Nucleon Spin Structure and Gauge Invariance X.S.Chen, X.F.Lu Dept. of Phys., Sichuan Univ. W.M.Sun, Fan Wang Dept. of Phys. Nanjing Univ.

Outline I.Introduction II.Gauge invariance and canonical angular momentum commutation relation of nucleon spin III.Energy, momentum, orbital angular momentum of hydrogen atom and the em multipole radiation IV. There is no proton spin crisis but quark spin confusion V. Summary

I. Introduction 1.It is still a quite popular idea that the quark spin measured in polarized deep inelastic lepton-nucleon scattering (DIS) 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 quark spin measured in DIS can be accomodated in CQM. 2.One has either gauge invariant or non-invariant decomposition of the total angular momentum operator of nucleon and atom, but up to now one has no decomposition which satisfies both gauge invariance and canonical angular momentum commutation relation. I will show the third decomposition where the gauge invariance and canonical angular momentum commutation relation are both satisfied. To use canonical spin and orbital angular momentum operators is important for the consistency between hadron spectroscopy and internal structure studies.

The question is whether the two fundamental requirements: gauge invariance, canonical commutation relation for J, i.e., angular momentum algebra for the individual component of the nucleon spin, can both be satisfied or can only keep one, such as gauge invariance, but the other one, the canonical commutation relation should be given up?

Our old suggestion: keep both requirements, the canonical commutation relation is intact; and the gauge invariance is kept for the matrix elements, but not for the operator itself. Other suggestion: keep gauge invariance only and give up canonical commutation relation. This is dangerous! One can not stay with it!

New solution We found a new decomposition of the angular momentum operator for atom (QED) and nucleon (QCD), both the gauge Invariance and angular momentum algebra are satisfied for individual components. In passing, the energy and momentum of hydrogen atom can also be gauge invariant. The key point is to separate the transverse and longitudinal components of the gauge field.

IV.Gauge Invariance and canonical Commutation relation of nucleon spin operators From QCD Lagrangian, one can get the total angular momentum by Noether theorem:

Each term in this decomposition satisfies the canonical angular momentum algebra, 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. Therefore the physical meaning is obscure.

One can have the gauge invariant decomposition,

However each term no longer satisfies the canonical angular momentum algebra except the quark spin, in this sense the second and third term is not the quark orbital and gluon angular momentum operator. The physical meaning of these operators is obscure too. 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. The photon is the same, but we have the polarized photon beam already.

How to reconcile these two fundamental requirements, the gauge invariance and canonical angular momentum algebra? One choice is to keep gauge invariance and give up canonical commutation relation.

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. Even the hydrogen energy is not an observable, neither the orbital angular momentum of electron nor the polarization (spin) of photon is observable either. It is totally unphysical!

New Solution A new decomposition: Gauge invariance and angular momentum algebra both satisfied for individual terms. Key point: to separate the transverse and Longitudinal part of gauge field.

New decomposition

Esential task:to define properly the pure gauge field and physical one

V.Hydrogen atom and em multipole radiation have the same problem Hydrogen atom is a U(1) gauge field system, where the canonical momentum, orbital angular momentum had been used about one century, but they are not the gauge invariant ones. Even the Hamiltonian of the hydrogen atom used in Schroedinger equation is not a gauge invariant one. After a time dependent gauge transformation, the energy of hydrogen will be changed. Totally unphysical and absurd!

Momentum operator in quantum mechanics Generalized momentum for a charged particle moving in em field: It is not gauge invariant, but satisfies the canonical momentum commutation relation. It is both gauge invariant and canonical momentum commutation relation satisfied.

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

Gauge transformation only affects the longitudinal part of the vector potential and time component it does not affect the transverse part, so is physical and which is used in Coulomb gauge.

Hamiltonian of hydrogen atom Coulomb gauge: Hamiltonian of a nonrelativistic particle Gauge transformed one

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 gauge invariant and physical.

Multipole radiation Multipole radiation analysis is based on the decomposition of em vector potential in Coulomb gauge. The results are physical and gauge invariant, i.e., gauge transformed to other gauges one will obtain the same results.

III.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:.

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. New measurement gave smaller strange contribution.

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.

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, 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. 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 creation and annihilation contribution, which also takes back the missing spin.

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.

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.

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,

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.

Thanks