# Presentation: 523246

## Xiangdong Ji
University of Maryland

- University of Maryland

**Gluons in the proton**

**BNL Nuclear Physics Seminar, Dec. 19, 2006**

## Outline 

- Introduction
- Gluons and proton mass:
- a “virial” theorem
- Gluons and proton spin:
- a model calculation
- Gluons and strange quark contribution to proton’s magnetic moment:
- an EFT analysis
- Summary

## The Glue

- Mediator of the strong interactions
- Without of it, the mass of the proton would be the sum of three quark masses!
- Determine all the essential features of strong interactions  (more important than quarks!)
- However, it is actually hard to “see” the glue in the low-energy world
- Does not couples to electromagnetism
- Gluon degrees of freedom “missing” in hadronic spectrum.
- ...

**“****Crouching quarks, hidden glue”**

## Gluon dominance in the vacuum

- QCD vacuum has interesting non-perturbative structures,
- Color confinement
- Chiral symmetry breaking
- These properties are large due to strong fluctuation gluon fields in the vacuum as in pure glue QCD.

- J.Negele et al

## Large Nc QCD  (t’ Hooft, Witten)

- Many features of QCD seem to be kept in a theory with Nc quark colors, where Nc is large.
- Gluon dominance become obvious in this limit. A scalar gluon operator of GG types goes like Nc2 in the vacuum, because there are Nc2-1 gluons (the majority wins!).

## Chiral symmetry breaking (CSB) 

- The left and right-handed quarks can be rotated independently in flavor space.
- This symmetry is broken by the zero-mode of instantons.
- CSB might generates a new mass scale for quarks and responsible for the success of quark models.
- CSB and Goldstone boson physics are critical to the low-energy properties of the proton.

## Color confinement

- An amazing property of the QCD vacuum! May be understood from the view of color flux tubes
- Flux tubes deplete the vacuum color fields and generate (QCD) strings of constant energy density  needs for an effective string theory?

**$1M from Clay Institute of Mathematics**

## Gluon and proton Mass

## Gluons in a proton

- The proton matrix element of the gluon operators goes like Nc.
- Introduction of valence quarks yields a relatively *small change* in the background gluon field!
- Gluon distribution in the nucleon goes like
-       g(x) = Nc2 f(Ncx)   : fraction of nucleon momentum carried by gluon is a constant!

## Gluon parton distribution

- Gluons become dominant in the proton at small x (gluon saturation)
- Gluons account for about 1⁄2 of the proton momentum.

## The proton mass 

- One can calculate the proton mass through the expectation value of the QCD hamiltonian,

- Quark energy

- Quark mass

- Gluon energy

- Trace anomaly

## A “virial theorem”

- The hamiltonian is just d3xT00   (stress-energy tensor of QCD)
- “Virial theorem” ( X. Ji, PRL70,1071,1995)
**The traceless part of the stress-energy tensor accounts for 3⁄4 of the proton mass, and the trace part accounts for 1⁄4.**** **
- True in the so-called MIT bag model, where the trace part is just the vacuum (dark) energy.

## MIT bag model  (K. Johnson et al., 1975) 

- Quarks are confined in a 3D cavity in which the vacuum gluon fields are depleted.
- The quarks inside the cavity obey the free Dirac equation. The pressure generated from mechanical motion balances the negative pressure from “cosmological constant” .

- “false vacuum” energy density B

## Other ingredients

- The traceless part of the quark contribution can be determined through nucleon momentum sum rule
- dx x q(x) = fraction of the momentum by quarks
- The quark mass contribution to the proton mass can be determined from
- Pion-N sigma term
- Chiral perturbation theory for masses of baryon octect

## Mass budget of baryonic matter

- Dark-Energy

- Energy density of the universe

## Color electric & magnetic fields

- One can solve the color electric and magnetic fields in the nucleon from the above
- 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 surprising feature that it almost cancels that field in the vacuum.
-    (A key to color confinement?)

## Gluons and proton spin 

## Spin of the proton in QCD

- The spin of the nucleon can be decomposed into contributions from quarks and gluons
- Decomposition of quark contribution
- Decomposition of gluon contribution

## Gluon helicity distribution 

- In a polarized proton, the gluon parton may have helicity ± 1. Introduce their densities g±(x)
- Gluon helicity distribution is
-              g(x) = g+(x) – g-(x)
- The total gluon helicity is   G = dx g(x)

**1/2**

**+1 or -1**

## Size of ΔG? 

- Thought to be large because of the possible role of axial anomaly –(αs/2)G (Altarelli & Ross, 1988)
- 2-4 units of hbar!
- Theoretical controversies...
- Of course, the gluon contribute the proton spin directly.
-       	Δq + ΔG +  Lz = 1/2
-      Naturalness?
-       if ΔG is very large, there must be a large negative Lz to cancel this---(fine tuning!)

## Theory difficulty

- In gauge invariant form, g(x) is a non-local operator which cannot be calculated in lattice QCD.
- Only in light-cone gauge, G reduces to a local operator.
- However, light-cone gauge cannot be implemented in lattice QCD calculation!

## Experimental measurements (I)

- Two leading-hadron production in semi-inclusive DIS

- Q-evolution in inclusive spin structure function g1(x)

## Experimental measurements (II)

-  production in polarized PP collision at RHIC

- Two jet production in polarized PP collision at RHIC

## Fit to data

- Generally depend on the function forms assumed.

- Hirai, Kumano, Saito, hep-ph/003213  (ACC)

- g = 0.31 ± 0.32     type-1
-      = 0.47 ± 1.0       type-2
-      = —0.56 ± 2.16  type-3

- Type-3 fit assumes
- gluon polarization
- is negative at small
- x.

## Large Nc limit

- The polarized gluon matrix elements go like Nc0  1 in the large limit. Thus the polarized part of the gluon field is
- 1/Nc suppressed relative to the measurable gluon field in the proton.
- 1/Nc2 suppressed relative to the gluon fields in the vacuum.
- Δg(x) =  Nc h(Ncx)
- The polarized gluon field represents a weak response of the gluon system to the proton polarization.

## Calculating Δg(x) in Models

- Since the quarks are the primary constituents of a proton,  and Δg(x) effect is small, the polarized gluons may be calculated from
-    where color current is generated by valence quarks
- The dominant gluon responsible for the motions of the quarks have no pol. effect.
- This is very much like small-x gluons whose sources are mostly from the valence quarks.

## G: positive? negative?

- There was a calculation by Jaffe (PRB365, 1996), showing a negative result for G in NR quark and MIT bag models (two-body contribution)
- However, there is also the one-body contribution
- Part of the one-body contribution cancels the two-body one, a positive residue remains.

- Barone et al., PRB431,1998

## x-dependence

- No model calculation for x-dependent Δg(x) has ever reported in the literature so far.
- A calculation has recently been made in MIT bag model (P. Chen, X. Ji)

## A bag model g(x)

- It is positive at all x! Similar to the correlation between the angular
- momentum and magnetic moment.

## Compare with the fit

- Compared with the AAC fit with positivity constraint.

## Non-relativistic quark model vs. the bag model

## Gluons and strange quark contribution to proton’s magnetic moment

## Magnetic moment of the proton

- Magnetic moment of the proton was measured in early 1930s, a first indication that proton has a nontrivial internal structure
- Individual quark-flavor contributions add
- Different contributions can be obtained from isospin symmetry and parity-violating electron scattering  (SAMPLE + HAPPEX + G0 + ...)

## World Data near Q2 ~0.1 GeV2

- World Data near Q2 ~0.1 GeV2

***Caution: the combined fit is approximate.  Correlated errors and assumptions not taken into account***

**Preliminary**

**G****M****s**** = 0.28 +/- 0.20**
**G****E****s**** = -0.006 +/- 0.016**
**~3% +/- 2.3% of proton magnetic moment**
**~0.2 +/- 0.5% of electric distribution**** **

**HAPPEX-only fit suggests something even smaller:**
**G****M****s**** = 0.12 +/- 0.24**
**G****E****s**** = -0.002 +/- 0.017**

## Strangeness Models (as/of 2000)

- Leading moments of form factors:
- s =  GMs (Q2=0)
- s =  GEs/ (Q2=0)

## A light strange quark

- Strange quark mass is about 100 MeV, neither light nor heavy.
- When a strange quark is considered light, QCD has an approximate SUL(3)XSUR(3) chiral symmetry. The spontaneous breaking of the symmetry leads to 8 massless Goldstone bosons.
- If so, the proton may sometimes be dissociated into a  and K+.
- Simple calculation shows that in this picture, the strange quark contribution to the proton’s magnetic moment is always negative!

## A heavy strange quark (an EFT calculation)

**Light-by-light scattering**

**Gluon matrix element**

## Muon contribution to electron’s MM

## Understanding the EFT calculation

## Proton matrix element

## Contribution is positive at large M

**Ji & Toublan**

## Where does the transition happen?

- It is difficult to estimate where the transition happen in QCD
- However, lattice calculation seems to indicate that sharp transition occurs at relatively small quark mass.

## Contribution is positive at large M

**Ji & Toublan**

## Conclusion

- Gluons play a critical role in QCD. They dominate in the vacuum and high-energy.
- Although they are less visible in hadron physics at low-energy, their contribution to the mass and spin of the proton is as important as the quarks.
- The polarized gluons effects in the proton are small. However, precision experimental data allow us to learn their effects through QCD analysis.