Phase Transitions in Hot and Dense QCD at Large N Ariel Zhitnitsky - PowerPoint PPT Presentation

Phase Transitions in Hot and Dense QCD at Large N Ariel Zhitnitsky University of British Columbia, Vancouver, Canada Based on hep-ph/0806.1736 (with Andrei Parnachev) hep-ph/0601057

I. Introduction At large the system in the deconfined phase T � Λ QCD At small the system in the confined (hadronic) phase T � Λ QCD At large the system in the deconfined phase µ � Λ QCD At small the system in the confined (hadronic) phase µ � Λ QCD It is clear: something drastic must be happening on the way when temperature (chemical potential ) varies Question we want to address: what are the most important vacuum configurations which are responsible for the transitions when varies ? µ ( T )

2- Basic technique and methods: ( N f � N ) Main object: Large N QCD Main technique-1: dual representation Main technique-2: holographic description Crucial element: -parameter Θ η � -field as a probe Basic trick: light of topological charges of the constituents

The basic Conjecture : The parameter suddenly changes its Θ behavior precisely at the same point T c where the phase transition happens � θ + 2 π k � E vac = N 2 min h , T < T c N k satisfies (0) = ′ (0) = 0. Eq.(6) can also E vac ∼ cos θ, T > T c

3. Support for the CONJECTURE from the holographic model of QCD • The large N QCD is known to have a holographic description; • Confined / deconfined phases in the holographic description can be studied in the standard way by analyzing the Polyakov’s loop; • Transition from confined to deconfined phase corresponds to the transition from one background metric to another at temperature ; T c • The behavior has been also studied in both phases with the result: Θ the confinement- deconfinement phase transition takes place precisely at where dependence drastically changes. T c Θ χ ( T ) ∼ ∂ 2 E vac • ∼ 1 , T < T c ∂θ 2 χ ( T ) ∼ ∂ 2 E vac ∼ 0 , T > T c ∂θ 2

0 . 7 4. Support for the CONJECTURE from the lattices: 0 . 6 c the ratio 0 . 5 R ≡ χ ( T = T c + � ) /χ ( T = T c − � ) in deconfined and confined phases at 0 . 4 T � T c B. Lucini, M. Teper, U. Wenger, 2004 0 . 3 c 0 . 2 0 . 1 c 0 c 2 3 4 5 6 7 8 9 N

1.0 0.8 R 0.6 0.4 N=4, L t =6 N=4, L t =8 0.2 N=6, L t =6 0.0 -0.1 0.0 0.1 t Support for the CONJECTURE from the lattices: the ratio as a function of reduced R ( T ) ≡ χ ( T ) /χ ( T = 0) temperature for N=4, 6, L.Del Debbio, et al.2004 t = T/T c − 1

Deconfined Phase, 5. T > T c • According to the Conjecture , one can study the confinement -deconfinement phase transition by analyzing the dependence θ rather than Polyakov’s loop. • The dependence for is determined by instantons. θ T > T c • Instanton expansion converges at T > T c � πT � � 11 V inst ( θ ) ∼ e − γN cos θ, � γ = 3 ln − 1 . 86 , Λ QCD • Critical temperature is determined by the condition � π T c � 11 � � γ = 3 ln − 1 . 86 = 0 ⇒ T c ( N = ∞ ) ≃ 0 . 53 Λ QCD , Λ QCD defined in the Pauli -Villars scheme.

Deconfined phase--continue • For any positive the instanton density is parametrically small γ > 0 and calculations are under complete theoretical control even in close vicinity of T c � T − T c � V ∼ cos θ · e − αN ( T − Tc Tc ) , 1 � � 1 /N. T c • Topological susceptibility obviously vanishes in χ ( T > T c ) ∼ e − N = 0 agreement with results from holographic QCD • One can compute for small chemical potential. T c ( µ ) = 3 N f µ 2 � � T c ( µ ) = T c ( µ = 0) 1 − , µ ≪ T c 2(2 N + N f ) π 2 T 2 c • As expected, there is no dependence on at large N, in agreement µ with the lattice results: Fodor, et al, 2004; Fodor, Katz, Schmidt, 2007. • The dependence may only experience drastic changes in the vicinity Θ where the instanton expansion breaks down, which explains our conjecture on connection between the two parameters.

6. Coulomb Gas Representation (CGR) • η � We introduce field as a probe to investigate the topological charges of the constituents in both phases. It appears in unique combination in both phases. ( ϕ − θ ) , η � ∼ f η � ϕ • η � The partition function for light (almost massless) field in deconfined phase is given by ∇ ϕ ) 2 e λ � λ = Λ 3 QCD · e − γN d 3 x ( � d 3 x cos( ϕ ( x ) − θ ) , 2 T f 2 D ϕ e − 1 R R = η ′ • Mapping between sine-Gordon theory and its CGR is well known � � ∞ � ( λ / 2) M P M − T a>b =0 Q a G ( x a − x b ) Q b d 3 x M e − i θ P M a =0 Q a e f 2 d 3 x 1 . . . Z = . η ′ M + ! M − ! M ± =0 1 G ( x a − x b ) = x b | . 4 π | � x a − � tation for the partition function (20):

7. Coulomb gas representation. Physical interpretation. • The charges were introduced in a formal way. • Physical interpretation of charges: they are topological charges as follows from identification e iθ � Q a • The following hierarchy of scales exists (size, ρ ) ≪ (distance, ¯ r ) ≪ (Debye, r D ) a ≡ e − γN � 1 1 1 1 ≪ ≪ √ a √ a Λ QCD 3 T Λ QCD �

• ρ ∼ T − 1 Typical size of the instantons • The average distance between the instantons r ∼ λ − 1 / 3 ∼ Λ − 1 QCD a − 1 / 3 ¯ • Charge is identified with an integer topological charge localized at Q a point . This by definition corresponds to a small instanton at x a x a • is the fugacity of the instanton gas in deconfined phase. λ ∼ a � 1 • The instanton-anti-instanton interaction at large distances is the same as instanton-instanton. They both are Coulomb-like interactions (in contrast with semiclassical picture). • η � The mass emerges as a result of Debye screening • The was defined as the phase of the det(..) which does not vanish η � even if chiral symmetry is unbroken. In holographic model the chiral symmetry is broken in deconfined phase in a small window above T c

The basic Question : We identified the point with the place T c where behavior drastically changes. Θ It implies that some topological configurations (which couple to )must be Θ responsible for these drastic changes. In deconfined phase they are nice dilute instantons. What happens to them at ? T < T c

8.Confined phase. Speculations. We want to speculate here on the fate of instantons when we cross the phase transition line from above The instanton expansion is not justified. We do not attempt to use semiclassical ideas in this region We argue that the instantons do not disappear from the system, but rather dissociate into the instanton quarks, the quantum objects with fractional topological charges 1/N. Instanton quarks carry the magnetic charges along with topological charges. the field will play a crucial role in identification η � of topological charges 1/N of the constituents.

9. Instanton quarks: few historical remarks. Instanton quarks originally appeared in 2d models. namely, using the resummation of exact n-instanton solution in 2d models, the original problem CP N − 1 was mapped into 2d system of pseudo -particles with fractional 1/N topological charges, Fateev et al, 79; Berg, Luscher, 79. The picture leads to elegant explanation of the confinement. Similar calculations in 4d is proven to difficult to carry out, Belavin et al, 79.

10. Confined phase. Lagrangian for η � • We want to use the same trick (tested in weak coupling regime) η � with as a probe of the topological charges of constituents. • Effective lagrangian has the form � ϕ − θ � L ϕ = 1 η ′ ( ∂ µ ϕ ) 2 + E vac cos 2 f 2 , N • It follows from the following (2k)-th correlators (Veneziano,79) � 2 k � ∂ 2 k E vac ( θ ) g 2 dx i � Q ( x 1 ) ...Q ( x 2 k ) � ∼ ( i 32 π 2 G µ ν � N ) 2 k , | θ =0 ∼ where Q ≡ G µ ν . ∂ θ 2 k i =1 • There are few additional arguments supporting the SG structure • It satisfies U(1) anomalous WI and in large N limit leads to the standard expression E vac ( θ/N ) 2 ∼ 1

11. Coulomb Gas Representation (CGR). Confined Phase • We want to use the trick to present the effective lagrangian in η � η � the dual form (CGR). The is a unique field which explicitly measures the topological charges of constituents. • Repeating all previous steps we arrive at CGR, ∞ ( E vac 2 ) M 1 P ( a>b =0 ,Qa = ± 1 /N ) Q a G ( x a − x b ) Q b � � − d 4 x M · e − i θ P M f 2 ( a =0 ,Qa = ± 1 /N ) Q a · e � d 4 x 1 . . . Z = , η ′ M + ! M − ! M ± =0 1 G ( x a − x b ) = 4 π 2 ( x a − x b ) 2 .