1. Color glass condensate in dense quark matter and off-diagonal long range order of gluons A. Iwazaki (Nishogakusha-u) Success of an effective theory of color glass condensate in high energy scattering The theory has revealed gluon’s states in nucleons and nuclei. Saturation in gluon distribution, Its universality, geometrical scaling, etc ( color glass condensate ) ( independent of mass number ) We apply the effective theory to dense quark matter in order to see what kind of state the gluons form in the matter. Color glass condensate shows an off-diagonal long range order of gluons which is exactly the same as that the gluons forming a quantum Hall state show in the dense quark matter. Morimatsu, Nishikawa, Ohtani and Iwazaki Phys. Rev. D71 (2005) 034014 2006/2/28, KEK

2. Notations in light cone formulation used in deep inelastic scattering (time coordinate) (longitudinal coordinate) (transverse coordinate) ( energy ) (longitudinal momentum) (transverse momentum) x3x3 x2x2 x1x1 radius, R nuclei; R ∝ number density of nucleons quark matter; R independent of number density of quarks momentum fraction x of partons with momentum k + momentum of nuclei or quark matter itself → ∞

3. Number density of gluons per unit area in quark matter or nuclei τ =log(Λ + /k + )=log( (Λ + /P + )(P + /k + ) )=log(Λ + /P + )+log(1/x) ( a solution in the gauge, ∂A=0 ) obeying BK equation † † number density of gluons with momentum, (k +, k 1, k 2 ), in quark matter or nuclei transverse momentum longitudinal momentum (Iancu, Leonidov and Mclerran;hep-ph/020227) Iancu,Itakura,Mclerran, Nucl.Phys A724 (2003)

4. Balitsky-Kovchegov equation ( BK equation ) evolution equation in y=log(1/x) ( gauge coupling ) c=a numerical constant † not exponential decay large r→∞ small r→0 r V(r) V(0) †

5. Momentum distribution of gluons Saturation momentum, Q s (τ), depends on baryon number involved in the system, representing specific features of the system. k2k2 1/x τ=log(1/x) Qs(τ)Qs(τ) region (1); small density region(2); density saturated region (1) region(2) Λ QCD Λ QCD < k due to confinement CGC Universality; weakly dependent on the saturation momentum # density of gluons is small ~Q 2 s /(g 2 k 2 ) # density is large ~1/g 2 log(Q 2 s /k 2 ) confinement region gluons produced mainly by valence quarks Q 2 s (τ)~(mass number) 1/3 (1/x) 0.3

6. Q s (τ,n 1 ) Q s (τ,n 2 ) k2k2 1/x ( Quark number density, n 1 < n 2 ) In dense quark matter no confinement region “CGC saturation” “small # density of gluons” All region in the figure becomes the region of color glass condensate in the limit of infinite quark number density Q 2 s (ז ) ∝ number density of quarks Saturation momentum; ( Integration over the region of small gluon density which is produced mainly by valence quarks ) cross section of gluons with saturation momentum, Q s,times number of gluons with k>Q s larger quark number density Q s (τ,n 1 ) < Q s (τ,n 2 )

7. The number density of gluons is equal to a gluon propagator In the saturation region with small k ( large r ); N τ (k) ∝ 1/g 2 log(Q 2 s/k 2 ), indicates an algebraic long range order of gluons in spatial two dimensions Note no exponential decay and no log r behavior associated with one gluon exchange. No Q s (τ) dependence, in other words, no quark number dependence In the unsaturated region; N(k) ∝ Q 2 s (τ) /k 2 in large k, one gluon exchange in two spatial dimensions ( effectively two dimensional correlation ) gluon fields

8. Color electric and magnetic charge screening in saturation region No massless pole in gluon propagator Equivalently, color charge screening color charge density This color charge corresponds to electric field, F + i,in the effective theory, where F –,i =0 in infinite momentum frame, that is, Therefore, electric screening as well as magnetic screening. It does not simply mean Debye screening of color charges and suggests an existence of some orders, or a symmetry breaking. But there are no explicit masses generation (no Meissner effect ). The limit can be taken only in the quark matter; not in nucleons due to confinement, |k|>Λ QCD

9. Our approach to dense quark matter; Morimatsu and Iwazaki, Phys. Lett.B571 (2003) 61 SU(2) gauge theory Morimatsu, Ohtani, Nishikawa and Iwazaki, Phys. Lett.B579 (2004) 347 some Gluons form a quantum Hall state around a color magnetic field generated spontaneously B ~σ 3 Gluons make cyclotron motion and form a fractional quantum Hall state of off-diagonal gluons (~σ 1,2 ) color space quantum Hall state of off-diagonal gluons with filling factor=1/4 † The existence of off-diagonal long range order has been known in this quantum Hall system. † independent on quark number density color magnetic field (~σ 1,2 )

10. unstable modes; off-diagonal gluons Maximal abelian gauge field If =Λ 0 /g ~ = similar to the quantum Hall state, then, Namely, B=F 3 1,2 =(Λ 0 ) 2 /g. This is exactly the same as that in quantum Hall state. Minimum of V (gB) is given by gB= Λ 2 One loop effective potential Although B=F 1,2 =0 in the effective theory of CGC where only +component color current, J +, is non vanishing in the leading order of infinite momentum frame, B may be generated in the next leading order. † color space SU(2) gauge theory

11. Similarity between CGC and quantum Hall state (QHS) of gluons Number of gluons are produced unlimitedly, but they are saturated due to nonlinear interaction, especially repulsive self-interaction; g 2 A 4 in CGC Unstable gluons,Φ, under color magnetic field grow unlimitedly, but they are saturated and form a quantum Hall state due to the repulsive self-interaction. The correlation in transverse direction of gluon fields indicates a off-diagonal long range order in CGC; ~ 1/(g 2 r 2 ) ; does not depend on Universality mass number of nuclei or quark number density. Universality The off-diagonal gluons in QHS shows the off diagonal long range order; ~ 1/(g 2 r 2 ) ; does not depend on the quark number density. Universality Locality in longitudinal direction, x -, ( ∝ δ(x - - y - ) ) in CGC with mean field approximation, implies correlation length is finite ( confinement scale ). The correlation length in the direction parallel to B is finite in the quark matter with quantum Hall states.

12. valence quark valence quark valence quark gluon gluon gluon gluon gluon gluon gluon gluon gluon gluon gluon gluon p+p+ p + /2 p + /4 p + /8 0 1/p + 0 2/p + 0 4/p + 0 8/p + ρ(p +, x - ) involving quarks with momentum up to P + ρ(p + /2, x - ) involving gluons with momentum up to P + /2 ρ(p + /4, x - ) involving gluons with momentum up to P + /4 ρ(p + /8, x - ) extension of source in x - direction X–X–

13. Population dynamics; a system in which small population is unstable dN(t) /dt = α( N(t) - N 2 (t) ) ; α > 0 BK eq. has a similar structure to this eq. N(t) = population ( number of gluons ) Higgs model; V=−μ 2 |Ф| 2 +λ|Ф| 4 /4 equation of motion of spatially uniform field, (number density, N( t ) = |Ф| 2, of Φ quantum ) This suggests that the gluon’s saturation is associated with a certain Higgs mechanism or an order of gluons. repulsive interaction Itakura; proceedings for ICHEP2004

14. Conclusion Color glass condensate in nucleons or nuclei observed in real experiments of high energy scattering, strongly(?) suggests that gluons in dense quark matter form an ordered state, namely, a fractional quantum Hall state under a color magnetic field generated spontaneously.