Derivation Of Nonlocal Chiral Lagrangian And IR-UV Duality For Dynamical Equations

As a powerful low energy effective theory of non-perturbative QCD physics,chi-ral effective Lagrangian originates from a generalization of the nonlinear a model targeting with the coset space of chiral spontaneous breaking.The only degrees of freedom for the original chiral Lagrangian are pseudo Nambu-Goldstone bosons,namely pseudo scalar mesons staying at the lowest level of hadron spectrums.As the UV cut-off of the effective field theories increases,more and more higher resonances and topological defects are involved,such as vector mesons and baryons.There are two methods to include vector mesons into chiral Lagrangian,the hid-den local symmetry model and 2-form matter field model.Since the former one exhibits a better convergence behavior and coincides with the phenomenological re-sults,it is more favored in the literatures.As for the baryon-included effective theories,nonrelativistic and relativistic models are proposed in various ways.In the large Nc limit,many-body quantum mechanics and Skyrme model both give the self-consistent Nc-counting rules for baryonic processes,while neither of them is Lorentz covariant.Applying functional derivation or graphs induction,one can always get quantum equations of motion,such as Dyson-Schwinger equations for correlation functions of fundamental fields and Bethe-Salpeter equations for mesonic bound-state ampli-tudes.Quantum equations of motion root in the first principle and therefore cover all the dynamics information from infrared to ultraviolet regimes.However in gener-al,they form an infinitely coupled system of equations and therefore are hard to be solved.It seems that a truncation approximation is necessary for a specific solution process.We start from QCD and finally derive a nonlocal effective Lagrangian with the help of functional techniques and large Nc limit approximation.Our result satisfies Lorentz covariance,chiral invariance and an extra hidden local symmetry,and fur-thermore permits the coexistence of nonlocal and local bound-states.If the nonlocal degrees of freedom are valued on shell,then the local chiral Lagrangian with known low energy coefficients is obtained.The competition of nonlocal and local degrees of freedom is expected to describe a transition analogous to BCS-BEC crossover.Our nonlocal chiral Lagrangian is semi-classical in the limit of large Nc,and its equations of motion reproduce Dyson-Schwinger equations for quark's propagator,Bethe-Salpeter equation for meson,and Faddeev equations for baryon and anti-baryon.Especially,the later two involve both the nonlocal and local fields.This deep connection between chiral Lagrangian and QCD quantum equations of motion can be thought as an IR-UV duality for hadronic dynamicsAs a bonus,our derivation proposes a constraint between mesonic and baryon-ic bound-state amplitudes.This implies that baryons and mesons with the same amount of constituent quarks and anti-quarks exhibit some equivalence in the large Nc limit,which can be understood as an ambiguity in defining global composite ex-citation in strong correlated systemsIn addition,various Nc counting rules from our result are checked and fixed,co-inciding with the traditional analysis from double-line graphic counting for meosons and many-body Hatree-Fock approximation for baryons.