is relevant in many ways in the context of metallic quantum criticality. Previous studies have found that near a quantum critical point, the system enters into a regime in which the fermionic self-energy is large near hot spots on the Fermi surface [points on the Fermi surface connected by the antiferromagnetic ordering vector ${q}_{\ensuremath{\pi}}=(\ensuremath{\pi},\ensuremath{\pi})$] and has much stronger dependence on frequency than on momentum. We show that in this regime, which we termed a critical FL, the conventional random-phase-approximation- (RPA) type approach breaks down, and to properly calculate the vertex function one has to sum up an infinite series of terms which were explicitly excluded in the conventional treatment. Besides, we show that, to properly describe the spin component of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ even in an ordinary FL, one has to add Aslamazov-Larkin (AL) terms to the RPA vertex. We show that the total ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ is larger in a critical FL than in an ordinary FL, roughly by an extra power of magnetic correlation length $\ensuremath{\xi}$, which diverges at the quantum critical point. However, the enhancement of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ is highly nonuniform: It holds only when, for one of the two momentum variables, the distance from a hot spot along the Fermi surface is much larger than for the other one. This fact renders our case different from quantum criticality at small momentum, where the enhancement of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ was found to be homogeneous. We show that the charge and spin components of the total vertex function satisfy the universal relations following from the Ward identities related to the conservation of the particle number and the total spin. We show that in a critical FL, the Ward identity involves ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ taken between particles on the FS. We find that the charge and spin components of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ are identical to leading order in the magnetic correlation length. We use our results for ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ and for the quasiparticle residue to derive the Landau parameters ${F}_{c}^{l=0}={F}_{s}^{l=0}$, the density of states, and the uniform ($q=0$) charge and spin susceptibilities ${\ensuremath{\chi}}_{c}^{l=0}={\ensuremath{\chi}}_{s}^{l=0}$. We show that the density of states ${N}_{F}$ diverges as $log\ensuremath{\xi}$; however, ${F}_{c,s}^{l=0}$ also diverge as $log\ensuremath{\xi}$, such that the total ${\ensuremath{\chi}}_{c,s}^{(l=0)}\ensuremath{\propto}{N}_{F}/(1+{F}_{c}^{l=0})$ remain finite at $\ensuremath{\xi}=\ensuremath{\infty}$. We show that at weak coupling these susceptibilities are parametrically smaller than for free fermions.

Coordinate-space singularities of massless gauge theories

Abstract: The structure of singularities in perturbative massless gauge theories is investigated in coordinate space. The pinch singularities in coordinate-space integrals occur at configurations of vertices which have a direct interpretation in terms of the physical scattering of particles in real space-time in the same way as for the loop momenta in the case of momentum-space singularities. In the analysis of vertex functions in coordinate space, the well-known factorization into hard, soft, and jet functions is found. By power-counting arguments, it is found that coordinate-space integrals of vertex functions have logarithmic divergences at worst.

Two-loop massive scalar three-point function in a dispersive approach

Abstract: We present a dispersion relation formalism to calculate a massive scalar two-loop vertex function. Such calculation is of direct relevance in the evaluation of the hadronic light-by-light contribution to the muon's anomalous magnetic moment due to meson poles. The discontinuity of the two-loop diagram is obtained by a sum of two- and three-particle cut contributions, which involve a phase space integration over the physical intermediate states. The real part of the vertex function is subsequently reconstructed through evaluation of a dispersion integral. We explicitly demonstrate that the dispersive formalism yields exactly the same result as the direct two-loop calculation.

Relevance of various Dirac covariants in hadronic Bethe-Salpeter wave functions in electromagnetic decays of ground state vector mesons

Abstract: In this work we have employed the Bethe-Salpeter equation (BSE) under covariant instantaneous ansatz to study electromagnetic decays of ground state equal mass vector mesons $\ensuremath{\rho}$, $\ensuremath{\omega}$, $\ensuremath{\phi}$, $\ensuremath{\psi}$, and $Y$ through the process $V\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\rightarrow}{e}^{+}+{e}^{\ensuremath{-}}$. We employ the generalized structure of the hadron-quark vertex function $\mathrm{\ensuremath{\Gamma}}$ that incorporates all Dirac structures from their complete set order by order in powers of the inverse of the meson mass. We have explicitly shown the derivation of this general form of this hadron-quark vertex function $\mathrm{\ensuremath{\Gamma}}$ (in terms of unknown coefficients) for a vector meson with the incorporation of all the Dirac structures (i.e., those dependent on external hadron momentum $P$, as well those dependent on internal hadron momentum $q$) as the solution of the full $4\ifmmode\times\else\texttimes\fi{}4$ Bethe-Salpeter equation. The unknown coefficients multiplying the various Dirac structures are calculated by reducing the $4\ifmmode\times\else\texttimes\fi{}4$ BSE to a determinantal form. These coefficients thus determined were also employed for the calculation of ${f}_{V}$ values and gave good agreement with data, as well as an acceptable solution of the full BSE.

Dual-fermion approach to interacting disordered fermion systems

Abstract: We generalize the recently introduced dual fermion (DF) formalism for disordered fermion systems by including the effect of interactions. For an interacting disordered system the contributions to the full vertex function have to be separated into elastic and inelastic scattering processes, and addressed differently when constructing the DF diagrams. By applying our approach to the Anderson-Falicov-Kimball model and systematically restoring the nonlocal correlations in the DF lattice calculation, we show a significant improvement over the Dynamical Mean-Field Theory and the Coherent Potential Approximation for both one-particle and two-particle quantities.

Linearized self-consistent GW approach satisfying the Ward identity

Abstract: We propose a linearized self-consistent GW approach satisfying the Ward identity. The vertex function derived from the Ward-Takahashi identity in the limit of $\mathbit{q}=0$ and $\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}^{\ensuremath{'}}=0$ is included in the self-energy and the polarization function as a consequence of the linearization of the quasiparticle equation. Due to the energy dependence of the self-energy, the Hamiltonian is a non-Hermitian operator and quasiparticle states are nonorthonormal and linearly dependent. However, the linearized quasiparticle states recover orthonormality and fulfill the completeness condition. This approach is very efficient, and the resulting quasiparticle energies are greatly improved compared to the nonlinearized self-consistent GW approach, although its computational cost is not much increased. We show the results for atoms and dimers of Li and Na compared with other approaches. We also propose convenient ways to calculate the Luttinger-Ward functional $\ensuremath{\Phi}$ based on a plasmon-pole model and calculate the total energy for the ground state. As a result, we conclude that the linearization improves overall behaviors in the self-consistent GW approach.

$\Delta(1232)$ Resonance Contribution to Two-Photon Exchange in Electron-Proton Scattering Revisited

Abstract: We revisit the question of the contributions of two-photon exchange with $\Delta(1232)$ excitation to the electron-proton scattering in a hadronic model. Three improvements over the previous calculations are made, namely, correct vertex function for $\gamma N\rightarrow\Delta$, realistic $\gamma N\Delta$ form factors, and coupling constants. The discrepancy between the values of $R\equiv \mu_p G_E/G_M$ extracted from Rosenbluth technique and polarization transfer method can be reasonably accounted for if the data of Andivahis {\it et al.} \cite{Andivahis94} are analyzed. However, substantial discrepancy remains if the data of Qattan {\it et al.} \cite{Qattan05} are used. For the ratio $R^\pm$ between $e^\pm p$ scatterings, our predictions appear to be in satisfactory agreement with the preliminary data from VEPP-3. The agreement between our model predictions and the recent measurements on single spin asymmetry, transverse and longitudinal recoil proton polarizations ranges from good to poor.

Exact Green's Function for the Bethe Lattice in the Hopping Model

Abstract: We calculated the exact self energy and the exact Green's function for the Bethe lattice in the hopping model using the Dyson's equation and from the fourth order of perturbation theory. The self energy is found from processes where an electron hops to its neighbors and includes terms where the particle hops twice to a next nearest neighbor (NNN) and then hops back. carried out by counting all non-self intersecting paths, leading to an implicit equation for the local Green function. By integrating out branches of the Bethe lattice, the same equation is obtained from a path integral approach for the partition function. In this study, the electron Green's function will be used to calculate the exact Green's function and self energy for the Bethe lattice in the hopping model. Though it is not one of the goals of our study here, it is pertinent to point out that the self energy can be used to obtain the vertex function which contains all the information on the kinetic and interaction behaviour of the particles in the system. Interestingly the vertex function can be decoupled into the charge vertex function and spin vertex function, thereby enabling the studying of either non-spin ordering systems or spin ordering system independently.

Two-loop Feynman integrals for $\phi^4$ theory with long-range correlated disorder

Abstract: Two-loop massive Feynman integrals for $\phi^4$ field-theoretical model with long-range correlated disorder are considered. Massive integrals for the vertex function $\Gamma^{(4)}$ including two or three massless propagators for generic space dimension and for any value of the correlation parameter are evaluated analytically applying Mellin-Barnes method as well as familiar representation for one-loop integrals. Obtained expressions are presented in the form of hypergeometric functions.

Electromagnetic annihilation into charged leptons and scattering off nucleons of spin-3/2 Majorana particles

Abstract: We compute the cross section for the electromagnetic annihilation into charged leptons, and the electromagnetic scattering off nucleons, of spin-$3/2$ self-conjugate (Majorana) particles using the general form of the electromagnetic vertex function that was obtained previously for such particles. In addition to the restrictions imposed by common principles such as electromagnetic gauge invariance and Hermiticity, the vertex function incorporates the restriction due to the Majorana condition as well as the particular properties related to the spinors in the Rarita--Schwinger representation and is the counterpart of the so-called anapole interaction of spin-$1/2$ Majorana particles. The formulas obtained for the cross sections share certain similarities with the corresponding results in the spin-$1/2$ case, but they also reveal some important differences which are pointed out and discussed. The results given here can be useful for applications involving the electromagnetic interactions of spin-$3/2$ or spin-$1/2$ Majorana particles in several contexts that have been of interest in the recent literature such as nucleosynthesis and dark matter.

One-loop calculation of the β-function in the modified formulation of the Yang-Mills theory

Abstract: In the one-loop approximation for the modified Yang-Mills theory, we calculate contributions to the two- and three-point Green’s functions of the gauge field A μ , to the Green’s function of the anticommuting fields e and b, and to the vertex function Γ Aeb. We find the renormalization constants Z 1 and Z 2 and the corresponding constants $$\bar Z_1$$ and $$\bar Z_2$$ for the anticommuting fields and show that they satisfy the same equation as in the standard Yang-Mills theory. We demonstrate that the β-functions in the standard and modified theories coincide.