Light-front quantization applications
The light-front quantization of quantum field theories provides a useful alternative to ordinary equal-time quantization. In particular, it can lead to a relativistic description of bound systems in terms of quantum-mechanical wave functions. The quantization is based on the choice of light-front coordinates, where plays the role of time and the corresponding spatial coordinate is. Here, is the ordinary time, is a Cartesian coordinate, and is the speed of light. The other two Cartesian coordinates, and, are untouched and often called transverse or perpendicular, denoted by symbols of the type. The choice of the frame of reference where the time and -axis are defined can be left unspecified in an exactly soluble relativistic theory, but in practical calculations some choices may be more suitable than others. The basic formalism is discussed elsewhere.
There are many applications of this technique, some of which are discussed below. Essentially, the analysis of any relativistic quantum system can benefit from the use of light-front coordinates and the associated quantization of the theory that governs the system.
Nuclear reactions
The light-front technique was brought into nuclear physics by the pioneering papers of Frankfurt and Strikman. The emphasis was on using the correct kinematic variables in making correct treatments of high-energy nuclear reactions. This sub-section focuses on only a few examples.Calculations of deep inelastic scattering from nuclei require knowledge of nucleon distribution functions within the nucleus. These functions give the probability that a nucleon of momentum carries a given fraction of the plus component of the nuclear momentum,,.
Nuclear wave functions have been best determined using the equal-time framework. It therefore seems reasonable to see if one could re-calculate nuclear wave functions using the light front formalism. There are several basic nuclear structure problems which must be handled to establish that any given method works. It is necessary to compute the deuteron wave function, solve mean-field theory for infinite nuclear matter and for finite-sized nuclei, and improve the mean-field theory by including the effects of nucleon-nucleon correlations. Much of nuclear physics is based on rotational invariance, but manifest rotational invariance is lost in the light front treatment. Thus recovering rotational invariance is very important for nuclear applications.
The simplest version of each problem has been handled. A light-front treatment of the deuteron was accomplished by Cooke and Miller, which stressed recovering rotational invariance. Mean-field theory for finite nuclei was handled Blunden et al. Infinite nuclear matter was handled within mean-field theory and also including correlations. Applications to deep inelastic scattering were made by Miller and Smith. The principal physics conclusion is that the EMC effect cannot be explained within the framework of conventional nuclear physics. Quark effects are needed. Most of these developments are discussed in a review by Miller.
There is a new appreciation that initial and final-state interaction physics, which is not intrinsic to the hadron or nuclear light-front wave functions, must be addressed in order to understand phenomena such as single-spin asymmetries, diffractive processes, and nuclear shadowing. This motivates extending LFQCD to the theory of reactions and to investigate high-energy collisions of hadrons. Standard scattering theory in Hamiltonian frameworks can provide valuable guidance for developing a LFQCD-based analysis of high-energy reactions.
Exclusive processes
One of the most important areas of application of the light-front formalism are exclusive hadronic processes. "Exclusive processes" are scattering reactions in which the kinematics of the initial state and final state particles are measured and thus completely specified; this is in contrast to "inclusive" reactions where one or more particles in the final state are not directly observed. Prime examples are the elastic and inelastic form factors measured in the exclusive lepton-hadron scattering processes such as In inelastic exclusive processes, the initial and final hadrons can be different, such as. Other examples of exclusive reactions are Compton scattering, pion photoproduction and elastic hadron scattering such as. "Hard exclusive processes" refer to reactions in which at least one hadron scatters to large angles with a significant change in its transverse momentum.Exclusive processes provide a window into the bound-state structure of hadrons in QCD as well as the fundamental processes which control hadron dynamics at the amplitude level. The natural calculus for describing the bound-state structure of relativistic composite systems, needed for describing exclusive amplitudes, is the light-front Fock expansion which encodes the multi-quark, gluonic, and color correlations of a hadron in terms of frame-independent wave functions. In hard exclusive processes, in which hadrons receive a large momentum transfer, perturbative QCD leads to factorization theorems which separate the physics of hadronic bound-state structure from that of the relevant quark and gluonic hard-scattering reactions which underlie these reactions. At leading twist, the bound-state physics is encoded in terms of universal "distribution amplitudes", the fundamental theoretical quantities which describe the valence quark substructure of hadrons as well as nuclei. Nonperturbative methods, such as AdS/QCD, Bethe–Salpeter methods, discretized light-cone quantization, and transverse lattice methods, are now providing nonperturbative predictions for the pion distribution amplitude. A basic feature of the gauge theory formalism is color transparency", the absence of initial and final-state interactions of rapidly moving compact color-singlet states. Other applications of the exclusive factorization analysis include semileptonic meson decays and deeply virtual Compton scattering, as well as dynamical higher-twist effects in inclusive reactions. Exclusive processes place important constraints on the light-front wave functions of hadrons in terms of their quark and gluon degrees of freedom as well as the composition of nuclei in terms of their nucleon and mesonic degrees of freedom.
The form factors measured in the exclusive reaction encode the deviations from unity of the scattering amplitude due to the hadron's compositeness. Hadronic form factors fall monotonically with spacelike momentum transfer, since the amplitude for the hadron to remain intact continually decreases. One can also distinguish experimentally whether the spin orientation of a hadron such as the spin-1/2 proton changes during the scattering or remains the same, as in the Pauli and Dirac form factors.
The electromagnetic form factors of hadrons are given by matrix elements of the electromagnetic current such as where is the momentum four-vector of the exchanged virtual photon and is the eigenstate for hadron with four momentum. It is convenient to choose the light-front frame where with The elastic and inelastic form factors can then be expressed as integrated overlaps of the light-front Fock eigenstate wave functions and of the initial and final-state hadrons, respectively. The of the struck quark is unchanged, and. The unstruck quarks have. The result of the convolution gives the form factor exactly for all momentum transfer when one sums over all Fock states of the hadron. The frame choice is chosen since it eliminates off-diagonal contributions where the number of initial and final state particles differ; it was originally discovered by Drell and Yan and by West. The rigorous formulation in terms of light-front wave functions is given by Brodsky and Drell.
Light-front wave functions are frame-independent, in contrast to ordinary instant form wave functions which need to be boosted from to, a difficult dynamical problem, as emphasized by Dirac. Worse, one must include contributions to the current matrix element where the external photon interacts with connected currents arising from vacuum fluctuations in order to obtain the correct frame-independent result. Such vacuum contributions do not arise in the light-front formalism, because all physical lines have positive ; the vacuum has only, and momentum is conserved.
At large momentum transfers, the elastic helicity-conserving form factors fall-off as the nominal power where is the minimum number of constituents. For example, for the three-quark Fock state of the proton. This "quark counting rule" or "dimensional counting rule" holds for theories such as QCD in which the interactions in the Lagrangian are scale invariant. This result is a consequence of the fact that form factors at large momentum transfer are controlled by the short distance behavior of the hadron's wave function which in turn is controlled by the "twist" of the leading interpolating operator which can create the hadron at zero separation of the constituents. The rule can be generalized to give the power-law fall-off of inelastic form factors and form factors in which the hadron spin changes between the initial and final states. It can be derived nonperturbatively using gauge/string theory duality and with logarithmic corrections from perturbative QCD.
In the case of elastic scattering amplitudes, such as, the dominant physical mechanism at large momentum transfer is the exchange of the quark between the kaon and the proton. This amplitude can be written as a convolution of the four initial and final state light-front valence Fock-state wave functions. It is convenient to express the amplitude in terms of Mandelstam variables, where, for a reaction with momenta, the variables are. The resulting "quark interchange" amplitude has the leading form which agrees well with the angular dependence and power law fall-off of the amplitude with momentum transfer at fixed CM angle. The behavior of the amplitude, at fixed but large momentum transfer squared, shows that the intercept of Regge amplitudes at large negative. The nominal power-law fall-off of the resulting hard exclusive scattering cross section for at fixed CM angle is consistent with the dimensional counting rule for hard elastic scattering, where is the minimum number of constituents.
More generally, the amplitude for a hard exclusive reaction in QCD can be factorized at leading power as a product of the hard-scattering subprocess quark scattering amplitude, where the hadrons are each replaced with their constituent valence quarks or gluons, with their respective light-front momenta, convoluted with the "distribution amplitude" for each initial and final hadron. The hard-scattering amplitude can then be computed systematically in perturbative QCD from the fundamental quark and gluon interactions of QCD. This factorization procedure can be carried out systematically since the effective QCD running coupling becomes small at high momentum transfer, because of the asymptotic freedom property of QCD.
The physics of each hadron enters through its distribution amplitudes, which specifies the partitioning of the light-front momenta of the valence constituents. It is given in light-cone gauge as, the integral of the valence light-front wave function over the internal transverse momentum squared ; the upper limit is the characteristic transverse momentum in the exclusive reaction. The logarithmic evolution of the distribution amplitude in is given rigorously in perturbative QCD by the ERBL evolution equation. The results are also consistent with general principles such as the renormalization group. The asymptotic behavior of the distribution such as where is the decay constant measured in pion decay can also be determined from first principles. The nonperturbative form of the hadron light-front wave function and distribution amplitude can be determined from AdS/QCD using light-front holography. The deuteron distribution amplitude has five components corresponding to the five different color-singlet combinations of six color triplet quarks, only one of which is the standard nuclear physics product of two color singlets. It obeys a evolution equation leading to equal weighting of the five components of the deuteron's light-front wave function components at The new degrees of freedom are called "hidden color". Each hadron emitted from a hard exclusive reaction emerges with high momentum and small transverse size. A fundamental feature of gauge theory is that soft gluons decouple from the small color-dipole moment of the compact fast-moving color-singlet wave function configurations of the incident and final-state hadrons. The transversely compact color-singlet configurations can persist over a distance of order, the Ioffe coherence length. Thus, if we study hard quasi elastic processes in a nuclear target, the outgoing and ingoing hadrons will have minimal absorption - a novel phenomenon called "color transparency". This implies that quasi-elastic hadron-nucleon scattering at large momentum transfer can occur additively on all of the nucleons in a nucleus with minimal attenuation due to elastic or inelastic final state interactions in the nucleus, i.e. the nucleus becomes transparent. In contrast, in conventional Glauber scattering, one predicts nearly energy-independent initial and final-state attenuation. Color transparency has been verified in many hard-scattering exclusive experiments, particularly in the diffractive dijet experiment at Fermilab. This experiment also provides a measurement of the pion's light-front valence wave function from the observed and transverse momentum dependence of the produced dijets.