QCD Lagrangian Research Articles

Large Nc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_c$$\\end{document} QCD phase diagram at μB=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu _B=0$$\\end{document}

Lattice studies suggest that at zero baryon chemical potential and increasing temperature there are three characteristic regimes in QCD that are connected by smooth analytical crossovers: a hadron gas regime at T<Tch∼155\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T < T_{ch}\\sim 155$$\\end{document} MeV, an intermediate regime, called stringy fluid, at Tch<T<∼3Tch\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{ch}< T < \\sim 3 T_{ch}$$\\end{document}, and a quark-gluon plasma regime at higher temperatures. These regimes have been interpreted to reflect different approximate symmetries and effective degrees of freedom. In the hadron gas the effective degrees of freedom are hadrons and the approximate chiral symmetry of QCD is spontaneously broken. The intermediate regime has been interpreted as lacking spontaneous chiral symmetry breaking along with the emergence of new approximate symmetry, chiral spin symmetry, that is not a symmetry of the Dirac Lagrangian, but is a symmetry of the confining part of the QCD Lagrangian. While the high temperature regime is the usual quark-gluon plasma which is often considered to reflect “deconfinement” in some way. This paper explores the behavior of these regimes of QCD as the number of colors in the theory, Nc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_c$$\\end{document}, gets large. In the large Nc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_c$$\\end{document} limit the theory is center-symmetric and notions of confinement and deconfinement are unambiguous. The energy density is O(Nc0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(N_c^0)$$\\end{document} in the meson gas, O(Nc1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {O}}}(N_c^1)$$\\end{document} in the intermediate regime and O(Nc2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {O}}}(N_c^2)$$\\end{document} in the quark-gluon plasma regime. In the large Nc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_c$$\\end{document} limit these regimes may become distinct phases separated by first order phase transitions. The intermediate phase has the peculiar feature that glueballs should exist and have properties that are unchanged from what is seen in the vacuum (up to 1/Nc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/N_c $$\\end{document} corrections), while the ordinary dilute gas of mesons with broken chiral symmetry disappears and approximate chiral spin symmetry should emerge.

Four-gluon vertex in collinear kinematics

To date, the four-gluon vertex is the least explored component of the QCD Lagrangian, mainly due to the vast proliferation of Lorentz and color structures required for its description. In this work we present a nonperturbative study of this vertex, based on the one-loop dressed Schwinger–Dyson equation obtained from the 4PI effective action. A vast simplification is brought about by resorting to “collinear” kinematics, where all momenta are parallel to each other, and by appealing to the charge conjugation symmetry in order to eliminate certain color structures. Out of the fifteen form factors that comprise the transversely-projected version of this vertex, two are singled out and studied in detail; the one associated with the classical tensorial structure is moderately suppressed in the infrared regime, while the other diverges logarithmically at the origin. Quite interestingly, both form factors display the property known as “planar degeneracy” at a rather high level of accuracy. With these results we construct an effective charge that quantifies the strength of the four-gluon interaction, and compare it with other vertex-derived charges from the gauge sector of QCD.

K± → π±a at next-to-leading order in chiral perturbation theory and updated bounds on ALP couplings

The weak decays K± → π±a offer a powerful probe of axion-like particles (ALPs). In this work, we provide a comprehensive analysis of these processes within chiral perturbation theory, extending existing calculations by including complete next-to-leading order (NLO) contributions and isospin-breaking corrections at first order in (md – mu). We show that the consistent incorporation of ALPs in the QCD and weak chiral Lagrangians requires a non-trivial extension of the corresponding operator bases, which we describe in detail. Furthermore, we show that in the presence of an ALP the so-called “weak mass term”, which is unobservable in the Standard Model, is non-redundant already at leading order. We find that NLO corrections associated with flavor-violating ALP couplings modify the leading-order result by a few percent, with negligible uncertainties. NLO corrections proportional to flavor-conserving ALP couplings lead to potentially larger corrections, which, however, are accompanied by sizable uncertainties mainly due to the currently limited knowledge of various low-energy constants. We study how these corrections impact bounds on the ALP couplings, first model independently, and then specializing to the case of an ALP with flavor-universal couplings in the UV. Our findings confirm that the decays K± → π±a provide the strongest particle-physics constraints for ma ≲ 300 MeV. In addition, we point out that these bounds have interesting implications for the ALP couplings to nucleons, which were so far only constrained by astrophysical measurements and non-accelerator experiments.

Chiral symmetry restoration at finite temperature in a model with manifest confinement

Multiple lattice evidences support the existence of a confining but chirally symmetric regime of QCD above the chiral symmetry restoration crossover at Tch≃155 MeV. This regime is characterised by an approximate chiral spin symmetry of the partition function, which is a symmetry of the colour charge and the confining electric part of the QCD Lagrangian. It is traditionally believed that confinement should automatically induce spontaneous breaking of chiral symmetry, which would preclude the existence of a confining but chirally symmetric regime of QCD at high temperatures. We employ a well-known solvable quark model for QCD in 3+1 dimensions that is chirally symmetric and manifestly confining and argue that while confinement indeed induces dynamical breaking of chiral symmetry at T=0, a chiral restoration phase transition takes place at some critical temperature Tch. Above this temperature, the spectrum of the model consists of chirally symmetric hadrons with approximate chiral spin symmetry.

Heavy axions from twin dark sectors with [formula omitted]-characterized mirror symmetry

The QCD Lagrangian contains a CP violating gluon density term with a physical coefficient θ¯. The upper bound on the electric dipole moment of neutron requires the value of θ¯ to be extremely small. The tiny θ¯ is commonly known as the strong CP problem. In order to solve this puzzle, we construct a θ¯-characterized mirror symmetry between a pair of twin dark sectors with respective discrete symmetries. By taking a proper phase rotation of dark fields, we can perfectly remove the parameter θ¯ from the full Lagrangian. In our scenario, the discrete symmetry breaking, which are responsible for the mass generation of dark colored fermions and dark matter fermions, can be allowed near the TeV scale. This means different phenomena from the popular axion models with high scale Peccei-Quinn global symmetry breaking.

A New Solution to the Strong CP Problem

We suggest a new solution to the strong CP problem. The solution is based on the proper use of the boundary conditions for the QCD-generating functional integral. We expand the perturbative boundary conditions to both perturbative and nonperturbative fields integrated into the QCD-generating functional integral. It allows us to nullify the CP odd term in the QCD Lagrangian and, thus, to solve the strong CP problem. The presently popular solution to the strong CP problem of introducing axions violates the principle of renormalizability of the Quantum Field Theory, which is very successful phenomenologically. Our solution obeys the principle of renormalizability of the Quantum Field Theory and does not involve new exotic particles like axions.

Solving the strong CP problem via a -characterized mirror symmetry* *Supported in part by the National Natural Science Foundation of China (12175038) and in part by the Fundamental Research Funds for the Central Universities

In the standard model QCD Lagrangian, a term of CP violating gluon density is theoretically expected to have a physical coefficient , which is typically on the order of unity. However, the upper bound on the electric dipole moment of the neutron enforces the value of to be extremely small. The significant discrepancy between theoretical expectations and experimental results in this context is widely recognized as the strong CP problem. To solve this puzzle in an appealing context of two Higgs doublets, we propose a -characterized mirror symmetry between two Higgs singlets with respective discrete symmetries. In our scenario, the parameter can completely disappear from the full Lagrangian after the standard model fermions take a proper phase rotation as well as the Higgs doublets and singlets. Moreover, all of new physics for solving the strong CP problem can be allowed near the TeV scale.

Rotationally Symmetric Yang-Mills Gauge Theory and Generalized Uncertainty Principle

Recently, it has been of considerably high interest to introduce the notion of minimal length in quantum mechanics and quantum field theories to get a proper description of quantum gravity. In this paper, we have attempted to unify the notion of minimal length with R×S^3 topological field theories which were introduced by Carmeli and Malin in 1985 [14]. We have collectively called this (R×S^3 )_H topological field theories where the “H” in the sub-script stands for Heisenberg’s notion of minimal length. A proper description for equations like Schrodinger’s equation, Klein-Gordon equation and QED, QCD Lagrangian on (R×S^3 )_H topology is derived. HIGHLIGHTS It has been of considerably high interest to incorporate existing theories with the notion of minimal length because it is thought to unify theories at large scale with the quantum world In this paper, we have done a similar unification by incorporating Rotaionally symmetric field theories with the notion of minimal length The derived equations work consistently under local and global gauge transformations

Emergence of mass in the gauge sector of QCD* *Supported by the grant PID2020-113334GB-I00/AEI/10.13039/501100011033 of the Spanish AEI-MICINN, and the grant Prometeo/2019/087 of the Generalitat Valenciana

It is currently widely accepted that gluons, while massless at the level of the fundamental QCD Lagrangian, acquire an effective mass through the non-Abelian implementation of the classic Schwinger mechanism. The key dynamical ingredient that triggers the onset of this mechanism is the formation of composite massless poles inside the fundamental vertices of the theory. These poles enter the evolution equation of the gluon propagator and nontrivially affect the way the Slavnov-Taylor identities of the vertices are resolved, inducing a smoking-gun displacement in the corresponding Ward identities. In this article, we present a comprehensive review of the pivotal concepts associated with this dynamical scenario, emphasizing the synergy between functional methods and lattice simulations and highlighting recent advances that corroborate the action of the Schwinger mechanism in QCD.

Complex paths around the sign problem

A promising path to solving QCD is on a computer by discretizing spacetime, rewriting the QCD Lagrangian to fit in that discretization, and then taking the appropriate continuum and infinite volume limits. A problem with this method is that the calculation of certain quantities of interest is numerically intractable because of a ``sign problem.'' A sign problem appears whenever large cancellations make it impractical to obtain reliable numerical results. This review discusses a potential solution: deformation of the manifold on which the problem is defined in order to reduce these instabilities.

Double copy for Lagrangians at trilinear order

We present a novel double-copy prescription for gauge fields at the Lagrangian level and apply it to the original double copy, couplings to matter and the soft theorem. The Yang-Mills Lagrangian in light-cone gauge is mapped directly to the mathcal{N} = 0 supergravity Lagrangian in light-cone gauge to trilinear order, and we show that the obtained result is manifestly equivalent to Einstein gravity at tree level up to this order. The application of the double-copy prescription to couplings to matter is exemplified by scalar and fermionic QCD and finally the soft-collinear effective QCD Lagrangian. The mapping of the latter yields an effective description of an energetic Dirac fermion coupled to the graviton, Kalb-Ramond, and dilaton fields, from which the fermionic gravitational soft and next-to-soft theorems follow.

Three regimes of QCD

While the QCD Lagrangian as the whole is only chirally symmetric, its electric part has larger chiral-spin [Formula: see text] and [Formula: see text] symmetries. This allows separation of the electric and magnetic interactions in a given reference frame. Artificial truncation of the near-zero modes of the Dirac operator results in the emergence of the [Formula: see text] and [Formula: see text] symmetries in hadron spectrum. This implies that while the confining electric interaction is distributed among all modes of the Dirac operator, the magnetic interaction is located at least predominantly in the near-zero modes. Given this observation one could anticipate that above the pseudocritical temperature, where the near-zero modes of the Dirac operator are suppressed, QCD is [Formula: see text] and [Formula: see text] symmetric, which means absence of deconfinement in this regime. Solution of the [Formula: see text] QCD on the lattice with a chirally symmetric Dirac operator reveals that indeed in the interval [Formula: see text] QCD is approximately [Formula: see text] and [Formula: see text] symmetric which implies that degrees of freedom are chirally symmetric quarks bound by the chromoelectric field into color-singlet objects without the chromomagnetic effects. This regime is referred to as a Stringy Fluid. At larger temperatures this emergent symmetry smoothly disappears and QCD approaches the Quark–Gluon Plasma regime with quasifree quarks. The Hadron Gas, the Stringy Fluid and the Quark–Gluon Plasma differ by symmetries, degrees of freedom and properties.

Conformal invariance of the one-loop all-plus helicity scattering amplitudes

The massless QCD Lagrangian is conformally invariant and, as a consequence, so are the tree-level scattering amplitudes. However, the implications of this powerful symmetry at loop level are only beginning to be explored systematically. Even for finite loop amplitudes, the way conformal symmetry manifests itself may be subtle, e.g. in the form of anomalous conformal Ward identities. As they are finite and rational, the one-loop all-plus and single-minus amplitudes are a natural first step towards understanding the conformal properties of Yang-Mills theory at loop level. Remarkably, we find that the one-loop all-plus amplitudes are conformally invariant, whereas the single-minus are not. Moreover, we present a formula for the one-loop all-plus amplitudes where the symmetry is manifest term by term. Surprisingly, each term transforms covariantly under directional dual conformal variations. We prove the formula directly using recursive techniques, and check that it has the correct physical factorisations.

Anomaly matching in QCD thermal phase transition

We study an ’t Hooft anomaly of massless QCD at finite temperature. With the imaginary baryon chemical potential at the Roberge-Weiss point, there is a ℤ2 symmetry which can be used to define confinement. We show the existence of a mixed anomaly between the ℤ2 symmetry and the chiral symmetry, which gives a strong relation between confinement and chiral symmetry breaking. The anomaly is a parity anomaly in the QCD Lagrangian reduced to three dimensions. It is reproduced in the chiral Lagrangian by a topological term related to Skyrmion charge, matching the anomaly before and after QCD phase transition. The effect of the imaginary chemical potential is suppresssed in the large N expansion, and we discuss implications of the ’t Hooft anomaly matching for the nature of QCD phase transition with and without the imaginary chemical potential. Arguments based on universality alone are disfavored, and a first order phase transition may be the simplest possibility if the large N expansion is qualitatively good.

Color Confinement and Supersymmetric Properties of Hadron Physics from Light-Front Holography

QCD is not supersymmetrical in the traditional sense – the QCD Lagrangian is based on quark and gluonic fields – not squarks or 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 of the same parity and twist as equal-mass members of the same 4-plet representation with a universal Regge slope. The pion eigenstate has zero mass for mq = 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 tool 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 ӄ appears which determines universal Regge slopes, hadron masses in the absence of the Higgs coupling. The predictions from light-front holography and superconformal algebra can also be extended to mesons, baryons, and tetraquarks with strange, charm and bottom quarks. Although conformal symmetry is strongly broken by the heavy quark mass, the basic underlying supersymmetric mechanism, which transforms mesons to baryons (and baryons to tetraquarks), still holds and gives remarkable mass degeneracy across the entire spectrum of light, heavy-light and double-heavy hadrons. One also predicts the form of the nonperturbative QCD running coupling: αs(Q2) ∝ e−Q2/4ӄ2, in agreement with the effective charge determined from measurements of the Bjorken sum rule. One also obtains viable predictions for spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions.

Supersymmetric and Conformal Features of Hadron Physics

The QCD Lagrangian is based on quark and gluonic fields—not squarks nor gluinos. However, one can show that its hadronic eigensolutions conform to a representation of superconformal algebra, reflecting the underlying conformal symmetry of chiral QCD. The eigensolutions of superconformal algebra provide a unified Regge spectroscopy of meson, baryon, and tetraquarks of the same parity and twist as equal-mass members of the same 4-plet representation with a universal Regge slope. The predictions from light-front holography and superconformal algebra can also be extended to mesons, baryons, and tetraquarks with strange, charm and bottom quarks. The pion q q ¯ eigenstate has zero mass for m q = 0 . A key tool is the remarkable observation of de Alfaro, Fubini, and Furlan (dAFF) 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 κ appears which determines universal Regge slopes, hadron masses in the absence of the Higgs coupling. One also predicts the form of the nonperturbative QCD running coupling: α s ( Q 2 ) ∝ e − Q 2 / 4 κ 2 , in agreement with the effective charge determined from measurements of the Bjorken sum rule. One also obtains viable predictions for spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions. The combination of conformal symmetry, light-front dynamics, its holographic mapping to AdS 5 space, and the dAFF procedure thus provide new insights, not only into the physics underlying color confinement, but also the nonperturbative QCD coupling and the QCD mass scale.

Electromagnetic Multipole Moments of Baryons

We calculate the charge quadrupole and magnetic octupole moments of baryons using a group theoretical approach based on broken SU(6) spin-flavor symmetry. The latter is an approximate symmetry of the QCD Lagrangian which becomes exact in the large color N_c limit. Spin-flavor symmetry breaking is induced by one-, two-, and three-quark terms in the electromagnetic current operator. Two- and three-quark currents provide the leading contributions for higher multipole moments, despite being of higher order in an 1/N_c expansion. Our formalism leads to relations between N --> N* transition multipole moments and nucleon ground state properties. We compare our results to experimental quadrupole and octupole transition moments extracted from measured helicity amplitudes.

Implications of Dilaton Couplings on the Axion Potential

In particle physics and cosmology, the dilaton is an hypothetical scalar field which can explain many physical phenomena. In this framework, we investigate an extended lagrangian of QCD which involves dilatonic degrees of freedom. Our approach is based on the relationship between the massive dilaton and the nonperturbative effects of QCD. We derive a general axion potential involving the dilaton properties and vacuum condensates.

Quarkonium radiative decays from the hadronic Paschen-Back effect

We study the radiative (E1 and M1) decays of P-wave quarkonia in a strong magnetic field based on the Lagrangian of potential nonrelativistic QCD. To investigate their properties, we implement a polarized wave function basis justified in the Paschen-Back limit. In a magnetic field stronger than the spin-orbit coupling, the wave functions of the P-wave quarkonia are drastically deformed by the Hadronic Paschen-Back effect. Such deformation leads to the anisotropy of the direction of decays from the P-wave quarkonia. The analytic formulas for the radiative decay widths in the nonrelativistic limit are shown, and the qualitative decay properties are discussed.

Chiralspin symmetry and QCD at high temperature

.It has been found very recently on the lattice that at high temperature at vanishing chemical potential QCD is increasingly SU(2)_{CS} and SU(2N_{F}) symmetric. We demonstrate that the chemical potential term in the QCD Lagrangian has precisely the same symmetry. Consequently the QCD matter beyond the chiral restoration line at high temperature on the T -μ plane is at least approximately SU(2)_{CS} and SU(2N_{F}) symmetric.