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Register a new userColor Confinement and Supersymmetric Properties of Hadron Physics from Light-Front Holography
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Stanley J. Brodsky
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I review applications of superconformal algebra. light-front holography, and an extended form of conformal symmetry to hadron spectroscopy and dynamics. QCD is not supersymmetrical in the traditional sense -- the QCD Lagrangian is based on quark and gluonic fields -- not squarks nor gluinos. However, its hadronic eigensolutions conform to a representation of superconformal algebra. and provide a unified Regge spectroscopy of meson, baryon, and tetraquarks with a universal Regge slope. The pion $q bar q$ eigenstate is composite but yet has zero mass for $m_q=0.$ Light-Front Holography also predicts the form of the nonperturbative QCD running coupling in agreement with the effective charge determined from measurements of the Bjorken sum rule. One also obtains viable predictions for hadron dynamics such as spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions. The combined approach of light-front holography and superconformal algebra also provides insight into the origin of the QCD mass scale and color confinement. A key tool is the dAFF principle which shows how a mass scale can appear in the Hamiltonian and the equations of motion while retaining the conformal symmetry of the action. When one applies the dAFF procedure to chiral QCD, a mass scale $kappa$ appears which determines the hadron masses in the absence of the Higgs coupling. The result is an extended conformal symmetry which has a conformally invariant action even though an underlying mass scale appears in the Hamiltonian. Although conformal symmetry is strongly broken by the heavy quark mass, the supersymmetric mechanism, which transforms mesons to baryons (and baryons to tetraquarks), still holds and gives remarkable mass degeneracies across the spectrum of light, heavy-light and double-heavy hadrons.
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The QCD light-front Hamitonian equation derived from quantization at fixed LF time provides a causal, frame-independent, method for computing hadron spectroscopy and dynamical observables. de Alfaro, Fubini, and Furlan (dAFF) have made an important observation that a mass scale can appear in the equations of motion without affecting the conformal invariance of the action if one adds a term to the Hamiltonian proportional to the dilatation operator or the special conformal operator. If one applies the dAFF procedure to the QCD light-front Hamiltonian, it leads to a color confining potential $kappa^4 zeta^2$ for mesons, where $zeta^2$ is the LF radial variable conjugate to the $q bar q$ invariant mass squared. The same result, including spin terms, is obtained using light-front holography if one modifies the AdS$_5$ action by the dilaton $e^{kappa^2 z^2}$ in the fifth dimension $z$. When one generalizes this procedure using superconformal algebra, the resulting light-front eigensolutions provide a unified Regge spectroscopy of meson, baryon, and tetraquarks, including remarkable supersymmetric relations between the masses of mesons and baryons and a universal Regge slope. The pion $q bar q$ eigenstate has zero mass at $m_q=0.$ The superconformal relations also can be extended to heavy-light quark mesons and baryons. AdS/QCD also predicts the analytic form of the nonperturbative running coupling in agreement with the effective charge measured from measurements of the Bjorken sum rule. The mass scale underlying hadron masses can be connected to the mass parameter in the QCD running coupling. The result is an effective coupling $alpha_s(Q^2)$ defined at all momenta. One also obtains empirically viable predictions for spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions.
Light-Front Quantization provides a physical, frame-independent formalism for hadron dynamics and structure. Observables such as structure functions, transverse momentum distributions, and distribution amplitudes are defined from the hadronic light-front wavefunctions. One obtains new insights into the hadronic spectrum, light-front wavefunctions, and the functional form of the QCD running coupling in the nonperturbative domain using light-front holography -- the duality between the front form and AdS$_5$, the space of isometries of the conformal group. In addition, superconformal algebra leads to remarkable supersymmetric relations between mesons and baryons of the same parity. The mass scale $kappa$ underlying confinement and hadron masses can be connected to the parameter $Lambda_{overline {MS}}$ in the QCD running coupling by matching the nonperturbative dynamics, as described by the effective conformal theory mapped to the light-front and its embedding in AdS space, to the perturbative QCD regime. The result is an effective coupling defined at all momenta. This matching of the high and low momentum transfer regimes determines a scale $Q_0$ which sets the interface between perturbative and nonperturbative hadron dynamics. The use of $Q_0$ to resolve the factorization scale uncertainty for structure functions and distribution amplitudes, in combination with the principle of maximal conformality (PMC) for setting the renormalization scales, can greatly improve the precision of perturbative QCD predictions for collider phenomenology. The absence of vacuum excitations of the front-form vacuum has important consequences for the cosmological constant. I also discuss evidence that the antishadowing of nuclear structure functions is flavor dependent, and why shadowing and antishadowing phenomena may be incompatible with sum rules for nuclear parton distribution functions.
A primary question in hadron physics is how the mass scale for hadrons consisting of light quarks, such as the proton, emerges from the QCD Lagrangian even in the limit of zero quark mass. If one requires the effective action which underlies the QCD Lagrangian to remain conformally invariant and extends the formalism of de Alfaro, Fubini and Furlan to light-front Hamiltonian theory, then a unique, color-confining potential with a mass parameter $kappa$ emerges. The actual value of the parameter $kappa$ is not set by the model - only ratios of hadron masses and other hadronic mass scales are predicted. The result is a nonperturbative, relativistic light-front quantum mechanical wave equation, the Light-Front Schrodinger Equation which incorporates color confinement and other essential spectroscopic and dynamical features of hadron physics, including a massless pion for zero quark mass and linear Regge trajectories with the identical slope in the radial quantum number $n$ and orbital angular momentum $L$. The same light-front equations for mesons with spin $J$ also can be derived from the holographic mapping to QCD (3+1) at fixed light-front time from the soft-wall model modification of AdS$_5$ space with a specific dilaton profile. Light-front holography thus provides a precise relation between the bound-state amplitudes in the fifth dimension of AdS space and the boost-invariant light-front wavefunctions describing the internal structure of hadrons in physical space-time. One can also extend the analysis to baryons using superconformal algebra - $2 times 2$ supersymmetric representations of the conformal group. The resulting fermionic LF bound-state equations predict striking similarities between the meson and baryon spectra. In fact, the holographic QCD light-front Hamiltonians for the states on the meson and baryon trajectories are identical if one shifts the internal angular...
A remarkable feature of QCD is that the mass scale which controls color confinement and hadron mass scales does not appear explicitly in the QCD Lagrangian. However, de Alfaro, Fubini, and Furlan have shown that a mass scale can appear in the equations of motion without affecting the conformal invariance of the action if one adds a term to the Hamiltonian proportional to the dilatation operator or the special conformal operator. Applying the same procedure to the light-front Hamiltonian leads to a unique confinement potential $kappa^4 zeta^2$ for mesons, where $zeta$ is the LF radial variable conjugate to the invariant mass. The same result, including spin terms, is obtained using light-front holography, the duality between the front form and AdS$_5,$ if one modifies the action by the dilaton $e^{kappa^2 z^2}$ in the fifth dimension $z$. Generalizing this procedure using superconformal algebra, leads to a unified Regge spectroscopy of meson, baryon, and tetraquarks, including remarkable supersymmetric relations between the masses of mesons and baryons of the same parity. One also predicts observables such as hadron structure functions, transverse momentum distributions, and the distribution amplitudes defined from the hadronic light-front wavefunctions. The mass scale underlying confinement and hadron masses can be connected to the mass parameter in the QCD running coupling by matching the nonperturbative dynamics to the perturbative QCD regime. The result is an effective coupling defined at all momenta and the determination of a momentum scale which sets the interface between perturbative and nonperturbative hadron dynamics. I also discuss evidence that the antishadowing of nuclear structure functions is non-universal, and why shadowing and antishadowing phenomena may be incompatible with sum rules for nuclear parton distribution functions.
QCD is not supersymmetrical in the traditional sense -- the QCD Lagrangian is based on quark and gluonic fields, not squarks nor gluinos. However, its hadronic eigensolutions conform to a representation of superconformal algebra, reflecting the underlying conformal symmetry of chiral QCD and its Pauli matrix representation. The eigensolutions of superconformal algebra provide a unified Regge spectroscopy of meson, baryon, and tetraquarks in the same 4-plet representation with a universal Regge slope. The pion $q bar q$ eigenstate has zero mass for $m_q=0.$ The superconformal relations also can be extended to heavy-light quark mesons and baryons. The combined approach of light-front holography and superconformal algebra also provides insight into the origin of the QCD mass scale and color confinement. A key observation is the remarkable dAFF principle which shows how a mass scale can appear in the Hamiltonian and the equations of motion while retaining the conformal symmetry of the action. When one applies the dAFF procedure to chiral QCD, a mass scale $kappa$ appears which determines universal Regge slopes, hadron masses in the absence of the Higgs coupling, and the mass parameter underlying the form of the nonperturbative QCD running coupling: $alpha_s(Q^2) propto exp{-{Q^2/4 kappa^2}}$, in agreement with the effective charge determined from measurements of the Bjorken sum rule. The mass scale $kappa$ underlying hadron masses can be connected to the parameter $Lambda_{overline {MS}}$ in the QCD running coupling by matching its predicted nonperturbative form to the perturbative QCD regime. One also obtains predictions for spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions.
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