PoS(CORFU2018)185 , , David g , f https://pos.sissa.it/ † , Luigi Rosa e ∗ [email protected] , , , Pablo Pais d gauge theory in the continuum formulation. Choosing the ) , Igor Justo c 2 , ( b SU [email protected] , [email protected] , , David Dudal a h ˇ ckách 2, 18000 Prague 8, Czech Republic [email protected] ˇ sovi Speaker. On leaving from KU Leuven Campus Kortrijk – Kulak, Department of Physics, Etienne Sabbelaan 53 bus 7657, We consider finite-temperature Landau gauge, the existing gauge copies are takenquantization into account scheme, by which means entails of the the Gribov-Zwanziger introductionrectly of a influencing dynamical mass the scale Green (GribovPolyakov mass) functions loop di- of (vacuum expectation the value) theory. andmizing the Gribov Here, vacuum mass we energy in determine with termsgap respect simultaneously equation. of to the The temperature, the main by Polyakov-loop result parameter mini- is andvisible that from the solving a Gribov the cusp mass occurring Gribov directly at feels the theFinally, same problems deconfinement temperature for transition, where the the pressure Polyakov loop at becomes low nonzero. temperatures are reported. † ∗ Copyright owned by the author(s) under the terms of the Creative Commons Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Universitá di Napoli Federico II INFN, Sezione di Napoli, Complesso Universitario di Monte S. Angelo Duy Tân University, Institute of Research and Development Instituto de Ciencias Físicas y Matemáticas Universidad Austral de Chile Faculty of Mathematics and Physics, Charles University Centro de Estudios Científicos (CECS) KU Leuven Campus Kortrijk – Kulak, Department of Physics Ghent University, Department of Physics and Astronomy c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). 8500 Kortrijk, Belgium. Complesso Universitario di Monte S. Angelo Via Cintia Edificio 6, 80126 Napoli,Via Italia Cintia Edificio 6, 80126 Napoli, Italia P809, 3 Quang Trung, Hải Châu,Email: Đà Nẵng, Vietnam Casilla 567, Valdivia, Chile V Hole [email protected] [email protected] Corfu Summer Institute 2018 "School and(CORFU2018) Workshops on Elementary Particle Physics and31 Gravity" August - 28 September, 2018 Corfu, Greece Vercauteren Casilla 1469, Valdivia, Chile Etienne Sabbelaan 53 bus 7657, 8500 Kortrijk, Belgium Krijgslaan 281-S9, 9000 Gent, Belgium Fabrizio Canfora Gribov horizon, Polyakov loop and finite temperature f e c g h a b d PoS(CORFU2018)185 ]). 1 ], this 11 Pablo Pais ], it was already 27 , 26 , ]). 25 5 ]. In [ 24 , 23 ] (see also [ ] Dell’Antonio and Zwanziger showed , 4 10 22 , , 9 21 ], corresponds to restricting the path integral , 8 , 20 7 , 1 , 6 19 , , 3 ] was proposed as an order parameter for the confine- 18 2 , 17 ]. , 13 16 , , 12 , presents this obstruction [ 15 1 , ], and present their main results. 14 28 , which is the region in the functional space of gauge potentials over which Yang-Mills gauge theories, it is observed that the asymptotic particle spectrum ) ], that represent an intrinsic overcounting of the gauge-field configurations which N 3 ( SU first Gribov region We shall work exclusively with the Landau gauge here. For all these reasons, it makes sense to compute the vacuum expectation value of the Polyakov The most effective method to eliminate Gribov copies, at leading order proposed by Gribov Within One of the most fascinating non-perturbative infrared effects is related to the appearance of 1 that all the orbits of theis theory lost intersect when the implementing Gribov this region,restriction restriction. indicating has that Even remarkable no though effects. physical this In information fact, regionthe due still gluon to contains propagator the copies is presence [ of suppressedgeneral, a while dynamical an the (Gribov) approach ghost mass in scale, propagator which iswhich the enhanced push gluon in the propagator the gluon is infrared.agreement out “dressed” with More of by the non-perturbative lattice the data corrections physical [ spectrum leads to propagators and glueball masses in loop when we eliminate thecomputations Gribov are copies available using using different the techniquesfinite Gribov–Zwanziger to temperature, (GZ) see cope approach. e.g. with [ nonperturbative Related propagators at to the the Faddeev-Popov operator is positive definite.Landau gauge, The so Faddeev–Popov it operator makes is sense Hermitian to in discuss its the sign. In [ himself, and refined later on by Zwanziger [ the perturbative gauge-fixing procedure isinduces unable the to existence take of care of. non-trivialpath zero integral The modes ill presence of of defined. the Gribovcondition, copies as Faddeev-Popov Soon operator, the after Landau which Gribov’s gauge make seminal the paper, Singer showed that any true gauge pointed out that the Gribov–Zwanziger quantization offersthe an typical interesting infrared way to problems illuminate for some finitewe of temperature recapitulate gauge the theories. paper Therefore, [ in the present work, does not contain theare elementary confined excitations into of color neutral quarksThe bound and best states: gluons. theoretical this is explanation These the wefects. color so-called have Different color criteria for charged confinement for confinement objects confinement phenomenon. is have been due proposed (see to the non-perturbative pedagogical infrared introduction [ ef- Gribov copies [ Gribov horizon, Polyakov loop and finite temperature 1. Introduction A very natural observationfining is theory. that gluons Hence, shoulda non-perturbative not way effects belong that should to dress the thespectrum the positivity physical anymore. perturbative conditions spectrum propagator are The in in violated. Polyakovment/deconfinement a loop such phase con- [ transition Therefore, via it its connection doesThe to importance not the to free clarify belong the energy to interplay ofGreen between a the function’s these (very behaviour two physical vs. heavy) different Polyakov quark. loop) points can of be viewin (nonperturbative understood principle, by observing infinitely that many while there differentpositivity are, ways conditions, to it write is down verythe a likely Polyakov criterion. that gluon only propagator few which of violates these the ways turns out to be compatible with PoS(CORFU2018)185 is α (2.2) (2.1) (2.3) A Pablo Pais ]). . Now, the , by adding b 30 , 29 , 2 ,  , b being the structure constants of c  , Faddeev–Popov ghosts ab , ) ) . The proposal of the gauge fixing abc α a µ c f U M A , are anticommuting fields, which are A µ α a ( , ¯ a c ∂ δα ) , c a G † µν , abc  µ c . gt µ ¯ c δ F A U , µ a D g f −  a a µν µ + exp A abc 1 − α b µ ∂ xF U c µ ∂ 4 Det g f ). In order to do that, we introduce the following ab − = µ a D d D δ A )) 2 g ¯ − c = † µ − xb 2.2 α Z µ 4 ∂ ) D A U ( ∂ d = 4 1 ( α µ Z ab = ) for A µ α G ∂ Z ( = ( δ A µ 0 δα = − G − 2.2 A
δ αδ = = ( YM µ gauge transformations → S D ) D ab µ ab µ µ ), we can write N Z D ( A ∂ M = 2.3 − SU 1 det is the gauge fixing condition (the Landau gauge in this case), while µ is a finite group element. A ) µ is to select one (and only one, if possible) representative of the gauge orbit, see N ∂ ( ] A [ SU Lie algebra. Using ( ] = G ) A ∈ [ . Thus, N 1 G ( U When we compute the propagators using path-integral formulation, or any observable of the We shall work in the Euclidean space through this work, therefore, we will do not distinguish if the Lorentz indices In this Section we give a brief overlook ofLet the us Gribov start ambiguity. with the The Yang–Mills (YM) interested action reader in can the Euclidean space 2 SU theory, we must read off the gauge redundance ( where which is invariant under local scalars under Lorentz transformation, andLandau received gauge the fixing name condition of can bethe implemented following with term the to help the of action an auxiliary field are up or down. the infinitesimal gauge transformation ( condition Figure the where is the Faddeev–Popov operator. We observe here that where where the covariant derivative is unity into the path integral find many excellent and fully detailed reviews about this topic (see, for instance, [ Gribov horizon, Polyakov loop and finite temperature 2. The Gribov Ambiguity Faddeev and Popov decided to represent the determinant of this quantity as PoS(CORFU2018)185 ]. 3 (2.7) (2.6) (2.5) (2.4) Pablo Pais , which are , i ]. Following 8 Ω )) 1 , ∂ − 7 2 , , N 6 ( ) 4 y 4 ( γ µ e V A − ) γ , . dec A f (  ) H b 4 y . c γ , x } + , given by the following non-local µ ab ( FP 0 D S ad ( µ  > − ) ∂ a γ ab ¯ c ( b e 1 + − D M ¯ µ a c A M horizon term D 0 ;  µ ) c ∂ = x a 3 ( D b µ a µ b  A A A x µ D 4 ∂ abc d , called the Z ; ) Z y f µ a A = 4 ( A − ]. { FP H S x d 31 4 − YM = d S b e Z Ω = 2 D g An sketched representation of the gauge-fixing condition. ¯ FP c is the set of all gauge field configurations fulfilling the Landau gauge, S D Faddeev–Popov action , the space configuration is divided by different horizons Ω . Namely, ) = c 2 0 γ , D . > A 0 A ( Figure 1: ab an additional term H D = Ω FP M ab Z S stands for the inverse of the FP operator. The partition function can then be written as = 1 M and − ]: 0 8 ¨ allén-Lehmann spectrum [ GZ , M Z = 7 The Gribov region A great deal of work has been done following the Gribov’s seminal paper, remarkably the con- Therefore, in order to avoid over-counting field configurations, we must compute the path As said before, it is desirable to have only one representative for each gauge orbit. However, , µ a 6 A , µ 3 [ where As is shown in Figure Gribov himself proposed a possible solution toof this problem the by Faddeev-Popov restricting (FP) the path-integral, domain ofghost showing integration propagators at with the a modified same infrared time behaviour.could that Moreover, be these this modified interpreted leads gluon as propagators to a a signalto gauge the of and K confinement, as their poles become complex and do not∂ belong these works, the restriction of the domainthe of FP integration action in the path integralexpression is achieved by adding to struction of a renormalizable action, known as Gribov-Zwanziger (GZ) action [ defined by integral of the following Gribov horizon, Polyakov loop and finite temperature this is not possible in the Landau gauge for a non-Abelian Lie algebra, as proven by Gribov [ PoS(CORFU2018)185 , Ω (2.9) (2.8) (2.10) ) can be Pablo Pais is the YM 2.7 ] are a pair of Φ , we have again ) [ 3 ab µ Ω GZ in eq.( ϕ S ], by looking at the , 3 GZ . In ab µ 0 ¯ Z ϕ ( = , while ab ϕ M , where ) , and ). An equivalent all-order proof of ab µ ¯ has the dimension of a mass and is , , ϕ ϕ γ γ ,
, 2.7 S , 1 ω ] ab µ + , ¯ Φ − ϕ [ 0 ¯ , 2 ω S GZ N , S ab µ b + − ω ]. e , , V c gf 4 32 ab Φ µ S , 4 ¯ ¯ c ω = ), is non-local, it can be cast in local form by means D + ( , , the FP operator fulfills i A Z 2.6 Ω GZ YM i ∂ = ) S γ , , eq.( = GZ ) ], given by A ( Z γ 8 , , GZ H A 7 h S ( , are anti-commuting. The partition function means that the vacuum expectation value of the horizon function 6 H . We must stress here the fact that this is not a free parameter of the ) , 3 GZ ab µ i ) ω , γ , ab µ A ¯ ( ω ( H . In the Gribov horizons h ] 0 8 , < 7 , ab Gribov parameter 6 M 2 accounts for the quantizing fields, The Gribov regions divided by their corresponding Gribov horizons. The first Gribov region is is the Euclidean space-time volume. The parameter . has to be evaluated with the measure defined in Eq.( Ω 0 ) can be given within the original Gribov no-pole condition framework [ Φ ) V γ > Although the horizon term , 2.8 A ab ( where action plus gauge fixing and Gribov–Zwanziger (GZ) terms, in its localized version, where the notation H exact ghost propagator in an external gauge field [ of the introduction of abosonic set fields, of while auxiliary fields Figure 2: while in eq.( rewritten as [ Gribov horizon, Polyakov loop and finite temperature M where known as the theory. It is a dynamical quantity, beingcalled determined the in horizon a condition self-consistent [ way through a gap equation PoS(CORFU2018)185 and (2.16) (2.17) (2.11) (2.13) (2.14) (2.15) (2.12) Pablo Pais ,  bc µ . Remarkably, it soft breaking ϕ ]. This important ]. Here, it suffices ) p 37 c 47 . . , ,  mp ν ) 46 36 1 D bc µ , , ω )( − µ a 45 2 35 ac µ , A ¯ , N ω ( ν 44 abc 34 V ∂ , 4 , ( , γ g f , 43 4 33 0 , , 0 amb , − ) + 0 8 = 42 = g f  bc µ , , = ) ¯ ab µ 7 ab + Gribov–Zwanziger action ϕ µ . ) takes the simpler form , a , bc 41 ¯ µ , ω ϕ bc ¯ c + µ , s , 6 s ∆ ϕ sb is the vacuum energy defined by 2.8 c 2 GZ ω 0 , b bc ) µ 40 γ + ) Z c γ b , ϕ = ( ab ν c bc 5 µ = v = )( abc , v , 2 39 D ϕ µ ab E v m E , ( , ν E c D ab g f µ ab µ GZ µ a ∂ ∂γ ∂ V a ¯ 38 2 1 b ϕ − ω A − µ am − sS e ) is connected with the restriction to the Gribov region ( D abc = = = = = ( ac µ ¯ a a µ a ω g f ) is known as the ¯ 2.16 ab ab abc µ µ c s sc  ¯ ϕ − ω sA x s g f s 4 bc µ 2.10 − d is of dimension two in the fields. As such, it is a ϕ  ) ∆ Z x ab ν 4 2 d γ D in eq.( ν Z ∂ = ) that the horizon condition ( GZ ]. γ − = S ( S 13 2.7 ∆ ac µ , ¯ ϕ 12  x 4 d Z ]. These correlation functions can be employed to obtain mass estimates on the spectrum = 48 0 S The local action ¨ . However, a set of BRST invariant composite operators whose correlation functions exhibit the allén-Lehmann spectral representation with positive spectral densities can be consistently intro- where Notice that the breaking term the ultraviolet divergences can be controlled ating the of quantum the level. BRST The symmetry properties ofintensive of the investigation the in GZ soft theory recent break- years, and see its [ relation with confinement have been object of has been shown to be renormalizable to all orders [ to mention that the broken identity ( of the glueballs [ Ω K duced [ and which is called the gap equation. The quantity property of the GZthe action quantum is corrections a in general consequencenilpotent and of BRST possible transformations an divergences in extenstive particular. set In of fact, Ward introducing identities the constraining it can immediately be checked thatas the summarized GZ by the action equation exhibits a soft breaking of the BRST symmetry, Gribov horizon, Polyakov loop and finite temperature with It can be seen from ( PoS(CORFU2018)185 . : N T = µ Z P a 0 A ¯ (3.4) (3.2) (3.3) (3.1) A . This with P Pablo Pais FT − e ∝ symmetry group. P ) N ( becomes diagonal and , we are in the “broken 0 and a quantum field 2 π SU ]. Within this framework, A . Since µ ¯ 0 A 49 < i 6= 0 i A generator is present, so that h P 3 t β h , g 2 1 E ) corresponds to the confinement phase, x , t ( π 0 only the , , = ) , r 2 0 dt A 2 (PL) order parameter, r ¯ 0 ( A β 0 belonging to the Cartan subalgebra of the gauge R β = cos 6 ig 0 g SU ]. As mentioned before, with the Polyakov gauge ¯ µ A = Pe = A ; otherwise, if 20 D µ 0 r , P ¯ D Tr = 19 then we are in the “unbroken symmetry phase” (confined , with i 1 N , 0 2 π µ the infinitesimal generators of the Polyakov loop P 18 = δ ], at leading order we then simply find, using the properties of h a = , the time-component becomes diagonal and time-independent. 0 , t ¯ µ i A 20 P ¯ 0 17 A center symmetry of a pure gauge theory (no dynamical matter in A , h N ) = 15 β x Z , with ( g µ µ 2 1 A ¯ A the free energy of a heavy quark, it is clear that in the confinement phase, + on the gauge field, such that, the time-component µ F ¯ A case, if becomes a good alternative choice for the order parameter instead of corresponds to deconfinement. With a slight abuse of language, we will refer to = ], see also [ ) i a 0 2 . As explained in [ π t ( 16 as the PL hereafter. 0 A ) ¯ h A x r < ( 3 SU r a µ a δ a ≤ the inverse temperature. Just like before, denoting path ordering, needed in the non-Abelian case to ensure the gauge invariance of gauge field theory in the presence of two static sources of (heavy) quarks. The standard way ≡ Polyakov gauge 0 ) β 3 0 P ) = Within this framework, it is convenient to define the gauge condition for the quantum field, In order to probe the phase transition in a quantized non-Abelian gauge field theory, we use the Besides the trivial simplification of the PL, when imposing the Polyakov gauge it turns out that In this Section we shall investigate the confinement/deconfinement phase transition of the In the 2 ¯ x A ( ( 3 a µ δ group. For instance, in the Cartan subalgebra of The BFG method is asymmetry becomes convenient easier approach, by since choosing the the Polyakov gauge tracking for of the background breaking/restoration field. of the In other words, we have imposed to the background field where we defined with the effective gauge field will bea defined as the sum of a classical field the Pauli matrices, symmetry phase” (deconfined or ordered phase),the equivalent temperature to and an infinite amount of energysymmetry would referred be to required is to the actually get a free quark. The broken/restored while Background Field Gauge (BFG) formalism, detailed in general in e.g. [ the quantity the quantity or disordered phase), equivalent to the fundamental representation). SU to achieve this goal is by probing the Gribov horizon, Polyakov loop and finite temperature 3. The Polyakov Loop with In this analytical description ofcalled the phase transition involving the PL,independent one of usually (imaginary) imposes time, meaning the that so- the gauge field belongs to the Cartan subalgebra. extra benefit can be provenexplained by in means of [ Jensen’s inequality for convex functions and is carefully PoS(CORFU2018)185 , 2 n 1 , − = (3.5) (4.1) (4.2) (3.6) is the  N µ 2 c . , we end ¯ A m a , where Pablo Pais o 2 b ) + abc q ~ 1 2 and dividing + q ~ g f − 2 is the covariant 2 2 + ) − λ , D 2 N µ 2 ) 2 ( dc µ ∂ 2 rsT is a scale parameter d D Λ ϕ Λ 4 ab ) 2 . − rsT + γ a δ ), Λ ( + ln − ) when the temperature is = nT bd ν d  is the isospin, given by π c Tr D µ ab λ nT 2.14 ) 2 s bc µ ¯ V π a ab ν D ¯ T ϕ ( ), and 2 2 ¯ D ( = ( d dependence for generality of the µ bd + ac µ 2 3.4  D ¯ − bc ) with respect to N µ ) is equivalent to the Landau gauge, ϕ D ln 4 µ ab ϕ + ]) as ¯ ε λ 4.1  D 3.4 d 4 q − a µ a ¯ c 53 ε 3 + c Λ ) ) A − 4 is the Gribov parameter, a 3 + π ( D d γ 2 abc 2 ( f , µ ) bd ln 4 2 ξ ]). D γ Z γ 2 ¯ 2 Tr DA 54 µ g ab ∞ ) ( ), the integration of the partition function gives us 7 ¯ ∞ 1 − D − Ng ∑ a + = − ¯ 2 3.6 − c n dc µ , the condition ( a d µν s µ = + ω ( ∑ F ) 4 A 2 a ]. As we are interested in the qualitative result, we shall V ) T a T λ µν ( 2 ξ F 52 = 2 ¯ bd ν DA + 1 4 ( 2 D 4 ( m ab ν λ ] as well (see also [ − ¯ x ) 2 D + 20 d 1 a µν Λ 2 ac 2 d µ F − ¯ D . ω 2 Z a 3 Ng µν − − 2 N F is the spacelike momentum component, and = ( ln 4 1 q d ~ case ( Tr ) − BFG 2 x ]. Actually dealing with the Gribov copies in the LDW gauge is something very V S 1 ( d = β d v 50 ] and more recently [ , SU E Z case was handled in [ 51 ) = 28 ) = T 3 , taken at the very end of each computation. , ( 0 s , for the is the Euclidean space volume, r SU , → 1 2 V ξ + m The The general trace is of the form Let us now investigate what happens to the Gribov parameter The LDW is also plagued by Gribov ambiguities, and the GZ procedure is applicable also in Considering only the quadratic terms of ( ( 3 GZ+PLoop I S , or Notice that, concerning the quantum field the following vacuum energy at one-loop order, defined according to ( where in order to regularizecolors, the although result. we will We frequently stress continueformulae. the to Using explicitly fact write the that usual in Matsubarais formalism, this the we work Matsubara have mode, we that are0 dealing with which will be computed numerically. nonzero. Taking the derivative of the effective potential ( background covariant derivative. After integrating outup the with (gauge fixing) the auxiliary following YM field action, background derivative in the adjoint representation defined in ( this instance [ delicate, see [ take the GZ action modified for the BFG framework (see [ Gribov horizon, Polyakov loop and finite temperature known as the Landau–DeWitt (LDW) gauge fixing condition, where yet the action still haslimit background center symmetry. The LDW gauge is actually recovered in the 4. The finite temperature effective action PoS(CORFU2018)185 π , (4.6) (4.7) (4.4) (4.5) (4.3)  ) , T . , 0 Pablo Pais 3 s , = r , . 2  ) λ 0 i ]. T , . Furthermore, we = − s s 28 ( ,  2 r ) I , m 2 T ∂ , ∂ λ r 0 i , Λ  ). After dividing through − 0 T − ( ) ( ) yields the gap equation for I r 2 4.3 T 1.0 0 equals to zero (upper line) and I ∂ ∂ , m r s ∂ ) = , ∂ r 1 at 2 accounts for terms independent of , − 2 d − . This is shown in Figure λ 0 , T ) 0 λ d i 4 λ 0 0.8 T . Figure taken from [ ( = ( . We porpose that the physical value of , 0 λ r s 2 s λ I − , 1 m + r ) ∂ : , 4 ∂ 0 with respect to its first argument. If we now T 2 ∂ , I .  λ r = i , 0.6 Tr 0 s ( , in units 2 2 ∑ T r and that 2 γ I i = 2 8 ) m i ∂ v Ng λ − T ∂ ( E , 1 1 I r 1  , the result is 2 d d − dr ∂ ∂ 0.4 2 − − s d ∑ , 2 Ng r = 2 1 2 , N ) + 2 i λ N 1 T = 3 m , ( d r − 1 I , 0.2 + d 2 and and background as a function of the temperature γ  2 1 4 i T ) = 4 ( λ 0 ) we have I r + = T λ , ∂ denotes the derivative of ∂ 4 : 1 d 0 1 + Λ 4.1  2  λ N 4 Λ ) + 1.5 1.0 0.5 m q 1 , 1 + r is found by minimizing the vacuum energy: ∂ 4 , (as we are not interested in the solution / − − r 2 ∂ I 2 4 d ) was obtained after summation over the possible values of m ∂ ( λ Tr I 2 Ng 4.7 1 = ( + / and setting 2 4 v Ng r d q λ E 1 2 ) ∂ ∂  / 1 d − 2 to be the solution to the gap equation at 4 The Gribov parameter ) − d q 0 4 2 Ng π λ 1 2 ) d 2 N 1 ( Let us now investigate the temperature dependence of ( = d − as a function of temperature 1 Z d (lower line), in units of the zero-temperature Gribov parameter From the vacuum energy ( The expression ( then we can subtract this equation from the general gap equation ( the background field used the fact that Figure 3: general number of colors ( where now all integrations are convergent.λ This equation can be easily solved numerically to yield Gribov horizon, Polyakov loop and finite temperature by where the notation define PoS(CORFU2018)185 ) T ( (5.1) (5.2) (4.8) λ , both 0 λ Pablo Pais 40 . develops a 0 ) is the Stefan– ≈ T , pointing to a ( T π λ 45 / and the interaction 6= . : 4 p 1 r . At T )  ± r π T ) ( 0 = 1 Λ λ  cos s T − 2 m 2 1.0 + T 2 N r q ~ , we have 0 √ λ = ( sin − ]. . e 2 40 κ m 2 . 28 + crit 0.8 0 T 2 T − q ~ , 2 √ ≈ = m  − , + 2 T 2 2 e MS renormalization prescription and choos- T π g q crit ~ 3 2 , where 32 T GZ √ 4 0.6 2 − Z as a function of temperature. From the dotted T 6 5 > − r . Here the plot of the pressure is given relative ln  κ e 9 v T − . One can get, whenever E V + e = 1 r 2 0 β , while the continuous line is 1 − ) λ SB  0.4 T = p = ( 3 = r ) p q p 2 3 π ¯ µ d 2 so that the zero temperature gap equation is satisfied, ( ¯ (left and right respectively). As usual the (density) pressure is µ 0.2 Z 5 T . Namely, after using the = )] ) 0 T , 2.5 2.0 1.5 1.0 0.5 3.0 = r , r T 2 ∂ ( v m ( E one can easily see that, for ], we can also extract an estimate for the (density) pressure I ∂ − 4 55 ) , shown in Figure T 4 ( v T E ) is finite, we can numerically obtain / The dotted line curve represents [ I − 4.7 Following [ ) = T ( , which are cancelled by the derivation w.r.t. p Boltzmann constant accounting forWe subtract all the degrees zero-temperature value, of such freedom that the of pressure the becomes zero system at at zero high temperature: temperature. which is related to the free energy by cusp-like behaviour exactly at the critical temperature 5. The equation of state issue measure to the Stefan–Boltzmann limit pressure: defined as ing the renormalization parameter Figure 4: Since ( deconfined phase, confirming the computationsis of plotted the previous in section. a In continuous the line. same figure, We observe very clearly that the Gribov mass curve in Figure Gribov horizon, Polyakov loop and finite temperature r curves clearly have a discontinuous derivative. Figure taken from [ PoS(CORFU2018)185 (5.4) (5.5) (5.3) case). ) 3 Pablo Pais ( ). Besides ] a lattice- 0 λ SU is exactly the 27 . Note that the I 0 ] for some more the entropy den- s 58 ) = , 0 , and  ( 4 ) ] for the ] where a quark model 48 0 p  T , ) (with T 56 0 57 , satisfy their gap equation. T 13 0 0 , , Ts , r λ 0 , the pressure becomes nega- , ( c + ] (but note that they plot the 12 0 I T v ( 3 4 I 27 and E 4 3 , Fig. 4], although no comment is stands for the Gribov parameter at ), λ − = 4 0 0 ) 27 − , 0 λ ε λ ) . 0 T , . µν T  0 , η 2 , r 4 , while p p 2 , π 0 0 T 9 2 0 λ λ −  64 i λ ν i ) and that T − u − ∂ ( − in their notation), while we use units ∂ µ I (  c 5 u 5.3 I 10 ) T 1 2 T ]. ε ) + 0 ) + − = 0 61 + T 2 , 0 , T p , 0 ] that, at least at leading order, the thermodynamic quanti- µµ λ r 38 , θ , 2 ln 0 57 2 = ( 0 = λ  λ i i I µν ( 2 4 ( 0 I θ I π  λ  is defined as the trace anomaly in units of 9 2 32 3 3 I the (Euclidean) metric of the space-time. Given the thermodynamic ], see also [ − = + µν 60 ) η , 4 0 T ( λ 59 p and ) accounts for the zero temperature subtraction, so that − ) 0 ] if the pressure and interaction energy were to be computed at lower tempera- 5.3 , 0 27 , 0 , 1 = ( being the internal energy density, which is defined as ) prime quantities stand for quantities in units of . u ε 0 However, we notice that at temperatures relatively close to our The interaction measure 5.3 = definitions of each quantity (energy, pressure and entropy), we obtain this, and the fact that we included the effect of PL on the Gribov parameter, in [ tures than shown there. In anyus case, a the warning presence that of complex aproperties masses certain as and done care their in consequences is [ gives needed when using GZ dynamics, also at the level of spectral with sity), Both quantities display atemperature behavior in units similar of the to critical temperature that ( presented in [ inspired effective coupling was introduced atperturbative finite expression, which temperature is while consistent we with used the the order exact of one-loop all the computationstive. made This here. is clearly anhigher unphysical temperatures, feature, the possibly situation related is to finewhat some and missing is the essential seen pressure physics. moreover in displays For lattice aA behaviour simulations similar similar for problem to the is nonperturbative present pressuremade in about (see one it. of [ Another the strange plots featuremeasure presented is at in the low [ temperatures. oscillating behaviour Something of similar both waswas already pressure employed observed with and in complex interaction [ conjugate quarkvelops mass. two It complex is conjugate well-known masses that in the GZ gluon quantization, propagator see de- e.g. [ ties develop an oscillatory behaviour. Webe expect this present oscillatory in behaviour [ would in principle also In ( coupling constant does not explicitly appear in ( trace of the stress-energy tensor, given by details, so we confirm the findings of [ Gribov horizon, Polyakov loop and finite temperature we have the following expression for the pressure (in units of T The last term of ( PoS(CORFU2018)185 ]. 0 Λ  T 21 5 Pablo Pais . 4 ). Right: GZ trace 4 (2011) 1 3 T (RGZ). The corrected ∼ 821 2 refined GZ 1 group, where Gribov copies arise. By Lect. Notes Phys. , which is not far from the lattice value , ) 4 2 T (  I 0.4 0.2 0.2 0.4 0.6 0.8 GeV - - - - SU 250 ]. For that reason, also quark propagators coming . 11 0 63 0 Λ =  T C 5 T ]. 28 4 ]. However, we must stress here that we pursuit the quality of the ], it is also investigated how this result is modified if we take into ]. We expect that a more detailed exploration about the trace anomaly 62 [ 28 63 , 3 57 GeV 295 2 , An introduction to the confinement problem 0 = . C 0 T Left: GZ pressure (relative to the Stefan-Boltzmann limit pressure 1 λ ) 2 40 . ( 0 P. P. acknowledges the warm hospitality of the Centro de Estudios Científicos (CECs) in Val- The thermodynamic pressure has a negative sector. This is an issue that does not only appear We have presented here one possible scenario for the confined-deconfined phase transition for In the original paper [ L SU ≈ T SB H p p [1] J. Greensite, C 0.4 0.2 0.8 0.6 Actually, the strange behaviour of thethe thermodynamic GZ variables propagator, is as related is tofrom explicitly the shown lattice in complex fitting, poles [ which of are characterizedmodynamic behaviour by [ complex poles, acquire the same pathological ther- divia, Chile, during different stagesfunded by of the this Chilean work. Government throughConicyt. The the Centers Centro of de Excellence Estudios Base Científicos Financing Program (CECs) of is References account the condensate of the extra fields, an approach known as for both from thermodynamic and hydrodynamic point of view, could shed some light onAcknowledgments this issue. in GZ or RGZ, but it is also shared by many different confinement scenarios, see for instance [ critical temperature in RGZ scenario is transition behaviour rather thantuned quantify as the the computation results, machine as capabilities are also improved. the lattice values are getting better a toy model theory, with an internal non-Abelian Figure 5: 6. Conclusions Gribov horizon, Polyakov loop and finite temperature anomaly. These plots were taken from [ implementing the PL to theT resulting GZ action, gives us a deconfinement critical temperature of PoS(CORFU2018)185 d 3 Nucl. Phys. , (1989) , (1978) Pablo Pais . Commun. B323 72B ) theories from , . Phys. Lett. N , (2013) 203 . ]. 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