Three-point Green Functions Research Articles

On the three-point order parameters of chiral symmetry breaking

The nonlocal order parameters of the QCD chiral symmetry breaking, namely the nonanomalous three-point Green functions of the quark bilinears belonging to the odd intrinsic parity sector, are studied within the framework of the Resonance chiral theory. The general form of these correlators is then matched with various high energy constraints: the leading and sub-leading OPE for all three momenta simultaneously large, the leading OPE for two momenta large and the leading and sub-leading Brodsky-Lepage limit for the corresponding transition form factor. In order to satisfy these constraints, the Resonance chiral theory has to be enlarged with additional resonance multiplets and with higher derivative operators as well. We discuss a minimal extension of this kind in detail and present corresponding correlators in a closed form. The remaining free parameters are then constrained from lattice data. As a phenomenological application, we discuss the pion-pole contribution to the muon g − 2 and the decay π0→ e+e−.

Towards perturbative renormalization ofϕ2(iϕ)ϵquantum field theory

In a previous paper it was shown how to calculate the ground-state energy density $E$ and the $p$-point Green's functions $G_p(x_1,x_2,...,x_p)$ for the $PT$-symmetric quantum field theory defined by the Hamiltonian density $H=\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\phi^2(i\phi)^\varepsilon$ in $D$-dimensional Euclidean spacetime, where $\phi$ is a pseudoscalar field. In this earlier paper $E$ and $G_p(x_1,x_2,...,x_p)$ were expressed as perturbation series in powers of $\varepsilon$ and were calculated to first order in $\varepsilon$. (The parameter $\varepsilon$ is a measure of the nonlinearity of the interaction rather than a coupling constant.) This paper extends these perturbative calculations to the Euclidean Lagrangian $L= \frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\mu^2\phi^2+\frac{1}{2} g\mu_0^2\phi^2\big(i\mu_0^{1-D/2}\phi\big)^\varepsilon-iv\phi$, which now includes renormalization counterterms that are linear and quadratic in the field $\phi$. The parameter $g$ is a dimensionless coupling strength and $\mu_0$ is a scaling factor having dimensions of mass. Expressions are given for the one-, two, and three-point Green's functions, and the renormalized mass, to higher-order in powers of $\varepsilon$ in $D$ dimensions ($0\leq D\leq2$). Renormalization is performed perturbatively to second order in $\varepsilon$ and the structure of the Green's functions is analyzed in the limit $D\to 2$. A sum of the most divergent terms is performed to {\it all} orders in $\varepsilon$. Like the Cheng-Wu summation of leading logarithms in electrodynamics, it is found here that leading logarithmic divergences combine to become mildly algebraic in form. Future work that must be done to complete the perturbative renormalization procedure is discussed.

OPE of Green functions of chiral currents

In this paper, we investigate the high-energy behavior of two-point and three-point Green functions of the QCD chiral currents and densities using the framework of the operator product expansion in the chiral limit. In detail, we study the contributions of the quark, gluon, quark-gluon and four-quark condensates to all the relevant non-vanishing three-point correlators.

OPE of Green Functions of Chiral Currents

A study of high-energy behavior of the three-point Green functions of chiral currents within the framework of the operator product expansion (OPE) is presented. In detail, we introduce an expansion of such correlators in terms of the QCD condensates up to dimension six. Next, we discuss some specifics of matching the OPE with calculations of the correlators within the chiral perturbation theory and resonance chiral theory, which can lead to important constraints of the coupling constants of the chiral/resonance Lagrangian.

Scalar-involved three-point Green functions and their phenomenology

We analyze within the framework of resonance chiral theory the $⟨S{A}_{\ensuremath{\mu}}{A}_{\ensuremath{\nu}}⟩$ and $⟨S{V}_{\ensuremath{\mu}}{V}_{\ensuremath{\nu}}⟩$ three-point Green functions, where $S$, ${A}_{\ensuremath{\mu}}$ and ${V}_{\ensuremath{\mu}}$ are short for scalar, axial-vector and vector $SU(3)$ hadronic currents. We construct the necessary Lagrangian such that the Green functions fulfill the asymptotic constraints, at large momenta, imposed by QCD at leading order. We study the implications of our results on the spectrum of scalars in the $\mathrm{large}\text{\ensuremath{-}}{N}_{C}$ limit, and analyze their decays.

Gauge dependence of the one-loop divergences in 6D, [formula omitted] abelian theory

We study the gauge dependence of the one-loop effective action for the abelian 6D, N=(1,0) supersymmetric gauge theory formulated in harmonic superspace. We introduce the superfield ξ-gauge, construct the corresponding gauge superfield propagator, and calculate the one-loop two- and three-point Green functions with two external hypermultiplet legs. We demonstrate that in the general ξ-gauge the two-point Green function of the hypermultiplet is divergent, as opposed to the Feynman gauge ξ=1. The three-point Green function with two external hypermultiplet legs and one leg of the gauge superfield is also divergent. We verified that the Green functions considered satisfy the Ward identity formulated in N=(1,0) harmonic superspace and that their gauge dependence vanishes on shell. Using the result for the two- and three-point Green functions and arguments based on the gauge invariance, we present the complete divergent part of the one-loop effective action in the general ξ-gauge.

Supersymmetric QCD on the lattice: An exploratory study

We perform a pilot study of the perturbative renormalization of a Supersymmetric gauge theory with matter fields on the lattice. As a specific example, we consider Supersymmetric ${\cal N}{=}1$ QCD (SQCD). We study the self-energies of all particles which appear in this theory, as well as the renormalization of the coupling constant. To this end we compute, perturbatively to one-loop, the relevant two-point and three-point Green's functions using both dimensional and lattice regularizations. Our lattice formulation involves the Wilson discretization for the gluino and quark fields; for gluons we employ the Wilson gauge action; for scalar fields (squarks) we use na\"ive discretization. The gauge group that we consider is $SU(N_c)$, while the number of colors, $N_c$, the number of flavors, $N_f$, and the gauge parameter, $\alpha$, are left unspecified. We obtain analytic expressions for the renormalization factors of the coupling constant ($Z_g$) and of the quark ($Z_\psi$), gluon ($Z_u$), gluino ($Z_\lambda$), squark ($Z_{A_\pm}$), and ghost ($Z_c$) fields on the lattice. We also compute the critical values of the gluino, quark and squark masses. Finally, we address the mixing which occurs among squark degrees of freedom beyond tree level: we calculate the corresponding mixing matrix which is necessary in order to disentangle the components of the squark field via an additional finite renormalization.

OPE of Green functions in the odd sector of QCD

A review of familiar results of the three-point Green functions of currents in the odd-intrinsic parity sector of QCD is presented. Such Green functions include very well-known examples of VVP , VAS or AAP correlators. We also present new results for VVA and AAA Green functions that have not yet been studied extensively in the literature before, more importantly with a phenomenological study and a discussion of the highenergy behaviour and its relation to the QCD condensates.

Three-point Green functions in the odd sector of QCD

A review of familiar results of the three-point Green functions of currents in the odd-intrinsic parity sector of QCD is presented. Such Green functions include very well-known examples of VVP , VAS or AAP correlators. We also shortly present some of the new results for VVA and AAA Green functions with a discussion of their high-energy behaviour and its relation to the QCD condensates.

Green functions of currents in the odd-intrinsic parity sector of QCD

In this talk, we will discuss all the three-point Green functions of currents, nontrivial for the odd-intrinsic parity sector. We briefly summarize their basic properties and high energy bahaviour via the OPE framework. A set of phenomenological studies is introduced. We also refer to a property that the vector field formalism is not compatible with the OPE, and for this reason the antisymmetric field formalism is a more suitable choice.

NEUTRON ELECTRIC DIPOLE MOMENT AT FIXED TOPOLOGY

We describe the finite volume effects of CP-odd quantities, such as the neutron electric dipole moment and the anapole moment in the θ-vacuum, under different topological sectors. We evaluate the three-point Green's functions for the electromagnetic current in a fixed nontrivial topological sector in order to extract these CP-odd observables. We discuss the role of zero modes in the CP-odd Green's function and show that, in the quenched approximation, there is a power divergence in the quark mass for CP-odd quantities at finite volume.

Transillumination fluorescence imaging in mice using biocompatible upconverting nanoparticles

We report on a systematic study of upconverting fluorescence signal generation within turbid phantoms and real tissues. An accurate three-point Green's function, describing the forward model of photon propagation, is established and experimentally validated. We further demonstrate, for the first time to our knowledge, autofluorescence-free transillumination imaging of mice that have received biocompatible upconverting nanoparticles. The method holds great promise for artifact-free whole-body visualization of optical molecular probes.

Three-point Green's function of massless QED in position space to lowest order

The transverse part of the three-point Green's function of massless QED is determined to the lowest order in position space. Taken together with the evaluation of the longitudinal part in Mitra (2008) (J. Phys. A: Math. Theor. 41 315401), this gives a relation for QED which is analogous to the star-triangle relation. We relate our result to conformal-invariant three-point functions.

On conformal invariant integrals involving spin one-half and spin-one particles

We consider the evaluation of D-dimensional conformal invariant integrals which involve spin one-half and spin-one particles. The star–triangle relation for the massless Yukawa theory is derived, and the longitudinal part of the three-point Green function of massless QED is determined to the lowest order in position space. The operator algebraic method of calculating massless Feynman integrals is used for the evaluation.

OPE-RχT matching at order αs: hard gluonic corrections to three-point Green functions

In this work we push the matching between the QCD operator product expansion (OPE) and resonance chiral theory (RχT) to order αs. To this end we compute two- and three-point QCD Green functions (GFs) in both theories and compare the results. GFs which are order parameters of chiral symmetry breaking make this matching more transparent and thus we concentrate on those. On the OPE side one needs to calculate the hard-gluon virtual corrections to the quark condensate, and in particular for three-point GFs this computation was hitherto missing. We also discuss the need for including the infinite tower of hadronic states in the hadronic representation of the GF when non-analytic terms such as logarithms are present in the OPE Wilson coefficients.

Form factors in the radiative pion decay

We perform an analysis of the form factors that rule the structure-dependent amplitude in radiative pion decay. The resonance contributions to π→eνeγ decays are computed through the proper construction of the vector and axial-vector form factors by setting the QCD driven asymptotic properties of the three-point Green functions 〈VVP〉 and 〈VAP〉, and by demanding the smoothening of the form factors at high transfer of momentum. A comparison between theoretical and experimental determination of the form factors is also carried out. We also consider and evaluate the role played by a non-standard tensor form factor. We conclude that, at present and due to the hadronic uncertainties, the search for new physics in this process is not feasible.

Three-point Green Functions in the resonance region : LEC's

Using the 1 / N C expansion scheme and truncating the hadronic spectrum to the lowest-lying multiplets of resonances we develop a procedure to implement QCD information on the non-perturbative regime populated by resonances through the use of three-point Green Functions of QCD currents. The process implies to enforce on these functions the constraints provided by their asymptotic behaviour ruled by QCD using their OPE or associated form factors. As a result we are able to determine several couplings of the O ( p 6 ) Chiral Perturbation Theory Lagrangian and, in consequence, to address phenomenological issues such as the SU ( 3 ) breaking in K l 3 decays.

Ghost and Gluon leading power corrections in Landau gauge

Gluon two- and three-point Green Functions computed in Landau gauge on the lattice have shown the existence of 1 / q 2 power corrections to the purely perturbative expressions that can be explained through the Operator Product Expansion as the influence of a gauge dependent mass dimension two condensate. The analysis of the ghost propagator within this framework of gauge-dependent power corrections is now performed.

Gauge-dependent O.P.E. power corrections on the QCD coupling

The running coupling constant can be estimated by computing gluon two- and three-point Green functions from the lattice. Computing in lattice implies working in a fixed gauge sector (Landau). A source of systematic uncertainty is then the contribution of non gauge-invariant condensates, as 〈 A 2〉, generating power corrections that might still be important at very high energies. We study the impact of this gluon condensate on the analysis of lattice data and the estimate of Λ MS . We finally try the relationship of this condensate with topological properties of QCD through the instanton picture.

An instanton picture O.P.E.〈A 2〉 condensate?

Gluon two- and three-point Green Functions computed in Landau gauge from the lattice show the existence of power corrections to the purely perturbative expressions, that can be explained through an Operator Product Expansion as the influence of a non gauge invariant mass dimension two condensate. The relationship of this condensate with topological properties of QCD, namely instantons, will be studied, giving a first estimate of the contribution of instantons to this condensate based in the direct lattice measure, after a cooling process, of the instanton liquid properties.