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Do gluons carry half of the nucleon momentum? X.S.Chen, X.F.Lu Dept. of Phys., Sichuan Univ. W.M.Sun, Fan Wang NJU and PMO Joint Center for Particle Nuclear Physics and Cosmology (J-CPNPC) T.Goldman T.D., LANL, USA

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Outline I.Introduction II.Conflicts between Gauge invariance and Canonical Quantization III.A new set of quark, gluon momentum, angular momentum, and spin operators III.0 A lemma:Decomposing the gauge field into pure gauge and physical parts III.1 Quantum mechanics III.2 QED III.3 QCD IV. Nucleon internal structure V. Summary

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I.Introduction A widely accepted nucleon internal momentum structure: quark and gluon carry half the nucleon momentum in the asymptotic limit. D.J.Gross and F. Wilczek, PRD9,980(1974); H.Georgi and H.D. Politzer, PRD9,416 (1974); H.D.Politzer, Phys.Rep. 14,129(1974). This is a distorted picture, because it is obtained based on a momentum decomposition where both quark and gluon momentum operators are not proper.

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A proper momentum operator must be gauge invariant and satisfy canonical momentum algebra. The nucleon internal spin structure has the same problem. The quark gluon angular momentum operators used in the nucleon spin sum rule is not proper either. The nucleon internal momentum and spin structure should be reexamined based on the proper quark and gluon momentum, orbital angular momentum and spin operators.

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II. Conflicts between gauge invariance and canonical quantization

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Quantum mechanics The classical canonical momentum of a charged particle moving in an electromagnetic field, an U(1) gauge field, is It is not gauge invariant! The gauge invariant one is

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Gauge is an internal degree of freedom, no matter what gauge is used, the canonical momentum of a charged particle is quantized as The orbital angular momentum is The Hamiltonian is

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Under a gauge transformation, the matrix elements transformed as They are not gauge invariant, even though the Schroedinger equation is.

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We call The physical momentum. It is neither the canonical momentum nor the mechanical momentum

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QED The canonical momentum and orbital angular momentum of electron are gauge dependent and so their physical meaning is obscure. The canonical photon spin and orbital angular momentum operators are also gauge dependent. Their physical meaning is obscure too. Even it has been claimed in some textbooks that it is impossible to have photon spin and orbital angular momentum operators. V.B. Berestetskii, A.M. Lifshitz and L.P. Pitaevskii, Quantum electrodynamics, Pergamon, Oxford, 1982. C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Photons and atoms, Wiley, New York,1987.

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Multipole radiation Multipole radiation measurement and analysis are the basis of atomic, molecular, nuclear and hadron spectroscopy. If the spin and orbital angular momentum of photon is gauge dependent and not measurable or even meaningless, then all determinations of the parity of these microscopic systems would be meaningless!

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Multipole field The multipole radiation theory is based on the decomposition of an em field into multipole radiation field with definite photon spin and orbital angular momentum quantum numbers coupled to a total angular momentum quantum number LM,

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QCD Because the canonical parton (quark and gluon) momentum is “gauge dependent”, so the present analysis of parton distribution of nucleon uses the covariant derivative operator instead of the canonical momentum operator ; uses the Poynting vector as the gluon momentum operator. They are not the proper momentum operators!

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The quark spin contribution to nucleon spin has been measured, the further study is hindered by the lack of gauge invariant quark orbital angular momentum, gluon spin and orbital angular momentum operators. The present gluon spin measurement is even under the condition that “there is not a gluon spin can be measured”.

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III. A New set of quark, gluon (electron, photon) momentum, orbital angular momentum and spin operators

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III.0 Decomposing the gauge field into pure gauge and physical parts There were gauge field decompositions discussed before, mainly mathematical. Y.S.Duan and M.L.Ge, Sinica Sci. 11(1979)1072; L.Fadeev and A.J.Niemi, Nucl. Phys. B464(1999)90; B776(2007)38. We suggest a new decomposition based on the requirement: to separate the gauge field into pure gauge and physical parts. X.S. Chen, X.F.Lu, W.M.Sun, F.Wang and T.Goldman, Phys. Rev. Lett. 100(2008)232012.

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U(1) Abelian gauge field

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Other solution

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Under a gauge transformation, The physical and pure gauge parts will be transformed as

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SU(3) non-Abelian gauge field

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The above equations can be rewritten as a perturbative solution in power of g through iteration can be obtained

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Under a gauge transformation,

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III.1 Quantum mechanics The classical canonical momentum of a charged particle moving in an electromagnetic field, an U(1) gauge field, is It is not gauge invariant! The gauge invariant one is

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Gauge is an internal degree of freedom, no matter what gauge is used, the canonical momentum of a charged particle is quantized as The orbital angular momentum is The Hamiltonian is

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New momentum operator The canonical momentum is, It satisfies the canonical momentum commutation relation, but its matrix element is not gauge invariant. The new momentum operator is, It satisfies the canonical momentum commutation relation and its matrix element is gauge invariant.

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We call The physical momentum. It is neither the canonical momentum nor the mechanical momentum

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Hamiltonian of hydrogen atom Coulomb gauge Gauge transformed one

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Follow the same recipe, we introduce a new Hamiltonian, which is gauge invariant, i.e., This means the hydrogen energy calculated in Coulomb gauge is physical.

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A rigorous derivation Start from a QED Lagrangian including electron, proton and em field, under the heavy proton approximation, one can derive a Dirac equation and a Hamiltonian for electron and proved that the time evolution operator is different from the Hamiltonian exactly as we obtained phenomenologically. The nonrelativistic one is the Schroedinger or Pauli equation.

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III.2 QED Different approach will obtain different energy-momentum tensor and four momentum, they are not unique: Noether theorem They are not gauge invariant. Gravitational theory (Weinberg) or Belinfante tensor It appears to be perfect, but individual part does not satisfy the momentum algebra.

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New momentum for QED system We are experienced in quantum mechanics, so we introduce They are both gauge invariant and momentum algebra satisfied. They return to the canonical expressions in Coulomb gauge.

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We proved the renowned Poynting vector is not the proper momentum of em field It includes photon spin and orbital angular momentum

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Electric dipole radiation field

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Each term in this decomposition satisfies the canonical angular momentum algebra, so they are qualified to be called electron spin, orbital angular momentum, photon spin and orbital angular momentum operators. However they are not gauge invariant except the electron spin. Therefore the physical meaning is obscure.

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However each term no longer satisfies the canonical angular momentum algebra except the electron spin, in this sense the second and third term is not the electron orbital and photon angular momentum operator. The physical meaning of these operators is obscure too. One can not have gauge invariant photon spin and orbital angular momentum operator separately, the only gauge invariant one is the total angular momentum of photon. The photon spin and orbital angular momentum had been measured !

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Dangerous suggestion It will ruin the multipole radiation analysis used from atom to hadron spectroscopy, where the canonical spin and orbital angular momentum of photon have been used. It is unphysical!

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New spin decomposition for QED system

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Multipole radiation Photon spin and orbital angular momentum are well defined now and they will take the canonical form in Coulomb gauge. Multipole radiation analysis is based on the decomposition of em vector potential in Coulomb gauge. The results are physical and these multipole field operators are in fact gauge invariant.

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III.3 QCD three decompositions of momentum threethree

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Three decompositions of angular momentum 1. From QCD Lagrangian, one can get the total angular momentum by Noether theorem:

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2. One can have the gauge invariant decomposition,

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3.New decomposition

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IV. Nucleon internal structure it should be reexamined! The present parton distribution is not the real quark and gluon momentum distribution. In the asymptotic limit, the gluon only contributes ~1/5 nucleon momentum, not 1/2 ! arXiv:0904.0321[hep-ph],Phys.Rev.Lett. in press. The nucleon spin structure should be reexamined based on the new decomposition and new operators.

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Consistent separation of nucleon momentum and spin

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Quantitative example: Old quark/gluon momentum in the nucleon

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Proper quark/gluon momentum in nucleon

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One has to be careful when one compares experimental measured quark gluon momentum and angular momentum to the theoretical ones. The proton spin crisis is mainly due to misidentification of the measured quark axial charge to the nonrelativistic Pauli spin matrix elements. D. Qing, X.S. Chen and F. Wang, Phys. Rev. D58,114032 (1998)

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To clarify the confusion, first let me emphasize that the DIS measured one is the matrix element of the quark axial vector current operator in a nucleon state, Here a 0 = Δu+Δd+Δs which is not the quark spin contributions calculated in CQM. The CQM calculated one is the matrix element of the Pauli spin part only.

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The axial vector current operator can be expanded as

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The quark orbital angular momentum operator can be expanded as,

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It is most interesting to note that the relativistic correction and the creation and annihilation terms of the quark spin and the orbital angular momentum operator are exact the same but with opposite sign. Therefore if we add them together we will have where the, are the non-relativistic part of the quark spin and angular momentum operator.

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The above relation tell us that the nucleon spin can be either solely attributed to the quark Pauli spin, as did in the last thirty years in CQM, and the nonrelativistic quark orbital angular momentum does not contribute to the nucleon spin; or part of the nucleon spin is attributed to the relativistic quark spin, it is measured in DIS and better to call it axial charge to distinguish it from the Pauli spin which has been used in quantum mechanics over seventy years, part of the nucleon spin is attributed to the relativistic quark orbital angular momentum, it will provide the exact compensation missing in the relativistic “quark spin” no matter what quark model is used. one must use the right combination otherwise will misunderstand the nucleon spin structure.

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Conventional and new construction of parton distribution functions (PDFs)

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The conventional gauge-invariant “quark” PDF The gauge link (Wilson line) restores gauge invariance, but also brings quark-gluon interaction, as also seen in the moment relation:

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The new quark PDF With a second moment:

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The conventional gluon PDF Relates to the Poynting vector:

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The new gluon PDF Relates to the new gauge-invariant gluon momentum

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Gauge-invariant polarized gluon PDF and gauge-invariant gluon spin

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To measure the new quantities The same experiments as to measure the conventional PDFs New factorization formulae and extraction of the new PDFs needed New quark and gluon orbital angular momentum can in principle be measured through generalized (off-forward) PDFs

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VII. Summary: general The gauge field can be separated into pure gauge and physical parts. The renowned Poynting vector is not the proper momentum operator of photon and gluon field. The canonical momentum, angular momentum operators of the Fermion part are not observables. The gauge invariant and canonical quantization rule both satisfied momentum, spin and orbital angular momentum operators of the individual part do exist. They had been measured in QM and QED. The Coulomb gauge is physical, operators used in Coulomb gauge, even with gauge potential, are gauge invariant, including the hydrogen atom Hamiltonian and multipole radiation field operators.

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special to nucleon internal structure The nucleon internal structure should be reanalyzed and our picture of it might be modified A new set of quark, gluon momentum, orbital angular momentum and spin operators for the study of nucleon internal structure is provided Gluon spin is indeed meaningful and measurable Gluons carry not much of the nucleon momentum, not ½ but 1/5

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Prospect Computation of asymptotic partition of nucleon spin Reanalysis of the measurements of unpolarized quark and gluon PDFs New factorization formulas are needed Reanalysis and further measurements of polarized gluon distributions. A lattice QCD calculation of gluon spin contribution to nucleon spin.

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For the quark (electron), gluon(photon) momentum and angular momentum operators the Lorentz covariance can be kept to what extent, the meaning of non Lorentz covariance. The possibility of the gauge non-invariant operator might have gauge invariant matrix element for special states should be studied further.

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Thanks

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Nucleon Internal Structure 1. Nucleon anomalous magnetic moment Stern’s measurement in 1933; first indication of nucleon internal structure. 2. Nucleon rms radius Hofstader’s measurement of the charge and magnetic rms radius of p and n in 1956; Yukawa’s meson cloud picture of nucleon, p->p+ ; n+ ; n->n+ ; p+.

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3. Gell-mann and Zweig’s quark model SU(3) symmetry: baryon qqq; meson q. SU(6) symmetry: B(qqq)=. color degree of freedom. quark spin contribution to nucleon spin, nucleon magnetic moments.

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There is no proton spin crisis but quark spin confusion The DIS measured quark spin contributions are: While the pure valence q 3 S-wave quark model calculated ones are:.

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It seems there are two contradictions between these two results: 1.The DIS measured total quark spin contribution to nucleon spin is about one third while the quark model one is 1; 2.The DIS measured strange quark contribution is nonzero while the quark model one is zero.

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To clarify the confusion, first let me emphasize that the DIS measured one is the matrix element of the quark axial vector current operator in a nucleon state, Here a 0 = Δu+Δd+Δs which is not the quark spin contributions calculated in CQM. The CQM calculated one is the matrix element of the Pauli spin part only.

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The axial vector current operator can be expanded as

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Only the first term of the axial vector current operator, which is the Pauli spin part, has been calculated in the non-relativistic quark models. The second term, the relativistic correction, has not been included in the non-relativistic quark model calculations. The relativistic quark model does include this correction and it reduces the quark spin contribution about 25%. The third term, creation and annihilation, will not contribute in a model with only valence quark configuration and so it has never been calculated in any quark model as we know.

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An Extended CQM with Sea Quark Components To understand the nucleon spin structure quantitatively within CQM and to clarify the quark spin confusion further we developed a CQM with sea quark components,

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Where does the nucleon get its Spin As a QCD system the nucleon spin consists of the following four terms,

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In the CQM, the gluon field is assumed to be frozen in the ground state and will not contribute to the nucleon spin. The only other contribution is the quark orbital angular momentum. One would wonder how can quark orbital angular momentum contribute for a pure S-wave configuration?

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The quark orbital angular momentum operator can be expanded as,

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The first term is the nonrelativistic quark orbital angular momentum operator used in CQM, which does not contribute to nucleon spin in a pure valence S-wave configuration. The second term is again the relativistic correction, which takes back the relativistic spin reduction. The third term is again the creation and annihilation contribution, which also takes back the missing spin.

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It is most interesting to note that the relativistic correction and the creation and annihilation terms of the quark spin and the orbital angular momentum operator are exact the same but with opposite sign. Therefore if we add them together we will have where the, are the non-relativistic part of the quark spin and angular momentum operator.

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The above relation tell us that the nucleon spin can be either solely attributed to the quark Pauli spin, as did in the last thirty years in CQM, and the nonrelativistic quark orbital angular momentum does not contribute to the nucleon spin; or part of the nucleon spin is attributed to the relativistic quark spin, it is measured in DIS and better to call it axial charge to distinguish it from the Pauli spin which has been used in quantum mechanics over seventy years, part of the nucleon spin is attributed to the relativistic quark orbital angular momentum, it will provide the exact compensation missing in the relativistic “quark spin” no matter what quark model is used. one must use the right combination otherwise will misunderstand the nucleon spin structure.

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VI. Summary 1.The DIS measured quark spin is better to be called quark axial charge, it is not the quark spin calculated in CQM. 2.One can either attribute the nucleon spin solely to the quark Pauli spin, or partly attribute to the quark axial charge partly to the relativistic quark orbital angular momentum. The following relation should be kept in mind,

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3.We suggest to use the physical momentum, angular momentum, etc. in hadron physics as well as in atomic physics, which is both gauge invariant and canonical commutation relation satisfied, and had been measured in atomic physics with well established physical meaning.

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