Hooft Equation Research Articles

ρ -meson spectroscopy and diffractive production using the holographic light-front Schrödinger equation and the ’t Hooft equation

We determine the mass spectroscopy and light-front wave functions (LFWFs) of the ρ-meson by solving the holographic Schrödinger equation of light-front chiral QCD along with the ’t Hooft equation of (1+1)-dimensional QCD in the large Nc limit. Subsequently, we utilize the obtained LFWFs in conjunction with the color glass condensate dipole cross section to calculate the cross sections for the diffractive ρ-meson electroproduction. Our spectroscopic results align well with the experimental data. Predictions for the diffractive cross sections demonstrate good consistency with the available experimental data at different energies from H1 and ZEUS collaborations. Additionally, we show that the resulting LFWFs for the ρ-meson can effectively describe various properties, including its decay constant, distribution amplitudes, electromagnetic form factors, charge radius, magnetic and quadrupole moments. Comparative analyses are conducted with experimental measurements and the available theoretical predictions. Published by the American Physical Society 2024

Exclusive diffractive J/ψ and ψ(2S) production in dipole model using a holographic AdS/QCD light-front wavefunction with longitudinal confinement

We use anti-de Sitter/quantum chromodynamics based holographic light-front wavefunction (LFWF) for vector meson, in conjunction with the dipole model to investigate the cross sections data for the diffractive and exclusive J/ψ and ψ(2S) production. We confront the experimental data using a new explicit form of the holographic LFWF, where the longitudinal confinement dynamics in the light-front Schrödinger equation is captured by the ’t Hooft equation of (1+1)-dim, in large Nc approximation, in addition to the transverse confinement dynamics governed by the confining mass scale parameter κ in vector mesons. We obtain the LFWF parameters from fitting to the exclusive J/ψ electroproduction data from the electron-proton collision at the Hadron Electron Ring Accelerator collider for mc=1.27 GeV. Our results suggest that the dipole model together with holographic meson LFWFs with longitudinal confinement is able to give a successful description for differential scattering cross section for exclusive J/ψ electroproduction for H1 and ZEUS data. We also predict the rapidity distributions of differential scattering cross section and total photoproduction of J/ψ and ψ(2S) states in proton-proton ultraperipheral collisions (UPCs) at center of mass energy s=7,13 TeV. Using the minimum set of parameters, our predictions for the UPCs are in good agreement with the recent experimental observations of UPCs at ALICE and LHCb Collaborations. Published by the American Physical Society 2024

Pion spectroscopy and dynamics using the holographic light-front Schrödinger equation and the 't Hooft equation

We show that the holographic Schrödinger equation of light-front chiral QCD, together with the 't Hooft equation of (1+1)-dimensional QCD in the large Nc limit, can simultaneously describe pion spectroscopy as well as the pion decay constant, charge radius, electromagnetic form factor, photon-to-pion transition form factor, Parton Distribution Function (PDF) and Distribution Amplitude (DA). Furthermore, the chiral-limit constraints, as encoded in the Gell-Mann-Oakes-Renner (GMOR) relation, are satisfied.

Longitudinal dynamics in light-front holographic QCD and hadron spectroscopy

We demonstrate that the ’t Hooft equation and the light-front holographic Schrodinger equation are complementary to each other in governing ¨ the longitudinal and the transverse dynamics of color confinement in mesons (quark-antiquark) and baryons (quark-diquark). Together, they describe remarkably well the spectroscopic data for the light-light, heavy-light and heavy-heavy hadrons. In all hadrons, the transverse dynamics of confinement is controlled by the universal emerging hadronic scale of the light-front holography, κ ∼ 0.5 GeV, which, in heavy-heavy hadrons, coincides with the ’t Hooft coupling that governs the longitudinal confinement. This reflects the restoration of the rotational symmetry in the non-relativistic limit.

Two Schrödinger-like Equations for hadrons

In this talk, based on [1, 2], I argue that the holographic Schrödinger Equation of (3 +1)-dim, conformal light-front QCD and the ’t Hooft Equation of (1+1)-dim, large Nc QCD, can be complementary to each other in providing a first approximation to hadron spectroscopy. Together, the two equations play a role in hadronic physics analogous that of the ordinary Schrödinger Equation in atomic physics.

Extending light-front holographic QCD using the 't Hooft Equation

We show the 't Hooft Equation and the light-front holographic Schrödinger Equation are complementary to each other in governing the transverse and longitudinal dynamics of colour confinement in quark-antiquark mesons. Together, they predict remarkably well the light, heavy-light and heavy-heavy meson spectroscopic data. The universal emerging hadronic scale of light-front holography, κ≈0.5 GeV, controls the transverse dynamics of confinement in all these mesons. In heavy-heavy mesons, it also coincides numerically with the 't Hooft coupling which governs longitudinal confinement, thus reflecting the restoration of manifest 3-dimensional rotational symmetry.

Hadron spectroscopy using the light-front holographic Schrödinger equation and the ’t Hooft equation

Light-front holographic QCD provides a successful first approximation to hadron spectroscopy in the chiral limit of $(3+1)$-dim light-front QCD, where a holographic Schr\"odinger-like equation, with an emerging confining scale, $\kappa$, governs confinement in the transverse direction. In its supersymmetric formulation, light-front holography predicts that each baryon has two superpartners: a meson and a tetraquark, with their degenerate masses being generated by the same scale, $\kappa$. In nature, this mass degeneracy is lifted by chiral symmetry breaking and longitudinal confinement. In this paper, we show that the latter can be successfully captured by the 't Hooft equation of $(1+1)$-dim, large $N_c$, QCD. Together, the holographic Schr\"odinger equation and the 't Hooft equation, provide a good global description of the data across the full hadron spectrum with a universal $\kappa$.

Quark-hadron duality for heavy meson mixings in the \u2019t Hooft model

We study local quark-hadron duality and its violation for the {D}^0-{overline{D}}^0 , {B}_d^0-{overline{B}}_d^0 and {B}_s^0-{overline{B}}_s^0 mixings in the ’t Hooft model, offering a laboratory to test QCD in two-dimensional spacetime together with the large-Nc limit. With the ’t Hooft equation being numerically solved, the width difference is calculated as an exclusive sum over two-body decays. The obtained rate is compared to inclusive one that arises from four-quark operators to check the validity of the heavy quark expansion (HQE). In view of the observation in four-dimensions that the HQE prediction for the width difference in the {D}^0-{overline{D}}^0 mixing is four orders of magnitude smaller than the experimental data, in this work we investigate duality violation in the presence of the GIM mechanism. We show that the order of magnitude of the observable in the {D}^0-{overline{D}}^0 mixing is enhanced in the exclusive analysis relative to the inclusive counterpart, when the 4D-like phase space function is used for the inclusive analysis. By contrast, it is shown that for the {B}_d^0-{overline{B}}_d^0 and {B}_s^0-{overline{B}}_s^0 mixings, small yet non-negligible corrections to the inclusive result emerge, which are still consistent with what is currently indicated in four-dimensions.

Partonic quasidistributions in two-dimensional QCD

As a sequel of our preceding work [1] we carry out a comprehensive comparative study between the quasi parton distribution functions (PDFs), distribution amplitudes (DAs) and their light-cone counterparts for various flavor-neutral mesons, in the context of the 't Hooft model, that is, the two-dimensional QCD in the large $N$ limit. In contrast to the original derivation via diagrammatic techniques exemplified by Dyson-Schwinger and Bethe-Salpeter equations, here we employ the Hamiltonian operator approach to reconstruct the celebrated 't Hooft equation in light-front quantization, and Bars-Green equations in equal-time quantization. The novelty of our derivation is to employ the soft momentum cutoff as the IR regulator. As a virtue of this operator approach, the functional form of the quasi distributions can be transparently built out of the Bars-Green wave functions and the Bogoliubov angle with the aid of bosonization technique. Equipped with various bound-state wave functions numerically inferred in [1], we then investigate how rapidly the quasi distributions approach their light-cone counterparts with the increasing meson momentum. We observe that, light mesons' quasi distributions approach the light-cone distributions in a slower pace than the heavy quarkonia. Curiously, lattice simulations of quasi distributions in four-dimensional QCD also discover this feature. Furthermore, we also compute the partonic light-cone PDF and quasi-PDF to one-loop order in perturbation theory, again employing the momentum cutoff as the IR regulator. We explicitly verify one of the backbones underlying the large momentum effective field theory (LaMET), namely, both quasi-PDFs and light-cone PDFs in ${\rm QCD}_2$ indeed possess the same IR behavior at leading order in $1/P^z$

Model of quark–antiquark interaction in quantum chromodynamics on the light front

We formulate a model of quark–antiquark interaction related to the limit transition to the light-front Hamiltonian in quantum chromodynamics. As ultraviolet regularization, we use a lattice in the space of transverse coordinates, and we additionally introduce a longitudinal light-front coordinate cutoff and also corresponding periodic boundary conditions. We regard the zero mode with respect to this coordinate as an independent dynamical variable. The state space of the model is limited to a quark and an antiquark that interact only via the zero mode of the gluon field on the light front. In this framework, we obtain a discrete mass spectrum of bound states. This spectrum is determined by an equation that with respect to the longitudinal coordinate turns out to be analogous to the’ t Hooft equation in two-dimensional quantum chromodynamics. The equation also contains a quark–antiquark potential that ensures confinement in the transverse space.

Exact solution of the’ t Hooft equation in the limit of heavy quarks with unequal masses

We consider the’ t Hooft equation for bound states in the two-dimensional quantum chromodynamics in the limit of an infinite number of colors. In the case of quarks with unequal masses tending to infinity, we obtain an approximation to the low-energy part of the spectrum and the corresponding wave functions. We show that as in the case of equal masses, the’ t Hooft equation in the first approximation reduces to the Schrodinger equation with a linear potential, i.e., to the equation for a particle in a “triangular” potential well. We also discuss the possibility of obtaining corrections to this approximation.

Limit transition to the light-front QCD and a quark–antiquark approximation

We consider a transition to the light-front quantum chromodynamics from theories quantized on spacelike planes that approach the light front. This limit transition differs for zero and nonzero modes, which leads to the appearance of a semiphenomenological parameter that can be used to describe confinement effects. As an illustration, we consider the problem of the bound states of a quark–antiquark pair in 2+1 dimensions. We use a lattice gauge-invariant regularization in the transverse space and consequently obtain an analogue of the’ t Hooft equation. We also discuss the possibility of calculating the spectrum of bound states in 3+1 dimensions.

Exact solution of the 't Hooft equation in the limit of heavy quarks with unequal masses

Рассматривается уравнение 'т Хоофта для связанных состояний в двумерной квантовой хромодинамике в пределе бесконечного числа цветов. Для случая кварков разной массы в пределе стремления их масс к бесконечности получено приближение для низкоэнергетической части спектра и соответствующих волновых функций. Показано, что, как и в частном случае равных масс, уравнение 'т Хоофта сводится в первом приближении к уравнению Шредингера с линейным потенциалом, т. е. к уравнению для частицы в "треугольной" потенциальной яме. Также обсуждается возможность получения поправок к этому приближению.

On analytic solutions of multi-parton 't Hooft equations

On analytic solutions of multi-parton 't Hooft equations

ASYMPTOTIC PROPERTIES OF MASS SPECTRUM IN 'T HOOFT'S MODEL OF MESONS

We study 't Hooft's equation for bound states in two-dimensional multicolor QCD. We consider the case of quarks with equal masses. We derive asymptotic expansions for the spectrum of mesons in different regimes and study their properties.

Volume dependence of two-dimensional large-NQCD with a nonzero density of baryons

We take a first step towards the solution of QCD in 1+1 dimensions at nonzero density. We regularize the theory in the UV by using a lattice and in the IR by putting the theory in a box of spatial size L. After fixing to axial gauge we use the coherent states approach to obtain the large-N classical Hamiltonian H that describes color neutral quark-antiquark pairs interacting with spatial Polyakov loops in the background of baryons. Minimizing H we get a regularized form of the `t Hooft equation that depends on the expectation values of the Polyakov loops. Analyzing the L-dependence of this equation we show how volume independence, a la Eguchi and Kawai, emerges in the large-N limit, and how it depends on the expectation values of the Polyakov loops. We describe how this independence relies on the realization of translation symmetry, in particular when the ground state contains a baryon crystal. Finally, we remark on the implications of our results on studying baryon density in large-N QCD within single-site lattice theories, and on some general lessons concerning the way four-dimensional large-N QCD behaves in the presence of baryons.

A transverse lattice QCD model for mesons

QCD is analysed with two light-front continuum dimensions and two transverse lattice dimensions. In the limit of large number of colours and strong transverse gauge coupling, the contributions of light-front and transverse directions factorise in the dynamics, and the theory can be analytically solved in a closed form. An integral equation is obtained, describing the properties of mesons, which generalises the 't Hooft equation by including spin degrees of freedom. The meson spectrum, light-front wavefunctions and form factors can be obtained by solving this equation numerically. These results would be a good starting point to model QCD observables which only weakly depend on transverse directions, e.g. deep inelastic scattering structure functions.

Mass formula for two-dimensional flavor non-singlet mesons

We analytically and numerically investigate the 't Hooft equations, the lowest order mesonic Light-Front Tamm-Dancoff equations for $\rm SU(N_C)$ and $\rm U(N_C)$ gauge theories, generalized to flavor non singlet mesons. We find the wave function can be well approximated by new basis functions and obtain an analytic and an empirical formulae for the mass of the lightest bound state. Its value is consistent with the precedent results.

Large N limit of SO(N) gauge theory of fermions and Bosons

In this paper we study the large Nc limit of SO(Nc) gauge theory coupled to a Majorana field and a real scalar field in 1+1 dimensions extending ideas of Rajeev [Int. J. Mod. Phys. A 9, 5583 (1994)]. We show that the phase space of the resulting classical theory of bilinears, which are the mesonic operators of this theory, is OSp1(H|H)/U(H+|H+), where ℋ|ℋ refers to the underlying complex graded space of combined one-particle states of fermions and bosons and H+|H+ corresponds to the positive frequency subspace. In the begining to simplify our presentation we discuss in detail the case with Majorana fermions only [the purely bosonic case is treated in Toprak and Turgut, J. Math. Phys. 43, 1340 (2002)]. In the Majorana fermion case the phase space is given by O1(H)/U(H+), where ℋ refers to the complex one-particle states and H+ to its positive frequency subspace. The meson spectrum in the linear approximation again obeys a variant of the ’t Hooft equation. The linear approximation to the boson/fermion coupled case brings an additonal bound state equation for mesons, which consists of one fermion and one boson, again of the same form as the well-known ’t Hooft equation.

Super-Grassmannian and large N limit of quantum field theory with bosons and fermions

We study a large Nc limit of a two-dimensional Yang–Mills theory coupled to bosons and fermions in the fundamental representation. Extending an approach due to Rajeev we show that the limiting theory can be described as a classical Hamiltonian system whose phase space is an infinite-dimensional super-Grassmannian. The linear approximation to the equations of motion and the constraint yields the ’t Hooft equations for the mesonic spectrum. Two other approximation schemes to the exact equations are discussed.