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Review on Nucleon Spin Structure X.S.Chen, Dept. of Phys., Sichuan Univ. X.F.Lu, Dept. of Phys., Sichuan Univ. W.M.Sun, Dept. of Phys., Nanjing Univ. Fan Wang, Dept. of Phys. Nanjing Univ. Outline I. Introduction II. Nucleon internal structure III. There is no proton spin crisis but quark spin confusion IV. Gauge invariance and canonical commutation relation of nucleon spin operators V. Summary I. Introduction • • There are various reviews on the nucleon spin structure, such as B.W. Filippone & X.D. Ji, Adv. Nucl. Phys. 26(2001)1. We will not repeat those but discuss two problems related to nucleon spin which we believe where confusions remain. 1.It is still a quite popular idea that the polarized deep inelastic lepton-nucleon scattering (DIS) measured quark spin invalidates the constituent quark model (CQM). I will show that this is not true. After introducing minimum relativistic modification, as usual as in other cases where the relativistic effects are introduced to the non-relativistic models, the DIS measured quark spin can be accomodated in CQM. 2.One has either gauge invariant or non-invariant decomposition of the total angular momentum operator of nucleon, a quantum gauge field system, but one has no gauge invariant and canonical commutation relation of the angular momentum operator both satisfied decomposition. Unharmony within three thematic melodies of 20th century physics • Three thematic melodies: Symmetry, quantization, phase Combined the phase and symmetry leads to gauge invariance. • When applied these three thematic melodies to atom and nucleon internal structures, one meets unharmony. II. 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+ 0 ; n+ ; n->n+ 0 ; p+ . • 3. Gell-mann and Zweig’s quark model SU(3) symmetry: baryon qqq; meson q q . SU(6) symmetry: 1 B(qqq)= . [ ms (q3 )ms (q3 ) ma (q3 )ma (q3 )] 2 color degree of freedom. quark spin contribution to nucleon spin, u 4 1 ; d ; s 0. 3 3 nucleon magnetic moments. • SLAC-MIT e-p deep inelastic scattering Bjorken scaling. quark discovered. there are really spin one half, fractional charge, colorful quarks within nucleon. c,b,t quark discovered in 1974, 1977,1997 complete the history of quark discovery. there are only three quark generations. III.There is no proton spin crisis but quark spin confusion The DIS measured quark spin contributions are: While the pure valence q3 S-wave quark model calculated ones are: . • 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. • 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 a0= 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. The axial vector current operator can be expanded as • 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, qq 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. 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, Where does the nucleon get its Spin • As a QCD system the nucleon spin consists of the following four terms, • 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 Lq . • One would wonder how can quark orbital angular momentum contribute for a pure S-wave configuration? • The quark orbital angular momentum operator can be expanded as, • 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 qq creation and annihilation contribution, which also takes back the missing spin. • It is most interesting to note that the relativistic correction and the qq 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. • 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. IV.Gauge Invariance and canonical Commutation relation of nucleon spin operator • Up to now we use the following decomposition, • Each term in this decomposition satisfies the canonical commutation relation of angular momentum operator, so they are qualified to be called quark spin, orbital angular momentum, gluon spin and orbital angular momentum operators. • However they are not gauge invariant except the quark spin. • We can have the gauge invariant decomposition, • However each term no longer satisfies the canonical commutation relation of angular momentum operator except the quark spin, in this sense the second and third term is not the quark orbital and gluon angular momentum operator. • One can not have gauge invariant gluon spin and orbital angular momentum operator separately, the only gauge invariant one is the total angular momentum of gluon. • How to reconcile these two fundamental requirements, the gauge invariance and canonical commutation relation? • One choice is to keep gauge invariance and give up canonical commutation relation. • This is a dangerous suggestion. Is this the unavoidable choice? Our solution • We found a third decomposition, where each term both the gauge invariance and the canonical commutation relation are satisfied. PRL, 100(2008) 232002 arXiv:0806.3166[hep-ph];0709.1284[hep-ph]; 0709.3649[hep-ph];0710.1427[hep-ph]; 0802.2891[hep-ph]. J QED Se Le ' ' S ' ' L ' ' QCD '' q '' g J QCD S q L S L Sq d 3 x '' q L d 3 x r S L '' g '' g d 3 a 2 D phy i xE A a phy 3 a a d xE r A i i phy '' g Non Abelian complication D phy ig A pure A pure T A a a pure A A pure A phy D phy A pure A pure ig A pure A pure 0 A phy E E A phy 0 • It is not a special problem for angular momentum; • But all of the fundamental operators of QM and QFT: Four momentum; Hamiltonian. Quantum Mechanics • The fundamental operators in QM r p i L r p r i ( p e A) 2 H e 2m Gauge transformation ' eie ( x ) , A A A , ' ' t , The matrix elements transformed as | p | | p | | e | , | L | | L | | er | , | H | | H | | et | , even though the Schroedinger equation is gauge invariant. New momentum operator in quantum mechanics Generalized momentum for a charged particle moving in em field: p mr q A mr q A q A// It is not gauge invariant, but satisfies the canonical momentum commutation relation. p q A// mr q A A 0, A// 0 It is both gauge invariant and canonical momentum commutation relation satisfied. We call D phy 1 p q A// q A// i i physical momentum. It is neither the canonical momentum 1 p mr q A i nor the mechanical momentum 1 p q A mr D i Gauge transformation ' eiq ( x ) , A' A ( x), only affects the longitudinal part of the vector potential A//' A// ( x), and time component ' t ( x), it does not affect the transverse part, A' A , so A is physical and which is used in Coulomb gauge. A // is unphysical, it is caused by gauge transformation. Hamiltonian of hydrogen atom Coulomb gauge: c // c A 0, A 0, A 0. c 0 c Hamiltonian of a nonrelativistic particle c 2 (p qA ) Hc q c . 2m Gauge transformed one c // c A// A ( x) ( x), A A , c t ( x) c 2 ( p q A) ( p q q A ) H q q c q t. 2m 2m 2 Follow the same recipe, we introduce a new Hamiltonian, H phy c 2 ( p q A// q A ) H q t ( x) q c 2m 2 A// which is gauge invariant, i.e., | H phy | c | H c | c This means the hydrogen energy calculated in Coulomb gauge is gauge invariant and physical. • Few days ago, we derived the Dirac equation and the Hamiltonian of electron in the presence of a massive proton from a ED Lagrangian with electron and proton and found that indeed the time translation operator and the Hamiltonian are different, exactly as we obtained phenomenologically before. QED Different approach will obtain different energy-momentum tensor and four momentum, they are not unique: Noether theorem P d 3 x{ E i Ai } i Gravitational theory (weinberg) 3 D P d x{ E B} i It appears to be perfect and has been used in parton distribution analysis of nucleon, but do not satisfy the momentum algebra. Usually one supposes these two expressions are equivalent, because the integral is the same. We are experienced in quantum mechanics, so we introduce D P d 3 x{ phy i E iAi } A A// A D phy ieA// They are both gauge invariant and momentum algebra satisfied. They return to the canonical expressions in Coulomb gauge. We proved the renowned Poynting vector is not the correct momentum of em field J d xr ( E B) d x E A d xr E A 3 3 3 photon spin and orbital angular momentum It includes i i VI. Summary • • • There is no proton spin crisis but quark spin confusion. The DIS measured quark spin is better to be called quark axial charge, it is not the quark spin calculated in CQM. 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, • The renowned Poynting vector is not the right momentum operator of em field. • The space-time translation and space rotation generators are not the observable momentum, Hamiltonian and angular momentum operator. • The gauge invariance and canonical quantization rule for momentum, spin and orbital angular momentum can be satisfied simultaneously. • The Coulomb gauge is physical, expressions in Coulomb gauge, even with vector potential, are gauge invariant, including the hydrogen atomic Hamiltonian and multipole radiation. • We suggest to use the physical momentum, angular momentum, etc. in hadron physics as have been used in atomic, nuclear physics so long a time, Thanks 谢谢 xie xie