arXiv subjects

Paul Hoyer

Publications and source records attributed to Paul Hoyer.

At least 19 recordsLinked to original sources

arXiv2026

Principles and Possibilities for Bound States in Gauge Theory

Bound states differ from scattering yet are not covered in textbooks on Quantum Field Theory. I discuss a perturbative method for QED and QCD based on canonical quantization. Fully fixing temporal gauge $A^0(t,\boldsymbol{x})=0$ imposes Gauss' law on physical states. As pointed out by Dirac, this implies that electron states include a longitudinal gauge field $\boldsymbol{A}_L$, which determines the instantaneous bound state potential. The situation is analogous for quarks and gluons in QCD. An instantaneous confining potential arises for color singlet $q\bar q$ states when a non-vanishing boundary condition on $\boldsymbol{A}_L^a(\boldsymbol{x}\to\infty)$ is specified in Gauss' constraint. As suggested by Gribov, $\alpha_s(Q^2)$ may freeze at a perturbative value when the confining potential dominates. Hadrons can then be calculated perturbatively. At vanishing quark mass there is a $j^{PC}=0^{++}$ state with zero energy which can mix with the perturbative vacuum, giving rise to a spontaneous breaking of chiral symmetry.

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arXiv2025

Dirac states from the 't Hooft model

The dynamics of a light fermion bound to a heavy one is expected to be described by the Dirac equation with an external potential. The potential breaks translation invariance, whereas the bound state momentum is well defined. Boosting the bound state determines the frame dependence of the light fermion dynamics. I study the Dirac limit of QCD$_2$ in the limit of $N_c \to \infty$. The light quark wave function turns out to be independent of the frame of the bound state, up to an irrelevant Lorentz contraction. The discrete bound state spectrum determines corresponding discrete energies of the Dirac equation, which for a linear potential allows a continuous spectrum.

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arXiv2025

't Hooft model in the temporal gauge

I consider QCD$_2$ in the $N_c \to \infty$ limit at fixed $g^2N_c$. The derivation starts from equal-time $q\bar q$ bound states in coordinate space and temporal ($A^0=0$) gauge, avoiding the use of quark and gluon propagators. The wave function is given analytically by a $_1F_1$ function with an explicit frame dependence. In the infinite momentum frame the Fourier transformed wave function satisfies the 't~Hooft equation, however with contributions also from quarks with negative kinetic energy. Such contributions are present also in the rest frame, and do not vanish under boosts.

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arXiv2024

Bound state basics

Perturbative expansions for atoms in QED are developed around interacting states, typically defined by the Schr\"odinger equation. Calculations are nevertheless done using the standard Feynman diagram expansion around free states. The classical $-\alpha/r$ potential is then obtained through an infinite sum of ladder diagrams. The complexity of this approach may have contributed to bound states being omitted from QFT textbooks, restricting the field to select experts. The confinement scale 1 fm of QCD must be introduced without changing the Lagrangian. This can be done via a boundary condition on the gauge field, which affects the bound state potential. The absence of confinement in Feynman diagrams may be due to the free field boundary condition. Poincar\'e invariance is realized dynamically for bound states, i.e., the interactions are frame dependent. Gauge theories have instantaneous interactions, due to gauge fixing at all points of space at the same time. In bound state perturbation theory each order must have exact Poincar\'e invariance. This is non-trivial even for atoms at lowest order. I summarize a perturbative approach to equal time bound states in QED and QCD, using a Fock expansion in temporal ($A^0=0$) gauge. The longitudinal electric field $E_L$ is instantaneous and need not vanish at spatial infinity for the constituents of color singlet states in QCD. Poincar\'e covariance determines the boundary condition for $E_L$ up to a universal scale, characterised by the gluon field energy density of the vacuum. A non-vanishing density contributes a linear term to the $q\bar{q}$ potential, while $qqq,\ q\bar{q}g$ and $gg$ color singlet states get analogous confining potentials.

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arXiv2023

QCD bound states in motion

I consider the frame dependence of QCD bound states in the presence of a confining, spatially constant gluon field energy density. The states are quantized at equal time in $A^0=0$ (temporal) gauge. I derive the frame dependence of the wave functions, and demonstrate the Lorentz covariance of the electromagnetic (transition) form factors for states of any spin. The wave functions of $J^{PC}=0^{-+}$ states with CM momentum $P \neq 0$ are considered in some detail, verifying their local normalizability and the expected frame dependence of the bound state energy.

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arXiv2021

Hadrons as QCD Bound States

Bound state perturbation theory is well established for QED atoms. Today the hyperfine splitting of Positronium is known to $O(α^7\logα)$. Whereas standard expansions of scattering amplitudes start from free states, bound states are expanded around eigenstates of the Hamiltonian including a binding potential. The eigenstate wave functions have all powers of $α$, requiring a choice in the ordering of the perturbative expansion. Temporal $(A^0=0)$ gauge permits an expansion starting from valence Fock states, bound by their instantaneous gauge field. This formulation is applicable in any frame and seems promising even for hadrons in QCD. The $O(α_s^0)$ confining potential is determined (up to a universal scale) by a homogeneous solution of Gauss' law.

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arXiv2021

Journey to the Bound States

Guided by the observed properties of hadrons I formulate a perturbative bound state method for QED and QCD. The expansion starts with valence Fock states ($e^+e^-,\ q\bar q,\ qqq,\ gg$) bound by the instantaneous interaction of temporal gauge ($A^0=0$). The method is tested on Positronium atoms at rest and in motion, including hyperfine splitting at $O(α^4)$, electromagnetic form factors and deep inelastic scattering. Relativistic binding is studied for QED in $D=1+1$ dimensions, demonstrating the frame independence of the DIS electron distribution and its sea for $x_{bj} \to 0$. In QCD a homogeneous solution of Gauss' constraint in $D=3+1$ implies $O(α_s^0)$ confining potentials for $q\bar q,\ q\bar qg,\ qqq$ and $gg$ states, whereas $q\bar q\,q\bar q$ is unconfined. Meson states lie on linear Regge trajectories and have the required frame dependence. A scalar bound state with vanishing four-momentum causes spontaneous chiral symmetry breaking when mixed with the vacuum. These lecture notes assume knowledge of field theory methods, but not of bound states. Brief reviews of existing bound state methods and Dirac electron states are included. Solutions to the exercises are given in the Appendix.

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arXiv2019

Bound states and perturbation theory

A perturbative expansion for QED and QCD bound states is formulated in $A^0=0$ gauge. The constituents of each Fock state are bound by their instantaneous interaction. In QCD an O($α_s^0$) confining potential arises from a homogeneous solution of Gauss' constraint. The potential is uniquely determined by the QCD action, up to a universal scale. The Cornell potential is reproduced for quarkonia, and corresponding ones found for higher Fock states, baryons and glueballs.

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arXiv2019

Bound states and QCD

The similarities of hadrons and atoms motivate a study of the principles of QED bound states and of their applicability to QCD. The power series in $α$ and $\logα$ of the binding energy is reflected in the Fock expansion of the bound state in temporal gauge ($A^0=0$). Gauss' constraint on physical states fixes the gauge for time independent transformations and determines the instantaneous interaction within each Fock state. Positronium atoms generate a classical (dipole) electric field, whereas there can be no color octet gluon field for color singlet hadrons. Hence the gluon field generated by each color component of a hadron need not vanish at spatial infinity. Gauss' constraint has a homogeneous solution with a single parameter $Λ$ that is compatible with Poincaré invariance. The corresponding potential is linear for $q\bar q$ and $gg$ Fock states, and confining also for other states ($q\bar qg,\,qqq$). This approach is consistent with the quarkonium phenomenology based on the Cornell potential at lowest order. The relativistic meson and glueball eigenstates of the QCD Hamiltonian with the $O(α_s^0)$ linear potential are determined. The states lie on linear Regge trajectories and their daughters. There are also massless bound states which allow to include a $J^{PC}=0^{++}$ condensate in the perturbative vacuum, thus breaking chiral symmetry spontaneously.

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arXiv2017

The Born approximation for bound states

Bound states are stationary in time and interact continuously. Even a first approximation of atomic wave functions in QED requires contributions of all orders in α. Bound state perturbation theory depends on the choice of this first approximation, just as the Taylor expansion of an ordinary function depends on the expansion point. Considering the expansion to be not in $α$ but in $\hbar$, i.e., in the number of loops, defines the perturbative expansion uniquely also for bound states. I show how the Schrödinger equation for Positronium with the classical potential $V(r)=-α/r$ corresponds to the Born, $O(\hbar^0)$ bound state approximation in QED. Standard perturbation theory is based on an expansion around $O(α^0)$ free states that have no overlap with bound states. Perturbing around bound states requires using interacting $in$ and $out$ states. For Born states the binding potential arises from a classical gauge field. In the absence of loops the QCD scale $Λ_{QCD}$ can originate from a boundary condition imposed on the solution of the classical gluon field equations. A perturbative expansion may be relevant even for hadrons, if their non-perturbative features such as confinement and chiral symmetry breaking are present already in the Born term.

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arXiv2017

Born Level Bound States

Bound state poles in the $S$-matrix of perturbative QED are generated by the {\em divergence} of the expansion in $α$. The perturbative corrections are necessarily singular when expanding around free, \order{α^0} $in$ and $out$ states that have no overlap with finite-sized atomic wave functions. Nevertheless, measurables such as binding energies do have well-behaved expansions in powers of $α$ (and $\logα$). It is desirable to formulate the concept of "lowest order" for gauge theory bound states such that higher order corrections vanish in the $α\to 0$ limit. This may allow to determine a lowest order term for QCD hadrons which incorporates essential features such as confinement and chiral symmetry breaking, and thus can serve as the starting point of a useful perturbative expansion. I discuss a "Born" (no loop, lowest order in $\hbar$) approximation. Born level states are bound by gauge fields which satisfy the classical field equations. Gauss' law determines a distinct field $A^0(\xv)$ for each instantaneous position of the charges. A Poincaré covariant boundary condition for the gluon field leads to a confining potential for $q\bar q$ and $qqq$ states. In frames where the bound state is in motion the classical gauge field is obtained by a Lorentz boost of the rest frame field.

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arXiv2016

Lectures on Bound states

Even a first approximation of bound states requires contributions of all powers in the coupling. This means that the concept of "lowest order bound state" needs to be defined. In these lectures I discuss the "Born" (no loop, lowest order in $\hbar$) approximation. Born level states are bound by gauge fields which satisfy the classical field equations. As a check of the method, Positronium states of any momentum are determined as eigenstates of the QED Hamiltonian, quantized at equal time. Analogously, states bound by a strong external field $A^μ(\boldsymbol{x})$ are found as eigenstates of the Dirac Hamiltonian. Their Fock states have dynamically created $e^+e^-$ pairs, whose distribution is determined by the Dirac wave function. The linear potential of $D=1+1$ dimensions confines electrons but repels positrons. As a result, the mass spectrum is continuous and the wave functions have features of both bound states and plane waves. The classical solutions of Gauss' law are explored for hadrons in QCD. A non-vanishing boundary condition at spatial infinity generates a constant \order{α_s^0} color electric field between quarks of specific colors. Poincaré invariance limits the spectrum to color singlet $q\bar q$ and $qqq$ states, which do not generate an external color field. This restricts the \order{α_s^0} interactions between hadrons to string breaking dynamics as in dual diagrams. Light mesons lie on linear Regge and parallel daughter trajectories. There are massless states which may be significant for chiral symmetry breaking. Since the bound states are defined at equal time in all frames they have a non-trivial Lorentz covariance.

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arXiv2014

Confinement with Perturbation Theory, after All?

I call attention to the possibility that QCD bound states (hadrons) could be derived using rigorous Hamiltonian, perturbative methods. Solving Gauss' law for $A^0$ with a non-vanishing boundary condition at spatial infinity gives an \order{α_s^0} linear potential for color singlet $q\bar q$ and $qqq$ states. These states are Poincaré and gauge covariant and thus can serve as initial states of a perturbative expansion, replacing the conventional free $in$ and $out$ states. The coupling freezes at $α_s(0)\simeq 0.5$, allowing reasonable convergence. The \order{α_s^0} bound states have a sea of $q\bar q$ pairs, while transverse gluons contribute only at \order{α_s}. Pair creation in the linear $A^0$ potential leads to string breaking and hadron loop corrections. These corrections give finite widths to excited states, as required by unitarity. Several of these features have been verified analytically in $D=1+1$ dimensions, and some in $D=3+1$.

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arXiv2014

Bound states -- from QED to QCD

These lectures are divided into two parts. In Part 1 I discuss bound state topics at the level of a basic course in field theory: The derivation of the Schrödinger and Dirac equations from the QED Lagrangian, by summing Feynman diagrams and in a Hamiltonian framework. Less well known topics include the equal-time wave function of Positronium in motion and the properties of the Dirac wave function for a linear potential. The presentation emphasizes physical aspects and provides the framework for Part 2, which discusses the derivation of relativistic bound states at Born level in QED and QCD. A central aspect is the maintenance of Poincaré invariance. The transformation of the wave function under boosts is studied in detail in D=1+1 dimensions, and its generalization to D=3+1 is indicated. Solving Gauss' law for $A^0$ with a non-vanishing boundary condition leads to a linear potential for QCD mesons, and an analogous confining potential for baryons.

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arXiv2013

Towards a Born term for hadrons

We study bound states of abelian gauge theory in D=1+1 dimensions using an equal-time, Poincare-covariant framework. The normalization of the linear confining potential is determined by a boundary condition in the solution of Gauss' law for the instantaneous A^0 field. As in the case of the Dirac equation, the norm of the relativistic fermion-antifermion (f\bar{f}) wave functions gives inclusive particle densities. However, while the Dirac spectrum is known to be continuous, we find that regular f\bar{f} solutions exist only for discrete bound state masses. The f\bar{f} wave functions are consistent with the parton picture when the kinetic energy of the fermions is large compared to the binding potential. We verify that the electromagnetic form factors of the bound states are gauge invariant and calculate the parton distributions from the transition form factors in the Bjorken limit. For relativistic states we find a large sea contribution at low Bjorken x. Since the potential is independent of the gauge coupling the bound states may serve as "Born terms" in a perturbative expansion, in analogy to the usual plane wave in and out states.

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arXiv2013

Are hadrons simpler than they seem?

I briefly review a systematic approximation scheme of QCD in which the quark model picture of hadrons emerges at lowest order. A linear A^0 potential arises if Gauss' law is solved with a non-vanishing boundary condition at spatial infinity. Similarly to the Dirac case one can describe relativistic states including any number of particle pairs (sea quarks) using valence wave functions, whose norms give {\em inclusive} probability densities. Provided α_s(Q^2) freezes in the infrared, perturbative corrections to the S-matrix can be calculated in the usual way, but with states bound by the linear \order{α_s^0} potential instead of plane waves in the in and out states.

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arXiv2012

Relativistic bound states at Born level

Theoretical and phenomenological studies indicate that the QCD coupling α_s(Q^2) freezes in the infrared. Hadrons may then be described by a perturbative expansion around "Born" states bound only by a confining potential. A linear potential results from the QCD equations of motion when Gauss' law for A^0 is solved with F_{μν}^a F^{μν}_a \neq 0 as boundary condition. The \order{α_s^0} Born states are Poincaré covariant and can serve as \ket{in} and \bra{out} states of scattering amplitudes. Their Dirac-type wave functions include f\bar f creation/annihilation effects giving sea-like partons at low x_bj.

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arXiv2012

Boosting equal time bound states

We present an explicit and exact boost of a relativistic bound state defined at equal time of the constituents in the Born approximation (lowest order in hbar). To this end, we construct the Poincaré generators of QED and QCD in D=1+1 dimensions, using Gauss' law to express A^0 in terms of the fermion fields in A^1=0 gauge. We determine the fermion-antifermion bound states in the Born approximation as eigenstates of the time and space translation generators P^0 and P^1. The boost operator is combined with a gauge transformation so as to maintain the gauge condition A^1=0 in the new frame. We verify that the boosted state remains an eigenstate of P^0 and P^1 with appropriately transformed eigenvalues and determine the transformation law of the equal-time, relativistic wave function. The shape of the wave function is independent of the CM momentum when expressed in terms of a variable, which is quadratically related to the distance x between the fermions. As a consequence, the Lorentz contraction of the wave function is proportional to 1/(E-V(x)) and thus depends on x via the linear potential V(x).

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