Polyakov loop potential from a massive (phenomenological) extension - PowerPoint PPT Presentation

Polyakov loop potential from a massive (phenomenological) extension of the Landau-deWitt gauge Urko Reinosa ∗ (Work in collaboration with J. Serreau, M. Tissier, N. Wschebor) ∗ Centre de Physique Théorique, Ecole Polytechnique, CNRS, Palaiseau, France Workshop on Non-Perturbative Methods in Quantum Field Theory October 8-10, 2014, Balatonfüred, Hungary

Motivation Motivation

Motivation Approaches to strongly interacting matter ● Need for methods to investigate infrared properties of QCD or related theories. ● Functional methods applied to: ∗ first principle actions; ∗ model actions.

Motivation Phenomenological model I ● We put forward a model based on the observation (made on the lattice) that, in the Landau gauge, the Euclidean gluon propagator G ( p ) behaves like a massive propagator at low momenta, while the ghost propagator F ( p )/ p 2 is massless: 14 9 8 12 7 10 6 G(p) 8 5 F(p) 4 6 3 4 2 2 1 0 0 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 8 p (GeV) p (GeV) ● We propose to add a gluon mass to the usual Landau-gauge action, as a phenomenological parameter: c a ( D µ c ) a + ih a ∂ µ A a S = ∫ x { 1 µν + ∂ µ ¯ µ + 1 µ } 4 F a µν F a 2 m 2 A a µ A a ⇒ extended Landau gauge model (eLG). In our approach m has to be fitted. Ex: SU(3), T = 0 propagators → m ≃ 500 MeV.

Motivation Phenomenological model II ● Most appealing feature of the model, its simplicity: ∗ Just one additional parameter as compared to YM theory. ∗ Simple modification of the Feynman rules. ∗ The model is perturbatively renormalizable. ∗ No IR Landau pole ⇒ perturbation theory in this model can be used in the IR! ● Lattice Landau-gauge correlators are qualitatively well reproduced by simple one-loop perturbative calculations in this model: 9 14 8 12 7 10 6 G(p) 8 5 F(p) 4 6 3 4 2 2 1 0 0 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 8 p (GeV) p (GeV) Tissier, Wschebor, Phys.Rev. D84 (2011); Peláez, Tissier, Wschebor, Phys.Rev. D88 (2013) and arXiv:1407.2005.

Motivation Phenomenological model III ● It is intriguing that one single additional phenomenological parameter allows to reproduce, using a perturbative expansion, several features of the correlation functions of QCD. ● We aim at: ∗ justifying the presence of the mass term from first principles: there are indications that it could originate from the Gribov ambiguity → Serreau, Tissier, Phys.Lett. B712 (2012) ∗ exploring other features of QCD: QCD phase diagram at finite temperature and density.

Motivation Phenomenological model IV ● We have tested the model at finite temperature. ● One loop, finite T, eLG ghost and chromo-magnetic propagators agree well with lattice results: 3 3 2.6 2 2.2 F (0 , k ) G T (0 , k ) 1.8 1 1.4 1 0 0 0 1 2 3 1 2 3 k (GeV) k (GeV) UR, J. Serreau, M. Tissier and N. Wschebor, Phys.Rev. D89 (2014) 105016. (Lattice data: A. Maas, J.M. Pawlowski, L. von Smekal, D. Spielmann, Phys.Rev. D85 (2012) 034037)

Motivation Phenomenological model IV ● We have tested the model at finite temperature. ● One loop, finite T, eLG ghost and chromo-magnetic propagators agree well with lattice results: 3 3 2.6 2 2.2 F (0 , k ) G T (0 , k ) 1.8 1 1.4 1 0 0 0 1 2 3 1 2 3 k (GeV) k (GeV) UR, J. Serreau, M. Tissier and N. Wschebor, Phys.Rev. D89 (2014) 105016. (Lattice data: A. Maas, J.M. Pawlowski, L. von Smekal, D. Spielmann, Phys.Rev. D85 (2012) 034037) ● The eLG model fails in reproducing the lattice chromo-electric propagator in the vicinity of the confinement/deconfinement phase transition: ∗ could signal a failure of the eLG model. ∗ could be related to the fact that the underlying symmetry is not manifest in the LG: → investigate massive extensions of other (more appropriate) gauges.

Motivation Outline I. Confinement/deconfinement phase transition: ● Center symmetry, Polyakov loop; ● Landau-deWitt gauge. ● Extended Landau-deWitt gauge model. II. LO Polyakov loop and effective potential. III. NLO results. IV. Thermodynamics.

Confinement/Deconfinement phase transition Confinement/Deconfinement phase transition

Confinement/Deconfinement phase transition Polyakov loop and center symmetry breaking ● Free-energy F for having an isolated static quark located somewhere e − β F = 1 0 t a ( a = 1 , . . . , N ) 0 d τ A 0 ( τ ) ⟩ ≡ ⟨ L ⟩ A 0 = A a N ⟨ tr P e ig ∫ β with ● The Yang-Mills action at finite T is invariant under twisted or center (gauge) transformations x ) = U ( 0 , ⃗ V ∈ SU ( N ) center = { e i 2 π k / N 1 ∣ k = 0 , . . . , N − 1 } U ( β, ⃗ x ) V with ● Under a center transformation ⟨ L ⟩ → ⟨ L ⟩ e i 2 π k / N : ∗ if center symmetry is manifest ⟨ L ⟩ = 0 and F = ∞ (confined phase); ∗ if center symmetry is broken ⟨ L ⟩ ≠ 0 and F < ∞ (deconfined phase).

Confinement/Deconfinement phase transition Polyakov loop and center symmetry breaking ● Free-energy F for having an isolated static quark located somewhere e − β F = 1 0 t a ( a = 1 , . . . , N ) 0 d τ A 0 ( τ ) ⟩ ≡ ⟨ L ⟩ A 0 = A a N ⟨ tr P e ig ∫ β with ● The Yang-Mills action at finite T is invariant under twisted or center (gauge) transformations x ) = U ( 0 , ⃗ V ∈ SU ( N ) center = { e i 2 π k / N 1 ∣ k = 0 , . . . , N − 1 } U ( β, ⃗ x ) V with ● Under a center transformation ⟨ L ⟩ → ⟨ L ⟩ e i 2 π k / N : ∗ if center symmetry is manifest ⟨ L ⟩ = 0 and F = ∞ (confined phase); ∗ if center symmetry is broken ⟨ L ⟩ ≠ 0 and F < ∞ (deconfined phase). The lattice predicts a 2nd/1st order breaking of center symmetry in the SU(2)/SU(3) case.

Confinement/Deconfinement phase transition Which gauge to use? ● The gauge fixing should not break the center symmetry explicitely. D µ ϕ ) a ≡ ∂ µ ϕ a + gf abc ¯ ● Choose a background ¯ µ , define ( ¯ A a A b ϕ c , and fix the gauge according A µ )) a = 0. In the limit ξ → 0, one obtains the Landau-deWitt gauge (LdWG): D µ ( A µ − ¯ to ( ¯ c ) a ( D µ c ) a + ih a ( ¯ A [ A ] = ∫ x { 1 µν + ( ¯ D µ ( A µ − ¯ A µ )) a } 4 F a µν F a S ¯ D µ ¯ A U [ A U ] = S ¯ A [ A ] and thus Γ ¯ A U [ A U ] = Γ ¯ A [ A ] . By construction S ¯ ● If one considers ˜ Γ [ ¯ A ] ≡ Γ ¯ A [ ¯ A ] , then ∗ the physics is obtained at the absolute minimum of ˜ Γ [ ¯ A ] ; ∗ center symmetry is manifest because ˜ Γ [ ¯ A U ] = Γ ¯ A U [ ¯ A U ] = Γ ¯ A [ ¯ A ] = ˜ Γ [ ¯ A ] .

Confinement/Deconfinement phase transition Which gauge to use? ● The gauge fixing should not break the center symmetry explicitely. D µ ϕ ) a ≡ ∂ µ ϕ a + gf abc ¯ ● Choose a background ¯ µ , define ( ¯ A a A b ϕ c , and fix the gauge according A µ )) a = 0. In the limit ξ → 0, one obtains the Landau-deWitt gauge (LdWG): to ( ¯ D µ ( A µ − ¯ c ) a ( D µ c ) a + ih a ( ¯ A [ A ] = ∫ x { 1 µν + ( ¯ D µ ( A µ − ¯ A µ )) a } 4 F a µν F a S ¯ D µ ¯ A U [ A U ] = S ¯ A [ A ] and thus Γ ¯ A U [ A U ] = Γ ¯ A [ A ] . By construction S ¯ ● The LdWG and ˜ Γ [ ¯ A ] have been studied in the framework of the functional RG: 0.4 0 0.3 0.2 β 4 V( β <Α 0 >) -0.1 β 4 V( β <Α 0 >) 0.1 0.3 0.5 0.7 0 -0.2 -0.1 -0.2 -0.3 -0.3 -0.4 -0.4 -0.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 β <A 0 >/(2 π ) β <A 0 >/(2 π ) J. Braun, H. Gies and J.M. Pawlowski, Phys.Lett. B684 (2010).

Confinement/Deconfinement phase transition The extended Landau-deWitt gauge model (eLdWG) ● Following our approach in the LG, we add a phenomenological mass term in the LdWG action. ● The mass term should not break center symmetry ( S ¯ A U [ A U ] = S ¯ A [ A ] ⇒ ˜ Γ [ ¯ A U ] = ˜ Γ [ ¯ A ] ): c ) a ( D µ c ) a + ih a ( ¯ A µ )) a + 1 A [ A ] = ∫ x { 1 µν + ( ¯ D µ ( A µ − ¯ 2 m 2 ( A a µ − ¯ µ )( A a µ − ¯ µ )} 4 F a µν F a A a A a D µ ¯ S ¯ ⇒ extended Landau-deWitt gauge model (eLdWG): ∗ Does the model show a confined phase at small temperatures? ∗ How well are the observables described (in particular in the confined phase)? ∗ How much is captured from a simple perturbative expansion? ● Feynman rules?

Confinement/Deconfinement phase transition Feynman rules: simplifying remarks ● We are interested in thermodynamical properties: ⇒ uniform background: ¯ µ ( τ, ⃗ x ) = ¯ A a A a µ . ⇒ effective potential: γ ( ¯ A ) = ˜ Γ [ ¯ A ]/( β V ) . ● We are interested in the Polyakov loop: ⇒ temporal background ¯ µ = ¯ A a A a 0 δ µ 0 . ● One can always choose ¯ A 0 in the Cartan sub-algebra: ⇒ SU(2): β g ¯ A 0 = r 3 σ 3 2 ⇒ SU(3): β g ¯ A 0 = r 3 2 + r 8 λ 3 λ 8 2