# Presentation: 202728

## Xiangdong Ji
University of Maryland

- University of Maryland

- —  RIKEN workshop on polarized gluon distributions,
-        Dec. 3, 2005 —

**Gluons in the proton**

## Outline 

- Introduction
- Gluons in the vacuum
- Unpolarized gluon distribution in the proton
- Polarized gluon distribution and model calculations
- Other information about the polarized gluons
- Summary

## Introduction

- The gluon is the integral part of QCD and, of course, the structure of the proton.
- Without gluons, the quarks in the proton will fly apart.
- Without gluons, the mass of the proton will be just the sum of quark masses.
- Dominant mechanism for SM Higgs production
- ...
- However, it is actually hard to see the glue in the proton directly
- Does not couple to electromagnetism
- Gluon degrees of freedom missing in hadronic spectrum.
- ...

**Where is the glue?**

## Gluons in the vacuum

- In the large Nc limit, a scalar gluon operator of GG types goes like Nc2 in the vacuum.
- There are a lot of gluons in the vacuum which generates chiral symmetry breaking (instanton liquid) and color confinement (stochastic gluon vacuum)

## Gluon Parton Distribution

- The most talked about is the gluon distribution in the proton and nuclei
- A lot of interesting things happen at small-x (parton saturation)

## Large Nc limit

- However, in the proton the matrix element goes like Nc.
- Introduction of valence quarks yields only a small change in the background gluon field.
- The conclusion might be different at very small x «1/Nc
- Gluon distribution in the nucleon goes like
-       g(x) = Nc2 f(Ncx)   : fraction of nucleon momentum carried by gluon is a constant!

## Nucleon Mass

- The hamiltonian of QCD comes from traceless and trace part of the energy momentum tensor
- The traceless part gluon energy contributes (~Nc) about 320 MeV/c2 to the mass.
- The trace part contributes (Nc) about 200 MeV/c2  (dominated by gluon condensate in large Nc limit)

- X. Ji, Phys.Rev.Lett.74:1071-1074,1995

## Color Electric & Magnetic Fields

- One can solve the color electric and magnetic field in the nucleon from the above info
- The color electric field is stronger in the proton than that in the vacuum  (strong coulomb field?)
- The color magnetic field produced by the motion of valence quark is also strong, with the interesting feature that it almost cancels that field in the vacuum.
-    (A key to color confinement?)

## Polarized gluon distribution Δg(x)

- The polarized gluon matrix elements go like Nc in the large limit. Thus the polarized part of the gluon field is
- 1/Nc suppressed relative to the measurable gluon field in the nucleon
- 1/Nc2 suppressed relative to the gluon fields in the vacuum.
- Δg(x) =  Nc h(Ncx)
- The total gluon helicity  in the light-cone gauge A+ = 0 is

## Polarized Gluon Distribution Δg(x) in Models

- In quark models, the gluons can be generated by quarks by solving color Maxwell’s equation in linear approximation
- Of course, there are other gluons which are  responsible for example, the potential between the quarks or bag confinement.
- However, one hopes that the correlation between the polarized gluon field and quark polarization can be generated correctly by models.
- Non-linear effects are ignored.

## Δg(x) in NR Quark Model

- In NR quark model, the spin-dependent color magnetic  field is generated from the magnetic dipole of the quarks.
- In MIT bag model, it is generated from the orbital motion of the nearly-massless quark.
- Two types of diagrams. The contribution from quark correlation and single quark.
- The first type  is negative. (PRB365, R. L. Jaffe, 1996)

## Δg(x) in NR quark model

- When the intermediate quark is taken to be at the ground state, the first type contribution cancel the second.
- Thus the most of the contribution comes from the contributions of the quarks in the excited states
- The contribution is always positive:  Δg>0

- Barone et al., PRB431,1998

- C0

- M1

## A more general argument about the sign

- If the polarized gluons are generated from polarized quark in linear approximation, the vector potential circulates around the spin according to the right-handed rule.
- Because the coulomb gluon is always along the direction of r, EXA is always aligned with the spin of the quark.

## Magnitude of  Δg

- There are arguments from the discrepancy of ΔΣfrom the quark model and EMC data and the QCD axial anomaly that Δg should be large: 1 to 2 units of hbar.
- The anomaly argument is at best controversial.
- Naturalness
-                  ΔΣ/2 + Δg +  Lz = 1⁄2
-     if Δg is very large, there must be a large negative Lz to cancel this---(fine tuning)
-               Δg <0.5?

## A QCD Sum Rule Calculation 

- The total gluon contribution to the angular momentum of the proton is
-                        ∫ rX(EXB)d3r
-     A QCD sum rule calculation found  that the contribution the contribution to the proton spin is about 0.25hbar
-  	I. Belitsky and X. Ji, PRL79, 1997
- Therefore if there is a cancellation, it must happen between Δg and orbital angular momentum of the gluon.

## Thus,  there are theoretical arguments to favor a positive Δg with a magnitude <0.5 hbar.

## Other probes of the polarized gluon fields

- Single-spin asymmetry
- Color-field polarizabilities
- Magnetic moment of the strange quark

## Single Spin Asymmetry

- In a polarized nucleon, the color electric field is polarized in a plane perpendicular to the direction of the polarization.
- This color electric field is the origin of the twist-three correlations
- The direction of the field depends on the type of quarks which the gluon field is  correlated--- M. Burkardt talk.

## Second moment of g2

## Strange contribution to electromagnetic form factors

- Strange quarks have a vector current contribute both to the electromagnetic and weak neutral currents
- Its contribution to the form factors
- Gs can be isolated from neutron & proton E&M form factors and weak current form factors, the latter can be extracted from parity-violating electron scattering.

## SAMPLE 

- Proposed around 1990 by R. McKeown et al.  Took a decade to finish.
- Data from both proton and deuteron targets.
- Cross section also depends on the axial correction

- E. Beise et.al, Nucl-ex/0412054

- Review:

##  Happex, PVA4, and G0 Results

- A4

- η~Q2

## GO conclusions

- Non-strange hypothesis is disfavored at 89% confidence.
- Initial rise from Q2 = 0.0 to 0.05 GeV2 is consistent with SAMPLE & HAPPEX & PVA4
- GE may be negative at Q2 = 0.3 GeV2 (PVA4)
- The combination is positive at large Q2. (HAPPEX)

## Magnetic moment as a function of mass

- When the mass of the strange quark is light, chiral-perturbation theory indicates that the  magnetic moment contribution is negative.
- Consider K+Λ components. K+ is in a p-wave with one-unit of orbital angular momentum.
- s-bar orbital contribution to the magnetic moment from K+ is negative.
- The spin contribution from Λ is also negative.

## Lattice QCD---An Indirect Approach

- D. B. Leinweber et. al.  PRL94, 212001 (2005)
- Find two expressions

-  They assume 0<R<1, and hence
- Gs must be negative!
-  From a model, they find
- R= 0.139, hence μs = -0.046!
-  However, R could be a number
- less than 0 or larger than 1.
- if R=−0.5,   Gs = 0.2!

- R is the ratio of
- strange to down
- quark sea contribution.

##  A heavy strage quark

- Consider a heavy quark contribution to the electromagnetic current

## Contribution to the magnetic moment

- In quark and bag model calculations, the heavy quark contribution is positive, similar to the argument leading to Δg is positive.
- Therefore the magnetic moment from strange quarks varies from negative to positive as mass runs from light to heavy compared to Λ.

## Magnetic moment as a function of mass

- Magnetic moment has a strong chiral dependence. The chiral behavior sets in at very small quark mass.

- μs

- ms

## Magnetic moment as a function of mass

- Magnetic moment has a strong chiral dependence. The chiral behavior sets in at very small quark mass.

## Conclusion

- The polarization of the proton leads to polarization of the color electric and magnetic field. There are many probes which allow us to establish a coherent picture
- Δg
- SSA
- g2, higher-twist correction to g1
- Magnetic moment