Searcharxiv⌕ Search

arXiv subjects

G. McCartor

Publications and source records attributed to G. McCartor.

14 recordsLinked to original sources
arXiv2005

Two-boson truncation of Pauli-Villars-regulated Yukawa theory

We apply light-front quantization, Pauli-Villars regularization, and numerical techniques to the nonperturbative solution of the dressed-fermion problem in Yukawa theory in 3+1 dimensions. The solution is developed as a Fock-state expansion truncated to include at most one fermion and two bosons. The basis includes a negative-metric heavy boson and a negative-metric heavy fermion in order to provide the necessary cancellations of ultraviolet divergences. The integral equations for the Fock-state wave functions are solved by reducing them to effective one-boson--one-fermion equations for eigenstates with J_z=1/2. The equations are converted to a matrix equation with a specially tuned quadrature scheme, and the lowest mass state is obtained by diagonalization. Various properties of the dressed-fermion state are then computed from the nonperturbative light-front wave functions. This work is a major step in our development of Pauli-Villars regularization for the nonperturbative solution of four-dimensional field theories and represents a significant advance in the numerical accuracy of such solutions.

hep-ph↗
arXiv2005

Spontaneously broken quark helicity symmetry

We discuss the origin of chiral symmetry breaking in the light-cone representation of QCD. In particular, we show how quark helicity symmetry is spontaneously broken in SU(N) gauge theory with massless quarks if that theory has a condensate of fermion lightcone zero modes. The symmetry breaking appears as induced interactions in an effective lightcone Hamiltonian equation based on a trivial vacuum. The induced interaction is crucial for generating a splitting between pseudoscalar and vector meson masses, which we illustrate with spectrum calculations in some 1+1-dimensional reduced models of gauge theory.

hep-ph↗
arXiv2004

Induced Operators in QCD

Light-cone quantization always involves the solution of differential constraint equations. The solutions to these equations include integration constants (fields independent of $x_-$). These fields are unphysical but when they are consistently removed from the dynamics, additional operators (induced operators), which would not be present if the integration constants were simply set to zero, are included in the dynamics. These induced operators can be taken to act in the usual light-cone subspace, for instance, the space used for DLCQ. Here, I shall give a derivation of two such operators. The operators are derived starting from the QCD Lagrangian but the derivation involves some guesses. The operators will provide for the linear growth of the pion mass squared with the quark bare mass and for the splitting of the pi and the rho at zero quark mass.

hep-th↗
arXiv2004

Adjoint $QCD_{1+1}$ in Light-cone Gauge, Quantized at Equal Time

SU(2) gauge theory coupled to massless fermions in the adjoint representation is quantized in light-cone gauge by imposing the equal-time canonical algebra. The theory is defined on a space-time cylinder with "twisted" boundary conditions, periodic for one color component (the diagonal 3- component) and antiperiodic for the other two. The focus of the study is on the non-trivial vacuum structure and the fermion condensate. It is shown that the indefinite-metric quantization of free gauge bosons is not compatible with the residual gauge symmetry of the interacting theory. A suitable quantization of the unphysical modes of the gauge field is necessary in order to guarantee the consistency of the subsidiary condition and allow the quantum representation of the residual gauge symmetry of the classical Lagrangian: the 3-color component of the gauge field must be quantized in a space with an indefinite metric while the other two components require a positive-definite metric. The contribution of the latter to the free Hamiltonian becomes highly pathological in this representation, but a larger portion of the interacting Hamiltonian can be diagonalized, thus allowing perturbative calculations to be performed. The vacuum is evaluated through second order in perturbation theory and this result is used for an approximate determination of the fermion condensate.

hep-th↗
arXiv2004

A nonperturbative calculation of the electron's magnetic moment

In principle, the complete spectrum and bound-state wave functions of a quantum field theory can be determined by finding the eigenvalues and eigensolutions of its light-cone Hamiltonian. One of the challenges in obtaining nonperturbative solutions for gauge theories such as QCD using light-cone Hamiltonian methods is to renormalize the theory while preserving Lorentz symmetries and gauge invariance. For example, the truncation of the light-cone Fock space leads to uncompensated ultraviolet divergences. We present two methods for consistently regularizing light-cone-quantized gauge theories in Feynman and light-cone gauges: (1) the introduction of a spectrum of Pauli-Villars fields which produces a finite theory while preserving Lorentz invariance; (2) the augmentation of the gauge-theory Lagrangian with higher derivatives. In the latter case, which is applicable to light-cone gauge (A^+ = 0), the A^- component of the gauge field is maintained as an independent degree of freedom rather than a constraint. Finite-mass Pauli-Villars regulators can also be used to compensate for neglected higher Fock states. As a test case, we apply these regularization procedures to an approximate nonperturbative computation of the anomalous magnetic moment of the electron in QED as a first attempt to meet Feynman's famous challenge.

hep-ph↗
arXiv2003

The mass renormalization of nonperturbative light-front Hamiltonian theory: An illustration using truncated, Pauli-Villars-regulated Yukawa interactions

We obtain analytic, nonperturbative, approximate solutions of Yukawa theory in the one-fermion sector using light-front quantization. The theory is regulated in the ultraviolet by the introduction of heavy Pauli-Villars scalar and fermion fields, each with negative norm. In order to obtain a directly soluble problem, fermion-pair creation and annihilation are neglected, and the number of bosonic constituents is limited to one of either type. We discuss some of the features of the wave function of the eigensolution, including its endpoint behavior and spin and orbital angular momentum content. The limit of infinite Pauli-Villars mass receives special scrutiny.

hep-th↗
arXiv2001

Exact solutions to Pauli-Villars-regulated field theories

We present a new class of quantum field theories which are exactly solvable. The theories are generated by introducing Pauli-Villars fermionic and bosonic fields with masses degenerate with the physical positive metric fields. An algorithm is given to compute the spectrum and corresponding eigensolutions. We also give the operator solution for a particular case and use it to illustrate some of the tenets of light-cone quantization. Since the solutions of the solvable theory contain ghost quanta, these theories are unphysical. However, we also discuss how perturbation theory in the difference between the masses of the physical and Pauli-Villars particles could be developed, thus generating physical theories. The existence of explicit solutions of the solvable theory also allows one to study the relationship between the equal-time and light-cone vacua and eigensolutions.

hep-th↗
arXiv2001

Application of Pauli-Villars regularization and discretized light-cone quantization to a single-fermion truncation of Yukawa theory

We apply Pauli-Villars regularization and discretized light-cone quantization to the nonperturbative solution of (3+1)-dimensional Yukawa theory in a single-fermion truncation. Three heavy scalars, including two with negative norm, are used to regulate the theory. The matrix eigenvalue problem is solved for the lowest-mass state with use of a new, indefinite-metric Lanczos algorithm. Various observables are extracted from the wave functions, including average multiplicities and average momenta of constituents, structure functions, and a form factor slope.

hep-ph↗
arXiv2000

Extended Hamiltonian Formalism of the Pure Space-Like Axial Gauge Schwinger Model

We demonstrate that pure space-like axial gauge quantizations of gauge fields can be constructed in ways which are free from infrared divergences. We begin by constructing an axial gauge formulation in auxiliary coordinates: $x^+=x^0\sinθ+x^1\cosθ, x^-=x^0\cosθ-x^1\sinθ$. For θless than π\over 4 we can take $x^-$ as the evolution parameter and construct a traditional canonical formulation of the temporal gauge Schwinger model in which residual gauge fields dependent only on $x^+$ are static canonical variables. Then we extrapolate the temporal gauge operator solution into the axial region, θ> π\over 4, where $x^+$ is taken as the evolution parameter. In the axial region we find that we have to change representations of the residual gauge fields from one realizing the PV prescription to one realizing the ML prescription in order for the infrared divergences resulting from $({\partial}_-)^{-1}$ to be canceled by corresponding ones resulting from the inverse of the hyperbolic Laplace operator. Finally, by taking the limit $θ\to\fracπ{2}-0$ we obtain an operator solution and the Hamiltonian of the axial gauge (Coulomb gauge )Schwinger model in ordinary coordinates. That solution includes auxiliary fields and the representation space is of indefinite metric, providing further evidence that ``physical'' gauges are no more physical than ``unphysical'' gauges.

hep-th↗
arXiv2000

Indispensability of Ghost Fields and Extended Hamiltonian Formalism in Axial Gauge Quantization of Gauge Fields

It is shown that ghost fields are indispensable in deriving well-defined antiderivatives in pure space-like axial gauge quantizations of gauge fields. To avoid inessential complications we confine ourselves to noninteracting abelian fields and incorporate their quantizations as a continuous deformation of those in light-cone gauge. We attain this by constructing an axial gauge formulation in auxiliary coordinates $x^μ= (x^+,x^-,x^1,x^2)$, where $x^+=x^0{\rm sin}θ+x^3{\rm cos}θ, x^-=x^0{\rm cos}θ-x^3{\rm sin}θ$ and $x^+$ and $A_-=A^0{\rm cos} θ+A^3{\rm sin}θ=0$ are taken as the evolution parameter and the gauge fixing condition, respectively. We introduce $x^-$-independent residual gauge fields as ghost fields and accomodate them to the Hamiltonian formalism by applying McCartor and Robertson's method. As a result, we obtain conserved translational generators $P_μ$, which retain ghost degrees of freedom integrated over the hyperplane $x^-=$ constant. They enable us to determine quantization conditions for the ghost fields in such a way that commutation relations with $P_μ$ give rise to the correct Heisenberg equations. We show that regularizing singularities arising from the inversion of a hyperbolic Laplace operator as principal values, enables us to cancel linear divergences resulting from $({\partial}_-)^{-2}$ so that the Mandelstam- Leibbrandt form of gauge field propagator can be derived. It is also shown that the pure space-like axial gauge formulation in ordinary coordinates can be derived in the limit $θ\to\fracπ{2}-0$ and that the light-cone axial gauge formulation turns out to be the case of $θ=\fracπ{4}$.

hep-th↗
arXiv1999

Consistent Perturbative Light Front Formulation of Yang-Mills Theories

It is shown how to obtain the consistent light front form quantization of a non-Abelian pure Yang-Mills theory (gluondynamics) in the framework of the standard perturbative approach. After a short review of the previous attempts in the light cone gauge $A_-=0$, it is explained how the difficulties can be overcome after turning to the anti light cone gauge $A_+=0$. In particular, the generating functional of the renormalized Green's functions turns out to be the same as in the conventional instant form approach, leading to the Mandelstam-Leibbrandt prescription for the free gluon propagator.

hep-th↗
arXiv1998

Pauli--Villars as a Nonperturbative Ultraviolet Regulator in Discretized Light-Cone Quantization

We propose a solution to the problem of renormalizing light-cone Hamiltonian theories while maintaining Lorentz invariance and other symmetries. The method uses generalized Pauli--Villars regulators to render the theory finite. We discuss the method in the context of Yukawa theory at one loop and for a soluble model in 3+1 dimensions. The model is studied nonperturbatively. Numerical results obtained with use of discretized light-cone quantization, special integration weighting factors, and the complex symmetric Lanczos diagonalization algorithm compare well with the analytic answers.

hep-th↗
arXiv1997

Vacuum Structure of Two-Dimensional Gauge Theories on the Light Front

We discuss the problem of vacuum structure in light-front field theory in the context of (1+1)-dimensional gauge theories. We begin by reviewing the known light-front solution of the Schwinger model, highlighting the issues that are relevant for reproducing the $θ$-structure of the vacuum. The most important of these are the need to introduce degrees of freedom initialized on two different null planes, the proper incorporation of gauge field zero modes when periodicity conditions are used to regulate the infrared, and the importance of carefully regulating singular operator products in a gauge-invariant way. We then consider SU(2) Yang-Mills theory in 1+1 dimensions coupled to massless adjoint fermions. With all fields in the adjoint representation the gauge group is actually SU(2)$/Z_2$, which possesses nontrivial topology. In particular, there are two topological sectors and the physical vacuum state has a structure analogous to a $θ$ vacuum. We formulate the model using periodicity conditions in $x^\pm$ for infrared regulation, and consider a solution in which the gauge field zero mode is treated as a constrained operator. We obtain the expected $Z_2$ vacuum structure, and verify that the discrete vacuum angle which enters has no effect on the spectrum of the theory. We then calculate the chiral condensate, which is sensitive to the vacuum structure. The result is nonzero, but inversely proportional to the periodicity length, a situation which is familiar from the Schwinger model. The origin of this behavior is discussed.

hep-th↗