Time-ordered Perturbation Theory Research Articles

Collinear parton distributions and the structure of the nucleon sea in a light-front meson-cloud model

The unpolarized, helicity and transversity parton distribution functions of the nucleon are studied within a convolution model where the bare nucleon is dressed by its virtual meson cloud. Using light-front time-ordered perturbation theory, the Fock states of the physical nucleon are expanded in a series involving a bare nucleon and two-particle, meson-baryon, states. The bare baryons and mesons are described with light-front wave functions (LFWFs) for the corresponding valence-parton components. Using a representation in terms of overlap of LFWFs, the role of the non-perturbative antiquark degrees of freedom and the valence quark contribution at the input scale of the model is discussed for the leading-twist collinear parton distributions. After introducing perturbative QCD effects through evolution to experimental scales, the results are compared with available data and phenomenological extractions. Predictions for the nucleon tensor charge are also presented, finding a very good agreement with recent phenomenological extractions.

Renormalization of the three-boson system with short-range interactions revisited

We consider renormalization of the three-body scattering problem in low-energy effective field theory of self-interacting scalar particles by applying time-ordered perturbation theory to the manifestly Lorentz-invariant formulation. The obtained leading-order equation is perturbatively renormalizable and non-perturbatively finite and does not require a three-body counter term in contrast to its non-relativistic approximation.

The Gauge-Field Propagator in Light-Cone Gauge: Which is the Correct One?

In the literature one can find two different expressions for the gauge-field propagator in light-cone gauge, containing the sum of three rather than two terms. The question of which of the two is the correct one has been a subject of debate. We propose a solution to this question by evaluating one-loop level processes in QED, both in the covariant approach in the light-cone gauge and in the light-front time-ordered perturbation theory (TOPT) approach, proving the equivalence between the two formulations of the theory. The form of the propagator turns out to be crucial in the proof, in particular as concerns its relation with the diagrams containing instantaneously propagating photons and instantaneous interactions. We show that the diagrams in light-front TOPT with instantaneous photons can be recovered in the covariant approach starting from the propagators with only two terms. Our proof of the equivalence clarifies which form should be used for the gauge-field propagator in the covariant approach. This result naturally applies to the QCD case also.

Revisiting the equivalence of light-front and covariant QED in the light-cone gauge

We discuss the equivalence between light-front time-ordered-perturbation theory and covariant quantum field theory in light-front quantization, in the case of quantum electrodynamics at one-loop level. In particular, we review the one-loop calculation of the vertex correction, fermion self-energy and vacuum polarization. We apply the procedure of integration by residue over the light-front energy in the loop to show how the perturbative expansion in covariant terms can be reduced to a sum of propagating and instantaneous diagrams of light-front time-ordered perturbation theory. The detailed proof of equivalence between the two formulations of the theory resolves the controversial question on which form should be used for the gauge-field propagator in the light-cone gauge in the covariant approach.

Two-nucleon scattering in a modified Weinberg approach with a symmetry-preserving regularization

We consider the nucleon-nucleon scattering problem by applying time-ordered perturbation theory to the Lorentz invariant formulation of baryon chiral perturbation theory. Using a symmetry preserving higher derivative form of the effective Lagrangian, we exploit the freedom of the choice of the renormalization condition and obtain an integral equation for the scattering amplitude with an improved ultraviolet behavior. The resulting formulation is used to quantify finite regulator artifacts in two-nucleon phase shifts as well as in the chiral extrapolations of the S-wave scattering lengths and the deuteron binding energy. This approach can be straightforwardly extended to analyze few-nucleon systems and processes involving external electroweak sources.

Nuclear axial currents in chiral effective field theory

Two-nucleon axial charge and current operators are derived in chiral effective field theory up to one loop. The derivation is based on time-ordered perturbation theory, and accounts for cancellations between the contributions of irreducible diagrams and the contributions due to non-static corrections from energy denominators of reducible diagrams. Ultraviolet divergencies associated with the loop corrections are isolated in dimensional regularization. The resulting axial current is finite and conserved in the chiral limit, while the axial charge requires renormalization. A complete set of contact terms for the axial charge up to the relevant order in the power counting is constructed.

Resumming double logarithms in the QCD evolution of color dipoles

The higher-order perturbative corrections, beyond leading logarithmic accuracy, to the BFKL evolution in QCD at high energy are well known to suffer from a severe lack-of-convergence problem, due to radiative corrections enhanced by double collinear logarithms. Via an explicit calculation of Feynman graphs in light cone (time-ordered) perturbation theory, we show that the corrections enhanced by double logarithms (either energy-collinear, or double collinear) are associated with soft gluon emissions which are strictly ordered in lifetime. These corrections can be resummed to all orders by solving an evolution equation which is non-local in rapidity. This equation can be equivalently rewritten in local form, but with modified kernel and initial conditions, which resum double collinear logs to all orders. We extend this resummation to the next-to-leading order BFKL and BK equations. The first numerical studies of the collinearly-improved BK equation demonstrate the essential role of the resummation in both stabilizing and slowing down the evolution.

Spectrum of the excitedN*andΔ*baryons in a relativistic chiral quark model

The spectrum of the SU(2) flavor baryons is studied in the frame of a relativistic chiral quark potential model based on the one-pion and one-gluon exchange mechanisms. It is argued that the ${N}^{*}$ and ${\mathrm{\ensuremath{\Delta}}}^{*}$ resonances strongly coupled to the $\ensuremath{\pi}N$ channel are identified with the orbital configurations $(1{S}_{1/2}{)}^{2}(nlj)$ with a single valence quark in the excited state ($nlj$). With the obtained selection rules based on the ``chiral constraint,'' we show that it is possible to construct a schematic periodic table of baryon resonances, consistent with the experimental data and yielding no ``missing resonances.'' A new original method for the treatment of the center of mass problem is suggested which is based on the separation of the three-quark Dirac Hamiltonian into the parts, corresponding to the Jacobi coordinates. The numerical estimations for the energy positions of the nucleon and delta baryons (up to and including F-wave ${N}^{*}$ and ${\mathrm{\ensuremath{\Delta}}}^{*}$ resonances), obtained within the field-theoretical framework by using time ordered perturbation theory, yield an overall good description of the experimental data at the level of the relativized constituent quark model of S. Capstick and W. Roberts without any fitting parameters. The only free parameter of the linear confinement potential was fitted previously by Th. Gutsche to reproduce the axial charge of the nucleon. The ground state $\mathrm{\ensuremath{\Delta}}(1232)$ is well reproduced. However, nucleon ground state and most of the radially excited baryon resonances (including Roper) are overestimated. On the contrary, the first band of the orbitally excited baryon resonances with a negative parity are underestimated. At the same time, the second band of the orbitally excited ${\mathrm{\ensuremath{\Delta}}}^{*}$ states with the negative parity are mostly overestimated, while the ${N}^{*}$ states are close to the experimental boxes. The theoretical estimations of the energy levels for the positive parity baryon resonances with $\mathrm{J}=5/2$, $7/2$ are close to the experimental data. At higher energies, where the experimental data are poor, we can extend our model schematically and predict an existence of seven ${N}^{*}$ and four ${\mathrm{\ensuremath{\Delta}}}^{*}$ new states with larger spin values.

Nuclear electromagnetic processes in ChEFT

We review our recent work on the derivation of the nuclear electromagnetic charge and current operators in chiral perturbation theory, based on time-ordered perturbation theory. We then discuss the strategies for fixing the relevant low-energy constants, and compare the resulting predictions for the electric and magnetic form factors of the deuteron and trinucleons with experimental data, using as input accurate nuclear wave functions derived with realistic potentials.

Cancellation of infrared divergences to all orders in light front QED

Coherent state approach has been proposed as an alternate way to deal with the true infrared divergences in light front field theory. We show that infrared divergences in fermion mass renormalization are eliminated to all orders in light front time ordered perturbation theory if one uses coherent state basis instead of the usual Fock basis to calculate the Hamiltonian matrix elements.

Nuclear Axial Currents in Chiral Effective Field Theory

Two-nucleon axial charge and current operators are derived in chiral eective field theory up to one loop. The derivation is based on time-ordered perturbation theory, and accounts for cancellations between the contributions of irreducible diagrams and the contributions due to non-static corrections from energy denominators of reducible diagrams. Ultraviolet divergencies associated with the loop corrections are isolated in dimensional regularization. The resulting axial current is finite and conserved in the chiral limit, while the axial charge requires renormalization. A complete set of contact terms for the axial charge up to the relevant order in the power counting is constructed.

Next-to-next-to-leading-order nucleon parton distributions from a light-cone quark model dressed with its virtual meson cloud

A detailed study of nucleon parton distribution functions is performed within a radiative Next-to-Next-to Leading Order (NNLO) parton model whose low-scale input is rigidly described by wave functions which include quarks and dressing meson-cloud. The light-front Hamiltonian dynamics fixes the three-quark wave functions and the meson cloud is introduced by means of high-order Fock component in time-ordered perturbation theory in the infinite momentum frame. Non-strange as well as strange mesons-baryon fluctuations are considered ($\pi$, $\rho$, $\omega$, $K$ and $K^*$, together with $N$, $\Delta$, $\Lambda$ and $\Sigma$) and the effects on strange and non-strange parton distributions investigated showing the large effects due to (non-strange) sea asymmetries and the delicate balance of the strange asymmetry. The non-strange and the strange components, the valence as well as the gluon distributions are compared with available experimental data confirming the need of both non-perturbative degrees of freedom and perturbative (NNLO) radiative effects.

Fermion self-energy correction in light-front QED using coherent state basis

We discuss IR divergences in lepton mass renormalization in Light-Front Quantum Electrodynamics (LFQED) in Feynman gauge. We consider LFQED with Pauli-Villars fields and using old-fashioned time ordered perturbation theory (TOPT), we show that our earlier result regarding cancellation of true IR divergences upto $O(e^4)$ in coherent state basis holds in Feynman gauge also.

Infrared divergences in light-front QED and coherent state basis

We present a next to leading order calculation of electron mass renormalization in Light-Front Quantum Electrodynamics (LFQED) using old-fashioned time ordered perturbation theory (TOPT). We show that the true infrared divergences in $\delta m^2$ get canceled up to $O(e^4)$ if one uses coherent state basis instead of fock basis to calculate the transition matrix elements.

Equivalence of covariant and light front QED: Generating instantaneous diagrams

One loop expressions for fermion self-energy, vacuum polarization, and vertex correction in light-front time ordered perturbation theory can be obtained from respective covariant expressions by performing ${k}^{\ensuremath{-}}$ integration. In an earlier work, we have shown that the third term in the doubly transverse gauge propagator is necessary to generate the diagrams involving instantaneous photon exchange both in the case of fermion self-energy as well as vertex correction. In this work, using the two-term photon propagator, we show that the instantaneous photon exchange diagrams in fermion self-energy as well as the IR singular terms in the propagating diagrams can be generated by taking the asymptotic limit of the covariant expression. It is further shown that this method reproduces the IR singular terms in propagating diagrams of vacuum polarization also.

Phases of augmented hadronic light-front wave functions

It is an important question whether the final/initial state gluonic interactions which lead to naive-time-reversal-odd single-spin asymmetries and diffraction at leading twist can be associated in a definite way with the light-front wave function hadronic eigensolutions of QCD. We use light-front time-ordered perturbation theory at the lowest nontrivial order to obtain augmented light-front wave functions which contain an imaginary phase which depends on the choice of advanced or retarded boundary condition for the gauge potential in light-cone gauge. We apply this formalism to the wave functions of the valence Fock states of nucleons and pions, and show how this illuminates the factorization properties of naive-time-reversal-odd transverse momentum dependent observables which arise from rescattering. In particular, one calculates the identical leading-twist Sivers function from the overlap of augmented light-front wave functions that one obtains from explicit calculations of the single-spin asymmetry in semi-inclusive deep inelastic lepton–polarized nucleon scattering where the required phases come from the final-state rescattering of the struck quark with the nucleon spectators.

Electromagnetic form factors and charge densities from hadrons to nuclei

A simple exact covariant model in which a scalar particle $\ensuremath{\Psi}$ is modeled as a bound state of two different particles is used to elucidate relativistic aspects of electromagnetic form factors $F({Q}^{2})$. The model form factor is computed using an exact covariant calculation of the lowest order triangle diagram. The light-front technique of integrating over the minus component of the virtual momentum gives the same result and is the same as the one obtained originally by Gunion et al. [Phys. Rev. D 8, 287 (1973)] by using time-ordered perturbation theory in the infinite-momentum frame. The meaning of the transverse density $\ensuremath{\rho}(b)$ is explained by providing a general derivation, using three spatial coordinates, of its relationship with the form factor. This allows us to identify a mean-square transverse size $\ensuremath{\langle}{b}^{2}\ensuremath{\rangle}=\ensuremath{\int}{d}^{2}b {b}^{2}\ensuremath{\rho}(b)=\ensuremath{-}4\frac{\mathit{dF}}{{\mathit{dQ}}^{2}}({Q}^{2}=0)$. The quantity $\ensuremath{\langle}{b}^{2}\ensuremath{\rangle}$ is a true measure of hadronic size because of its direct relationship with the transverse density. We show that the rest-frame charge distribution is generally not observable by studying the explicit failure to uphold current conservation. Neutral systems of two charged constituents are shown to obey the conventional lore that the heavier one is generally closer to the transverse origin than the lighter one. It is argued that the negative central charge density of the neutron arises, in pion-cloud models, from pions of high longitudinal momentum that reside at the center. The nonrelativistic limit is defined precisely, and the ratio of the binding energy $B$ to the mass $\mathcal{M}$ of the lightest constituent is shown to govern the influence of relativistic effects. It is shown that the exact relativistic formula for $F({Q}^{2})$ is the same as the familiar one of the three-dimensional Fourier transform of a square of a wave function for very small values of $B/\mathcal{M}$, but this only occurs for values of $B/\mathcal{M}$ less than about $0.001$. For masses that mimic the quark-diquark model of the nucleon we find that there are substantial relativistic corrections to the form factor for any value of ${Q}^{2}$. A schematic model of the lowest $s$ states of nuclei is developed. Relativistic effects are found to decrease the form factor for light nuclei but to increase the form factor for heavy nuclei. Furthermore, these lowest $s$ states are likely to be strongly influenced by relativistic effects that are of the order of $15%\text{\ensuremath{-}}20%$.

Electromagnetic two-body currents of one- and two-pion range

Nuclear electromagnetic currents are derived in time-ordered perturbation theory within an effective-field-theory framework including explicit nucleons, \ensuremath{\Delta} isobars, and pions up to one loop, or next-to-next-to-next-to-leading order (${\mathrm{N}}^{3}$LO). The currents obtained at next-to-next-to-leading order (${\mathrm{N}}^{2}$LO), i.e., ignoring loop corrections, are used in a study of neutron radiative captures on protons and deuterons at thermal energies, and of $A=2$ and 3 nuclei magnetic moments. The wave functions for $A=2$ are derived from solutions of the Schr\"odinger equation with the Argonne ${v}_{18}$ (AV18) or CD-Bonn (CDB) potentials, while those for $A=3$ are obtained with the hyperspherical-harmonics-expansion method from a realistic Hamiltonian including, in addition to the AV18 or CDB two-nucleon, a three-nucleon potential. With the strengths of the \ensuremath{\Delta}-excitation currents occurring at ${\mathrm{N}}^{2}$LO determined to reproduce the $n\text{\ensuremath{-}}p$ cross section and isovector combination of the trinucleon magnetic moments, we find that the cross section and photon circular polarization parameter, measured in $n\text{\ensuremath{-}}d$ and $\stackrel{\ensuremath{\rightarrow}}{n}\text{\ensuremath{-}}d$ processes, are underpredicted by theory; for example, the cross section is underpredicted by 11--38% as the cutoff is increased from 500 to 800 MeV. A complete analysis of the results, in particular their large cutoff dependence, is presented.

On general properties of Lorentz-invariant formulation of noncommutative quantum field theory

We study general properties of certain Lorentz-invariant noncommutative quantum field theories proposed in the literature. We show that causality in those theories does not hold, in contrast to the canonical noncommutative field theory with the light-wedge causality condition. This is the consequence of the infinite nonlocality of the theory getting spread in all spacetime directions. We also show that the time-ordered perturbation theory arising from the Hamiltonian formulation of noncommutative quantum field theories remains inequivalent to the covariant perturbation theory with usual Feynman rules even after restoration of Lorentz symmetry.

Can QFT on Moyal-Weyl Spaces Look as on Commutative Ones?

We sketch a natural affirmative answer to the question based on a joint work with J. Wess [G. Fiore and J. Wess, Phys. Rev. D 75 (2007), 105022; hep-th/0701078]. There we argue that a proper enforcement of the “twisted Poincaré” covariance makes any differences (x-y)µ of coordinates of two copies of the Moyal-Weyl deformation of Minkowski space like undeformed. Then QFT in an operator approach becomes compatible with (minimally adapted) Wightman axioms and time-ordered perturbation theory, and physically equivalent to ordinary QFT, as observables involve only coordinate differences.