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### Review on Nucleon Spin Structure

X.S.Chen, Dept. of Phys., Sichuan Univ.

T.Goldman, TD, LANL

X.F.Lu, Dept. of Phys., Sichuan Univ.

D.Qing, CERN

Fan Wang, Dept. of Phys. Nanjing Univ.

Outline

- Introduction
- There is no proton spin crisis but quark spin confusion
- Gauge invariance and canonical commutation relation of nucleon spin operators
- Hydrogen atom has the same problem
- 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.

The question is that do we have to give up the two fundamental requirements,

- gauge invariance and canonical commutation relation for the individual component of the nucleon spin,
- to be satisfied together and can only keep one, such as gauge invariant, but the other one, the canonical commutation relation is violated
- Or both requirements can be kept somehow?

Our solution:

One can keep both requirements,

- the canonical commutation relation is intact;
- The gauge invariance is kept for the matrix elements, which is enough for what is measurable, but not for the operator itself.

II.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.

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 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 reduction.
- The third term is again the creation and annihilation contribution, which also takes back the missing ones.

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.

III.Gauge Invariance and canonical Commutation relation of nucleon spin operator

- Up to now we use the following decomposition,

Each term in this decompositon 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.

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 invariant and canonical commutation relation?

- One choice is to keep gauge invariance and give up canonical commutation relation.
- Is this the unavoidable choice?

Our Solution

- Keep the canonical commutation relation intact and use the gauge non-invariant decomposition.
- To prove the gauge non-invariant operator, the quark orbital, gluon spin and orbital angular momentum operator, has gauge invariant matrix elements for selected states and so they are measurable.

An introductory remark

- It is well known that the following operator is SO(3) rotational non-invariant,

however its matrix element in a J=0 state,

is SO(3) rotational invariant.

Take the quark orbital angular momentum operator as an example, after SU(3) color gauge transformation U=exp(-iω), the third component of L3q gets an additional gauge dependent part,

Where

and

have been used in the derivation. Taking into account that the nucleon ground state has fixed J, we have

This confirms our qualitative argument that the matrix element for a color singlet nucleon ground state of the gauge variant part is identically zero and therefore the matrix element of the gauge non-invariant quark orbital angular momentum operator for a nucleon ground state is gauge invariant.

The gluon spin and orbital angular momentum can be proved to have gauge invariant matrix element for a color singlet nucleon state in a similar manner.

- Only in this sense one can talk about the gluon spin.

IV.Hydrogen atom has the same problem

- Hydrogen atom is a U(1) gauge field system, where we always use the canonical momentum, orbital angular momentum, they are not the gauge invariant ones. Even the Hamiltonian of the hydrogen atom used in Schroedinger

equation is not a gauge invariant one.

- One has to understand their physical meaning in the same manner as we suggested here.
- The quark momentum distribution of nucleon discussed up to now is not the canonical quark momentum distribution.

V. 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 canonical momentum, angular momentum, etc.

in hadron physics as have been used

in atomic, nuclear physics so long a time, but should be noted that only their matrix elements for selected states of these operators are gauge invariant and measurable ones.

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