V Gogokhia
email: gogohia@rmki.kfki.hu

M Vasúth
email: vasuth@rmki.kfki.hu

The Non-Perturbative Analytical Equation of State for the Gluon Matter. I

Keywords: numbers: 11.10.Wx, 12.38.Mh, 12.38.Lg, 12.38.Aw

The effective potential approach for composite operators is generalized to non-zero temperatures in order to derive the equation of state for pure SU (3) Yang-Mills fields. In the absence of external sources, this is nothing but the vacuum energy density. The key element of this derivation is the introduction of a temperature dependence into the expression for the bag constant. The non-perturbative analytical equation of state for gluon matter does not depend on the coupling constant, but instead introduces a dependence on the mass gap. This is responsible for the largescale structure of the QCD ground state. The important thermodynamic quantities, such as the pressure, energy and entropy densities, etc., have been calculated. We show explicitly that the pressure may vary smoothly around T c = 266.5 MeV, whereas all other thermodynamic quantities undergo drastic changes at this point. The proposed analytical approach makes it possible to control for the first time the thermodynamics of gluon matter at low temperatures, below T c . We reproduce the properties of the socalled "fuzzy bag"-type models through the presence of the mass gap in our equation of state. An analytic calculation of the gluon condensate is obtained as a function of temperature.

Introduction

The prediction of a possible existence of the quark-gluon plasma (QGP) created in the relativistic heavy ion collisions is one of the most interesting theoretical achievements of Quantum Chromodynamics (QCD) at non-zero temperatures and densities. A fairly full list of the relevant pioneering papers is given in Refs. [1,2,[START_REF] Csernai | Introduction to relativistic Heavy Ion Collisions[END_REF]) and the present status of the investigations of the properties of QCD matter is described in Refs. [START_REF] Plasma | Contributions from the RBRC Workshop[END_REF][START_REF] Alam | [END_REF].

The equation of state (EoS) for the QGP has been derived analytically up to the order g 6 ln(1/g 2 ) by using the perturbation theory (PT) expansion for the evaluation of the corresponding thermodynamic potential term by term ( [6,[START_REF] Kapusta | Finite-Temperature Field Theory[END_REF] and references therein). However, the most characteristic feature of the thermal PT expansion is its non-analytical dependence on the coupling constant g 2 , which means that PT QCD is not applicable at finite temperatures. The problem is not the poor convergence of this series [6,[START_REF] Kapusta | Finite-Temperature Field Theory[END_REF][START_REF] Braaten | [END_REF]9] but rather the fact that a radius of convergence cannot even be defined; any next calculated term can be bigger than the previous one. This is an in-principle problem which cannot be overcome. From the strictly mathematical point of view, fourdimensional QCD at non-zero temperatures effectively becomes a three-dimensional theory. At the same time, three-dimensional QCD has more severe infrared singularities [10] and its coupling constant becomes dimensional. It is as a consequence of this that the dependence becomes non-analytical when using the dimensionless coupling constant g 2 . One also needs to introduce three different scales, T , gT and g 2 T , where T is the temperature, in order to try somehow to understand the dynamics of the QGP within the thermal PT QCD approach.

At present, the only practical method to investigate the problem is lattice QCD at finite temperature and baryon density, which has recently shown rapid progress ( [START_REF] Plasma | Contributions from the RBRC Workshop[END_REF][START_REF] Alam | [END_REF]11,12,[START_REF] Karsch | 2007 PoS CPOD07 026[END_REF][START_REF] Cheng | [END_REF]15] and references therein). However, lattice QCD, being a very specific regularization scheme, is primary aimed at obtaining well-defined corresponding expressions in order to get realistic numbers for physical quantities. One may therefore get numbers and curves without understanding what the physics is behind them. Such understanding can only come from the dynamical theory, which is continuous QCD. For example, any description of the QGP has to be formulated within the framework of a dynamical theory. The need for an analytical EoS remains, but, of course, it should be essentially non-perturbative (NP), approaching the so-called Stefan-Boltzmann (SB) limit only at very hight temperatures. Thus the approaches of analytic NP QCD and lattice QCD to finite-temperature QCD do not exclude each other, but, on the contrary, should be complementary. This is especially true at low temperatures where the thermal QCD lattice calculations suffer from big uncertainties [START_REF] Plasma | Contributions from the RBRC Workshop[END_REF][START_REF] Alam | [END_REF]11,12,[START_REF] Karsch | 2007 PoS CPOD07 026[END_REF][START_REF] Cheng | [END_REF]15]. On the other hand, any analytic NP approach has to correctly reproduce thermal lattice QCD results at high temperatures (see papers cited above). There already exist interesting analytical approaches based on quasi-particle picture [16,17,18,19,20,21,22,23,24,25] (and references therein) to analyze results of SU(3) lattice QCD calculations for the QGP EoS.

The main purpose of this paper is to derive the NP analytical EoS for the gluon matter (GM), i.e., a system consisting purely of Yang-Mills (YM) fields without quark degrees of freedom. The formalism we use to generalize it to non-zero temperatures is the effective potential approach for composite operators [26]. In the absence of external sources it is nothing but the vacuum energy density (VED). The approach is NP from the very beginning, since it deals with the expansion of the corresponding skeleton vacuum loop diagrams. The key element is the extension of our initial work [27] to non-zero temperatures. This makes it possible to introduce the temperature-dependent bag constant (pressure) as a function of the mass gap. It is this which is responsible for the large-scale structure of the QCD ground state. The confining dynamics in the GM will therefore be nontrivially taken into account directly through the mass gap and via the temperature-dependent bag constant itself, but other NP effects will be also present. Let us note that the temperature-dependent bag constant within the thermal PT QCD has been introduced into the Gibbs equilibrium criteria for a phase transition [28] (see also Ref. [16]), and the effective potential approach has been already used in order to study the structure of QCD at very large baryon density for an arbitrary number of flavors [29].

The present paper is organized as follows. In section 2 the effective potential approach for composite operators is discussed in general terms. An explicit expression is obtained for the VED for pure YM fields. In section 3 the expression for the gluon pressure is derived taking into account the expression for the bag constant at zero temperature as a function of the mass gap [27]. Taken together this makes it possible to derive a formula for the pressure that is suitable for the generalization to non-zero temperatures. This consists of the two independent parts, describing the NP and PT contributions to the gluon pressure. In section 4 the generalization to non-zero temperatures is performed using the imaginary-time formalism. All the analytic results for the gluon pressure as a function of temperature are collected together in section 5. In section 6, expressions are given for the main thermodynamic quantities, such as the entropy and energy densities, the heat capacity, etc., as functions of the pressure. In section 7 we discuss all the numerical results for the NP part of the gluon pressure. Section 8 concludes our discussion. Some explicit expressions for the summation of the thermal logarithms over the Matsubara frequencies are present in appendix A, while in appendix B a scale-setting scheme of our calculations is formulated.

The Vacuum Energy Density

The quantum part of the VED is determined by the effective potential approach for composite operators [26]. It is given in the form of the skeleton loops expansion, containing all the types of the QCD full propagators and vertices (for its pictorial representation, see Ref. [27]). Each vacuum skeleton loop is itself a sum of an infinite number of the corresponding PT vacuum loops (i.e., containing the point-like vertices and free propagators). Thus the effective potential approach makes it possible to calculate the VED from first principles, i.e., using only fundamental constituents of QCD: gluons and quarks and their interactions or, more precisely, their propagators and vertices of their interactions. The number of the vacuum skeleton loops is equal to the power of the Planck constant, h, so that a series expansion for the effective potential is nothing other than the semiclassical WKB loop expansion, which, in general, is an asymptotic series. It has been widely used in quantum field theory [30]. In QCD the instantons have been discovered by using just this method [31] (and references therein).

It is instructive to begin with some general expressions at zero temperature in this formalism. The gluon part of the VED to leading order (the so-called log-loop level ∼ h, whose infinite series is shown in Fig. 1) is analytically given by the effective potential for composite operators [26] as follows:

V (D) = i 2 d 4 q (2π) 4 Sp{ln(D -1 0 D) -(D -1 0 D) + 1}, (1)
where D(q) is the full gluon propagator and D 0 (q) is its free counterpart. The traces over space-time and color group indices are assumed. It is clear from this that the effective potential is normalized to the free PT vacuum to be zero, i.e., V (D 0 ) = 0. We note that the YM bag constant has been calculated to this order in Ref. [27].

The two-point Green's function, describing the full gluon propagator, is

D µν (q) = -i T µν (q)d(-q 2 ; ξ) + ξL µν (q) 1 q 2 , ( 2
)
where ξ is the gauge-fixing parameter and

T µν (q) = g µν - q µ q ν q 2 = g µν -L µν (q). (3)
Its free counterpart D 0 ≡ D 0 µν (q) is obtaining by replacing the full gluon propagator's Lorentz structure d(-q 2 ; ξ) in Eq. ( 2) by unity, i.e.,

D 0 µν (q) = -i {T µν (q) + ξL µν (q)} 1 q 2 . (4)
In order to evaluate the effective potential (1) with Eq. ( 2), we use the well-known identity [START_REF] Golub | Matrix Computations[END_REF] Sp Going over to four-dimensional Euclidean space in Eq. ( 1), one obtains ( g = V (D)) g = -16

ln(D -1 0 D) = 8 × 4 ln det(D -1 0 D) = 32 ln[(3/4)d(-q 2 ; ξ) + (1/4)]. ( 5
) = D + + + ...
d 4 q (2π) 4 ln[1 + 3d(q 2 ; ξ)] - 3 4 d(q 2 ; ξ) + a , (6)
where the constant a = (3/4) -2 ln 2 and an integration from zero to infinity over q 2 is assumed. The VED g derived in Eq. ( 6) is already a colorless (color-singlet) quantity, since it has been summed over the color indices. It also does not depend explicitly on the unphysical (longitudinal) part of the full gluon propagator due to the product (D -1 0 D) which, in turn, comes from the above-mentioned normalization of the free PT vacuum to zero. Thus it is worth emphasizing that only the transversal ("physical") degrees of freedom only of the gauge bosons contribute to this equation. As a consequence, in the effective potential approach to leading order there is no need for the ghost degrees of freedom from the very beginning in order to cancel the longitudinal ("unphysical") component of the full gluon propagator. This role is played by the normalization condition. However, this does not work for the higher order vacuum skeleton loops. In this case and beyond the PT at any gauge (i.e., in the general case) the cancelation of unphysical gluon modes should proceed with the help of ghosts [27].

An overall numerical factor 1/2 has been introduced into Eq. ( 1) in order to make the gluon degrees of freedom equal to 32/2 = 16 = 8 × 2, where 8 color of gluons times 2 helicity (transversal) degrees of freedom (see Eqs. ( 5) and ( 6

)).

In what follows the Lorentz structure d(q 2 ) ≡ d(q 2 ; ξ) will be called the full effective charge ("running") or the gluon propagator invariant function (its form factor), for convenience. For the generalization of Eq. ( 6) to non-zero temperatures, the most important thing is to introduce correctly the above-mentioned bag constant.

The gluon pressure at zero temperature

The vacuum of QCD is a very complicated confining medium and its dynamical and topological complexity means that its structure can be organized at various levels: classical and quantum (see, for example Refs. [START_REF] Rubakov | Classical Gauge Fields[END_REF][START_REF]Duality, and Nonperturbative Aspects of QCD[END_REF][START_REF] Aoki | Structure of the QCD Vacuum[END_REF] and references therein). It is mainly NP by origin, character and magnitude, since the corresponding fine structure constant is large. However, the virtual gluon field configurations and excitations of the PT magnitude due to asymptotic freedom (AF) [36] are also present there.

One of the main dynamical characteristics of the true QCD ground state is the bag constant B. Its name has come from the famous bag models for hadrons [37,38], but its present understanding (and thus modern definition) has nothing to do with the hadron properties. It is defined as the difference between the PT and the NP VEDs [39,40,41,42]. We can symbolically write B = V ED P T -V ED, where the V ED is NP but "contaminated" by the PT contributions (i.e., this is a full V ED like a full gluon propagator). Rewriting this as

B = V ED P T -V ED = V ED P T - [V ED -V ED P T + V ED P T ] = V ED P T -[V ED T NP + V ED P T ] = -V ED T NP > 0,
since the VED is always negative. The bag constant is nothing but the truly NP (TNP) VED, apart from the sign, by definition, and thus is free of the PT contributions ("contaminations"). In order to consider it also as a physical characteristic of the true QCD ground state, the bag constant correctly calculated should satisfy some other requirements such as colorlessness and dependence on the physical degrees of freedom, finiteness, gauge-independence, no imaginary part (stable vacuum), etc.

In our previous work [27], we have already derived an expression for the bag constant, which satisfies the above conditions:

B Y M = 16 q 2 ef f d 4 q (2π) 4 ln[1 + 3d T NP (q 2 )] - 3 4 d T NP (q 2 ) , (7)
where symbolically shown q 2 ef f is the effective scale squared, separating the soft momenta from the hard ones in the integration over q 2 . This is a general expression for any TNP effective charge in order to calculate the bag constant from first principles. It is defined as the special function of the TNP effective charge integrated out over the NP region (soft momenta region, 0 ≤ q 2 ≤ q 2 ef f ). Adding the bag constant to the both sides of Eq. ( 6), and introducing the gluon pressure P g = g + B Y M , one obtains

P g = B Y M -16 d 4 q (2π) 4 ln[1 + 3d(q 2 )] - 3 4 d(q 2 ) + a . (8)
The next step is to establish the relation between the full effective charge d(q 2 ) and its TNP counterpart d T NP (q 2 ). In our previous work [43] it has been proven that the full effective charge depends explicitly and regularly on the scale ∆ 2 (the so-called mass gap) responsible for the NP dynamics in QCD, i.e., d(q 2 ) ≡ (q 2 ; ∆ 2 ). The abovementioned symbolic subtraction at the fundamental gluon level can then be defined as [27,43]: d T NP (q 2 ; ∆ 2 ) = d(q 2 ; ∆ 2 ) -d(q 2 ; ∆ 2 = 0) = d(q 2 ; ∆ 2 ) -d P T (q 2 ). In this way the separation between the TNP effective charge and its PT counterpart becomes exact, but not unique. On how to make this separation exact and unique at the same time see subsection below. Evidently, both the TNP part and its PT counterpart are valid throughout the whole energy/momentum range, i.e, they are not asymptotics. Let us also emphasize the principle difference between d(q 2 ) and d T NP (q 2 ). The former is the NP quantity "contaminated" by PT contributions, while the latter one, being also NP, is free of them.

However, this is not the whole story yet. Since the PT part d P T (q 2 ) contains the free gluon Lorentz structure d 0 (q 2 ) = 1, it should be extracted explicitly as follows, where we have dropped the explicit dependence on ∆ 2 :

d(q 2 ) = d T NP (q 2 ) + d P T (q 2 ) = d T NP (q 2 ) + 1 + d AF (q 2 ), (9)
and d AF (q 2 ) describes the part responsible for AF in the PT gluon effective charge. This procedure is necessary in order to maintain the normalization of the free PT vacuum to zero. In this connection, let us stress that the second equality in this relation does not imply any violation of AF in the full gluon effective charge d(q 2 ), provided by the PT effective charge d P T (q 2 ) in the first equality of the same relation. Extracting d 0 = 1 explicitly, we thereby subtract the divergent contribution associated with the constant a in Eq. ( 6), and thus the above-mentioned normalization condition will be automatically satisfied. We now replace all the effective charges as follows: d T NP (q 2 ) ≡ α T NP s (q 2 ) and d AF (q 2 ) ≡ α AF (q 2 ). The explicit expression for the effective charge responsible for AF, that's α AF (q 2 ), will be given in part II of our work, since its explicit expression is not used here.

Substituting the decomposition (9) into Eq. ( 8), lengthy algebra leads to

P g = P NP + P P T = B Y M + P Y M + P P T , (10)
i.e., P NP = B Y M + P Y M . In Eq. ( 10) B Y M is given in Eq. ( 7), while

P Y M = -16 d 4 q (2π) 4 ln[1 + 3 4 α T NP s (q 2 )] - 3 4 α T NP s (q 2 ) , (11)
and

P P T = -16 d 4 q (2π) 4 ln[1 + 3α AF (q 2 ) 4 + 3α T NP s (q 2 ) ] - 3 4 α AF (q 2 ) . (12)
P Y M in Eq. ( 11) depends exclusively on the TNP effective charge. Together with the bag constant ( 7) it forms the NP part of the gluon pressure (10). P P T in Eq. ( 12) contains the contribution which is mainly determined by the AF part of the PT effective charge, though the dependence on the TNP effective charge is also present (it is logarithmically suppressed in comparison to the pure AF term). If the interaction is switched off, i.e., putting formally α AF (q 2 ) = α T NP s (q 2 ) = 0, then P g = 0 in accordance with the initial normalization of the free PT vacuum to zero. Evidently, just the right-hand-side of Eq. ( 10) is to be generalized to non-zero temperature, on account of the explicit expressions [START_REF] Kapusta | Finite-Temperature Field Theory[END_REF], (11) and (12).

Confining effective charge

The only problem remaining is the explicit expression for the TNP effective charge. Evidently, it has to be consistent with our initial work [27], where the bag constant was evaluated at zero temperature, i.e.,

α T NP s (q 2 ) → α INP s (q 2 ) = ∆ 2 q 2 , ( 13
)
where the superscript "INP" stands for the intrinsically NP effective charge. Here ∆ 2 ≡ ∆ 2 JW is the Jaffe-Witten (JW) mass gap, mentioned above, which is responsible for the large-scale structure of the QCD vacuum, and thus for its INP dynamics [START_REF] Jaffe | Yang-Mills Existence and Mass Gap[END_REF]. Let us note that how the mass gap appears in QCD has been shown in our work [43].

A few additional remarks are in order. In Ref. [START_REF] Gogokhia | [END_REF] it has been shown that the TNP part of the full gluon propagator as a function of the mass gap also contains a term that is regular at origin. It is for this reason that it is not uniquely separated from the PT gluon propagator, which effective charge that is always regular at origin. We distinguish between the INP and the PT effective charges not only by the presence of the mass gap, but by the character of the IR singularities as well [START_REF] Gogokhia | [END_REF]46]. So only after the replacement (13) the INP effective charge is uniquely and exactly separated from its PT counterpart, and the obtained expression for the bag constant [START_REF] Kapusta | Finite-Temperature Field Theory[END_REF] becomes now free of all the types of the PT contributions ("contaminations").

In Ref. [46] we have shown that the so-called INP gluon propagator is, by construction, purely transversal in a gauge invariant way.

It exactly converges to the gluon propagator, whose effective charge is Eq. ( 13), after the renormalization programme of the regularized mass gap is performed. This result has been obtained in the most general way, i.e., without making any truncations/approximations/assumptions or choosing a special gauge. Hence the expression ( 13) is not an ansatz, and thus it is mathematically well justified. However, it should also be physically well justified. The problem is that the gluon equation of motion is highly nonlinear, so the number of independent exact solutions is not fixed a priori (this number may be even bigger, depending on the different truncations/approximations/assumptions and the concrete gauge choice made). They should be considered on equal footing from the very beginning. In our previous work [START_REF] Gogokhia | [END_REF], at least the two different general types of exact solutions for the full gluon propagator have been found as a function of the regularized mass gap: the first is smooth at small gluon momentum, allowing for the gluons to acquire an effective gluon mass (the socalled massive solution). The second is singular in the q 2 → 0 limit, so that the gluons always remain massless (the so-called nonlinear iteration solution). Only this solution has an affective charge (13) after the renormalization is completed. For the detailed arguments which makes its choice justified from the physical point of view as well (i.e., emphasizing its confining nature) see our paper [27]. In addition, let us note that the above-mentioned massive solution is not confining.

Generalization to non-zero temperatures

In the imaginary-time formalism [START_REF] Kapusta | Finite-Temperature Field Theory[END_REF]47], all the four-dimensional integrals can be easily generalized to non-zero temperatures T according to the prescription (note that there is already Euclidean signature)

dq 0 (2π) → T +∞ n=-∞ , q 2 = q 2 + q 2 0 = q 2 + ω 2 n = ω 2 + ω 2 n , ω n = 2nπT, (14)
i.e., each integral over q 0 of a loop momentum is to be replaced by the sum over the Matsubara frequencies labeled by n, which obviously assumes the replacement q 0 → ω n = 2nπT for bosons (gluons). In frequency-momentum space the INP effective charges (13) becomes

α INP s (q 2 ) = α INP s (q 2 , ω 2 n ) = α T NP s (ω 2 , ω 2 n ) = ∆ 2 ω 2 + ω 2 n , ( 15
)
and it is also convenient to introduce the following notations:

T -1 = β, ω = q 2 , (16)
α AF (q 2 ) = α AF (q 2 , ω 2 n ) = α AF (ω 2 , ω 2 n )
, where, evidently, in all the expressions here and below q 2 is the square of the three-dimensional loop momentum, in complete agreement with the relations [START_REF] Cheng | [END_REF].

Introducing the temperature dependence into the right-hand-side of the relation (10), we finally obtain

P g (T ) = P NP (T ) + P P T (T ) = B Y M (T ) + P Y M (T ) + P P T (T ). ( 17
) 4.1. Derivation of B Y M (T )
It is convenient to begin with Eq. ( 7) for the bag constant. In frequency-momentum space the temperature-dependent YM bag pressure becomes

B Y M (T ) = 16 d 3 q (2π) 3 T +∞ n=-∞ ln[1 + 3α T NP s (q 2 , ω 2 n )] - 3 4 α T NP s (q 2 , ω 2 n ) .( 18
)
After the substitution of the expression from Eq. ( 15), one obtains

B Y M (T ) = 16 d 3 q (2π) 3 T +∞ n=-∞ ln ω 2 + ω 2 n ω 2 + ω 2 n - 3 4 ∆ 2 ω 2 + ω 2 n , (19)
where we introduced the notations:

ω = q 2 + 3∆ 2 = ω 2 + m 2 ef f , m ef f = √ 3∆. (20)
One of the attractive features of the confining effective charge (15) is that it allows for an exact summation over the Matsubara frequencies (see appendix A). Substituting all our results of the summation into Eq. ( 19), dropping the β-independent terms [START_REF] Kapusta | Finite-Temperature Field Theory[END_REF], and performing almost trivial integration over angular variables, one gets

B Y M (T ) = - 8 π 2 ω ef f 0 dω ω 2 3 4 ∆ 2 1 ω 1 e βω -1 -2β -1 ln 1 -e -βω 1 -e -βω . (21)
The ω ef f , which is the three-dimensional analog of q ef f in Eq. ( 7), is discussed in appendix B, where numerical values are given.

It is convenient to present the integral (21) as a sum of several terms

B Y M (T ) = - 6 π 2 ∆ 2 B (1)
Y M (T ) -

16 π 2 T B (2) Y M (T ) -B (3) Y M (T ) , (22)
where the explicit expressions of all the terms are given as

B (1) Y M (T ) = ω ef f 0 dω ω e βω -1 , B (2)
Y M (T ) =

ω ef f 0 dω ω 2 ln 1 -e -βω , B (3)
Y M (T ) =

ω ef f 0 dω ω 2 ln 1 -e -βω . (23)
Here and below N = (e βω -1) -1 is the Bose-Einstein distribution (the so-called gluon mean number [START_REF] Kapusta | Finite-Temperature Field Theory[END_REF]).

B YM P YM Figure 2.
The bag constant (22) and the NP YM part (27) in GeV 4 units as functions of temperature T in GeV units. The bag constant at T = 1 GeV is zero, so up to this value it is responsible for the NP vacuum contributions to the pressure (17) and hence (29).

Derivation of P Y M (T )

In frequency-momentum space Eq. ( 11) becomes

P Y M (T ) = -16 d 3 q (2π) 3 T +∞ n=-∞ ln[1 + 3 4 α T NP s (q 2 , ω 2 n )] - 3 4 α T NP s (q 2 , ω 2 n ) , (24)
thus determining the temperature-dependent YM part of the NP pressure of Eq. ( 17).

After substituting of the expression (15), one obtains

P Y M (T ) = -16 d 3 q (2π) 3 T +∞ n=-∞ ln ω2 + ω 2 n ω 2 + ω 2 n - 3 4 ∆ 2 ω 2 + ω 2 n , (25)
where we introduced the following notations:

ω = q 2 + 3 4 ∆ 2 = ω 2 + m2 ef f , meff = √ 3 2 ∆ = 1 2 m ef f . (26)
Comparing Eqs. (19) and ( 25) one can write down the final result directly. For this purpose, in the system of Eqs. ( 22)-( 23) one must change the overall sign, replace ω by ω and integrate from zero to infinity. Thus, one obtains

P Y M (T ) = 6 π 2 ∆ 2 P (1) Y M (T ) + 16 π 2 T P (2)
Y M (T ) -P

Y M (T ) ,

where

P (1) Y M (T ) = ∞ 0 dω ω e βω -1 = π 2 6 T 2 , P (2)
Y M (T ) = ∞ 0 dω ω 2 ln 1 -e -βω , P (3)
Y M (T ) = ∞ 0 dω ω 2 ln 1 -e -β ω . ( 28
)
The bag pressure (22) and the NP YM part (27) in GeV 4 units as functions of temperature T are shown in Fig. 2. The scale-setting scheme of all our numerical calculations in this paper is presented in appendix B.

The gluon pressure at non-zero temperatures

Summing up all the expressions and integrals ( 22)-( 23) and ( 27)-( 28), the gluon pressure (17) becomes

P g (T ) = P NP (T ) + P P T (T ), (29)
where

P NP (T ) = B Y M (T ) + P Y M (T ) = 6 π 2 ∆ 2 P 1 (T ) + 16 π 2 T [P 2 (T ) + P 3 (T ) -P 4 (T )], (30)
and

P 1 (T ) = ∞ ω ef f dω ω e βω -1 , (31)
while

P 2 (T ) = ∞ ω ef f dω ω 2 ln 1 -e -βω , P 3 (T ) = ω ef f 0 dω ω 2 ln 1 -e -βω , P 4 (T ) = ∞ 0 dω ω 2 ln 1 -e -β ω . ( 32
)
Let us recall once more that in all these integrals β = T -1 , ω ef f along with the mass gap ∆ 2 is fixed (appendix B), while ω and ω are given by the relations ( 20) and ( 26), respectively. In the formal PT ∆ 2 = 0 limit it follows that ω = ω = ω and the combination P 2 (T ) + P 3 (T ) -P 4 (T ) becomes identically zero. Thus the NP part (30) of the gluon pressure (29) in this limit vanishes.

In frequency-momentum space the PT part (12) of the gluon pressure ( 29) is

P P T (T ) = - 8 π 2 ∞ 0 dω ω 2 T +∞ n=-∞ ln 1 + 3α AF (ω 2 , ω 2 n ) 4 + 3α T NP s (ω 2 , ω 2 n ) - 3 4 α AF (ω 2 , ω 2 n ) ,( 33
)
where the trivial integration over angular variables has already been carried out.

Unfortunately, one cannot perform analytically (i.e., exactly) the summation over the Matsubara frequencies in this integral, using the AF expression for α AF (ω 2 , ω 2 n ) [36,46]. At this stage the PT part [START_REF] Rubakov | Classical Gauge Fields[END_REF] remains undetermined and will be numerically evaluated elsewhere. Note that, since it becomes zero when the interaction is switched formally off (α AF (ω 2 , ω 2 n ) = 0), both terms (30) and [START_REF] Rubakov | Classical Gauge Fields[END_REF] and hence the gluon pressure (29) satisfy the initial normalization condition.

Confining dynamics in Eq. ( 29) is implemented via the bag pressure (22) and the mass gap itself. However, other NP effects are also present via P Y M (T ) and P P T (T ), given in Eqs. ( 27) and [START_REF] Rubakov | Classical Gauge Fields[END_REF], respectively, though in Eq. ( 33) they are logarithmically suppressed.

Main thermodynamic quantities

Together with the pressure P (T ), the main thermodynamic quantities are the entropy density s(T ) and the energy density (T ). The general formulae which connect them are [START_REF] Kapusta | Finite-Temperature Field Theory[END_REF]

s(T ) = ∂P (T ) ∂T , (T ) = T ∂P (T ) ∂T -P (T ) = T s(T ) -P (T ) ( 3 4 )
for pure YM fields, i.e., when the chemical potential vanishes. Let us note that in quantum statistics the pressure P (T ) is up to a sign equal to the thermodynamic potential Ω(T ), i.e., P (T ) = -Ω(T ) > 0 [START_REF] Kapusta | Finite-Temperature Field Theory[END_REF].

Other thermodynamic quantities of interest are the heat capacity c V (T ) and the velocity of sound squared c 2 s (T ), which are defined as follows:

c V (T ) = ∂ (T ) ∂T = T ∂s(T ) ∂T , (35)
and

c 2 s (T ) = ∂P (T ) ∂ (T ) = s(T ) c V (T ) , (36)
i.e., they are defined through the second derivative of the pressure. The so-called conformity

C(T ) = P (T ) (T ) (37)
mimics the behavior of the speed of sound squared (36) but without involving such a differentiation.

The thermodynamic quantity of a special interest is the thermal expectation value of the trace of the energy momentum tensor. This trace anomaly relation measures the deviation of the difference

(T ) -3P (T ) ( 3 8 )
from zero at finite temperatures, in the high temperature limit it must vanish. As a consequence it is very sensitive to the NP contributions to the EoS. It also assists in the temperature dependence of the gluon condensate [START_REF] Leutwyler | Proceedings of the Conference QCD -20 Y EARS LAT ER[END_REF] (see Ref. [START_REF] Boyd | [END_REF]), namely where < G 2 > 0 ≡< G 2 > T =0 denotes the gluon condensate at zero temperature, whose numerical value is discussed in appendix B. As mentioned above, the main purpose of this paper is to investigate analytically and calculate the NP part (30) of Eq. ( 29), while the corresponding investigation of the PT part (33) will be the subject of subsequent work. This will make it possible to complete the investigation, derivation and final calculation of the full GM EoS.

< G 2 > T =< G 2 > 0 -[ (T ) -3P (T )], (39)

SB limit

The high-temperature behavior of all the thermodynamic quantities is governed by the SB ideal gas limit, when the matter can be described in terms of non-interacting massless particles (gluons). In this limit these quantities satisfy special relations (see, for example Ref. [START_REF] Kapusta | Finite-Temperature Field Theory[END_REF])

3P SB (T ) T 4 = SB (T ) T 4 = 3s SB (T ) 4T 3 = c V (SB) (T ) 4T 3 = 24 45 π 2 ≈ 5.26, T → ∞,( 40
)
and

C SB (T ) = c 2 s(SB) (T ) = 1 3 , T → ∞, (41)
on account of the previous relations and their definitions in Eqs. (36) and (37). The trace anomaly relation (38) also satisfies the SB limit, namely

SB (T ) -3P SB (T ) = 0, T → ∞, (42)
as it comes out from the relations (40). In what follows its right-hand-side will be called the general SB number.

Numerical results and discussion

All our numerical results obtained for the thermodynamical quantities discussed in the previous section and calculated with the help of the NP pressure (30) are shown in Figs.

3, 4 and 5. The NP gluon pressure has a maximum at some finite ("characteristic") temperature, T = T c = 266.5 MeV. This means that it may change continuously its regime in the near neighborhood of this point in order for its full counterpart to achieve the thermodynamic SB limit (40) at high temperatures. However, this is not possible for the other NP thermodynamic quantities, as is seen from the curves shown in Fig. 3 (none of the power-type fall off around T c can be smoothly transformed into the almost constant behavior at high temperatures). In order to achieve the corresponding thermodynamic SB limits at high temperatures their full counterparts would have to undergo drastic changes around this point. As we already know from the thermodynamics of SU(3) lattice QCD [START_REF] Boyd | [END_REF] (see Refs. [50,51,52]) the energy and entropy densities have discontinuities (jump or, equivalently, step discontinuities) at a temperature of about T c = 260 -270 MeV. Our characteristic temperature T c = 266.5 MeV is surprisingly very close to this value. If we were not aware of the thermal lattice QCD results then we would be able to predict them. Since we are aware of them, these lattice results confirm our expectation of a sharp changes in the behavior of the entropy and energy densities in the region where the pressure is continuous. One of the interesting features of our calculations is seen in Fig. 3. At the characteristic temperature T c the NP pressure, the entropy and energy densities satisfy a SB-type relations, namely

3P NP (T c ) T 4 c = NP (T c ) T 4 c = 3s NP (T c ) 4T 3 c = 12 45 π 2 ≈ 2.63, (43)
where, obviously, the right-hand-side is half of the general SB number (40). In turn this yields

C NP (T c ) = 1 3 , NP (T c ) -3P NP (T c ) = 0, (44)
so that the NP trace anomaly relation approaches zero from below in the T → ∞ (β → 0) limit (see Fig. 4), i.e, satisfying the SB relation (42) in this limit as well.

Since the NP entropy and energy densities satisfy the SB-type relations (43), we expect for their full counterparts to have step discontinuities at T c , as discussed above. The NP heat capacity and the velocity of sound squared do not satisfy the SB-type relations (43), as can be seen in Figs. 3 and4. We expect, therefore, for the full heat capacity to have an essential discontinuity at T c , since, in general, [c V (T c )] -1 = 0, and hence c 2 s (T c ) = 0 due to the relation (36). They are defined through derivatives of the entropy and energy densities in Eqs. ( 35)- (36), i.e., they involve the second derivatives of the pressure. That is a reason why these thermodynamic quantities are too sensitive to the dynamical structure of the GM in the near neighborhood of T c .

Moreover, due to the SB-type relations ( 43)-( 44), and on account of the SB relations (40), for example it follows that

P SB (T ) -2P NP (T ) T =Tc = 0, ∂ ∂T P SB (T ) -2P NP (T ) T =Tc = 0. ( 45
)
Combining these relations again with the SB-type relations ( 43)-( 44) it is possible to derive some other exact relations between different thermodynamic quantities at T c . Their importance will be explicitly shown in part II of our investigation, where they will be extensively exploited.

From Fig. 3 it clearly follows that the temperature range for the NP thermodynamic quantities can be divided into the three different intervals:

(i). The low-temperatures interval up to its upper bound of about 0.6T c . It is defined by the first maximum of the heat capacity which appears in this region before temperature reaches T c . This region is obviously dominated by the NP contributions (30) to the gluon pressure (29). The behavior of all the NP thermodynamic quantities are shown in Figs. 5, 6 and 7. We do not expect any serious changes in the behavior of their full counterparts in this region. However, whatever changes may occur they will be under our control, since the NP part ( 30) is exactly calculated. The important observation is that all the NP thermodynamic quantities which have been properly scaled go down exponentially as temperature approaches zero. This is a general feature of the behavior of all the full thermodynamic quantities far below T c ; otherwise their zero temperature limit would not be realized. The almost constant behavior of the NP velocity of sound squared and conformity seen in Fig. 7 is due to the fact that they are the ratios of the corresponding thermodynamic quantities, see Eqs. ( 36)- (37). It seems that for the first time it is possible to predict the behavior of all the important thermodynamic quantities in this interval. We shall refer to the GM in this region as a "confining phase" (there are no lattice data for this region at all).

(ii). The transition interval (0.6 -1)T c . In this region all the thermodynamic quantities should undergo sharp changes in their regimes, so that their full counterparts will be able to achieve their corresponding SB limits at high temperatures. The full GM pressure remains continuous in this region. However, the existence of maxima in the behavior of the NP thermodynamic quantities is expected in this interval, since they should "die" at high temperatures. The NP pressure is a part of the full GM pressure, which should be a smoothly growing function of T throughout the whole temperature range. All other full thermodynamic quantities are smoothly growing functions of T only below and above T c . Within our approach they are expected to have discontinuities of different character at T c , as mentioned above (apart from the gluon condensate). Following the authors of Ref. [53], we shall refer to the GM in the transition region as a "mixed phase". The changes in the dynamical structure of the GM just in this region will determine the nature of the phase transition at T c .

(iii). The temperatures interval starting at T c . This interval itself can be clearly subdivided into two intervals, since in the integrals ( 31)-( 32) ω ef f = 3.75T c = 1 GeV: (a) The moderate-temperatures (1 -3.75)T c , when the NP effects are still significant and (b) The high-temperatures starting at 3.75T c , when the NP effects become small. Therefore, even above T c , the GM can be understood as being in the two different forms. The first one can be considered as strongly coupled GM, where the NP effects are still important. The second one can be considered as weakly coupled GM, where the NP effects become already negligible. The typical temperature which is of about 3.75T c = 1 GeV for the GM, is too high to be reached even at LHC. For QCD matter this temperature should be substantially decreased, and therefore be accessible at RHIC, and especially at LHC. The general feature for the behavior of all the properly scaled NP thermodynamic quantities above T c is their power-type fall off, while their full counterparts should show a power-type rise at high temperatures. Apparently, we may refer to the GM in the moderate temperature region as an "extended mixed phase".

The behavior of all the full thermodynamic quantities in both mixed phases is governed by the NP pressure (30) and by the first term in the PT pressure [START_REF] Rubakov | Classical Gauge Fields[END_REF]. We shall refer to the GM in the high temperatures region as an "AF phase", since the behavior of all the full thermodynamic quantities will be determined by the second term in the PT pressure [START_REF] Rubakov | Classical Gauge Fields[END_REF] and the free gluons contribution. A few remarks in advance are here in order. From our discussion it clearly follows that the approximation of the gluon pressure (29) by its NP part (30) only (though exactly calculated in the present investigation) breaks down at T c , when the NP pressure achieves its maximum. For the derivatives of the pressure, this approximation breaks down even earlier, starting from about 0.6T c , as discussed above. This explains the sign change in the interaction measure and the supersonic value of the speed of sound squared both at T c as seen in Fig. 4. The rapid rise of conformity and the speed of sound squared above T c in Fig. 4 is also unphysical. This also explains why the gluon condensate in Fig. 5 increases drastically above T c . Taking into account the NP (30), PT [START_REF] Rubakov | Classical Gauge Fields[END_REF], and free gluons contributions to the full GM pressure, all these unphysical effects will disappear in the full thermodynamic quantities, thus allowing them to achieve their corresponding SB limits. This will allow one to compare with lattice data above T c [START_REF] Boyd | [END_REF]52] as well.

The mass gap and a "fuzzy bag" term

On account of the first of Eqs. (28) and Eq. ( 31), it is instructive to re-write the NP gluon pressure (30) equivalently as follows: 48)) as a function of T/T c . It clearly approaches a finite constant in the high temperature limit (compare with Fig. 4).

P NP (T ) = ∆ 2 T 2 - 6 π 2 ∆ 2 P 1 (T ) + 16 π 2 T [P 2 (T ) + P 3 (T ) -P 4 (T )], (46)
where

P 1 (T ) = ω ef f 0 dω ω e βω -1 . (47)
It scales as T ω ef f in the T → ∞ limit, while the integrals P 2 (T ), P 3 (T ) and P 4 (T ) remain unchanged, see Eqs. [START_REF] Golub | Matrix Computations[END_REF]. Let us emphasize the explicit presence of the NP mass gap term in Eq. ( 46), namely ∆ 2 T 2 . It will be present in the full EoS as well, since in the PT part of the gluon pressure the mass gap contribution is logarithmical suppressed (see Eq. ( 33)), and hence its contribution cannot be canceled. This is in full agreement with the so-called "fuzzy bag" model [52,54,55,56] and with the massless Boltzmann stringy model [57]. The presence of the mass gap ∆ 2 in our approach from the very beginning naturally explains the properties of such models. The NP trace anomaly relation scaled by T 2 , t.e.,

NP (T ) -3P NP (T ) T 4 ×T 2 = NP (T ) -3P NP (T ) T 2 -→ -const., T → ∞(48)
is shown in Fig. 8. It clearly goes to a finite constant (having the dimensions of ∆ 2 ) in the high temperature limit, as it should in accordance with the previous discussion and the thermal lattice QCD results [START_REF] Boyd | [END_REF]52,55,58].

This effect immediately shows up in the NP gluon condensate (39). Scaled by T 4 , analytically it becomes

< G 2 > T T 4 = < G 2 > 0 T 4 - [ NP (T ) -3P NP (T )] T 2 × 1 T 2 ∼ const. T 2 , T → ∞.( 49
)
The difference in the fall off at high temperatures between the zero temperature gluon condensate and minus trace anomaly relation (as it enters Eq. ( 49)), both scaled by T 4 , is clearly seen in Fig. 9. Evidently, this difference is due to the asymptotic behavior of Eq. [START_REF] Leutwyler | Proceedings of the Conference QCD -20 Y EARS LAT ER[END_REF]. At temperatures below T c there is no difference between < G 2 > T /T 4 and < G 2 > 0 /T 4 , which is as it should be (see, for example again Fig. 5). At temperatures above T c the difference between these two quantities is explicitly determined by Eq. ( 49) and is clearly shown in Fig. 10. The presence of the mass gap contribution in Eq. ( 46), and hence in the trace anomaly relation (38) scaled by T 4 , explains that the way it goes down is not 1/T 4 [START_REF] Boyd | [END_REF]58] but, as follows from our consideration, it decreases as 1/T 2 . The inclusion of the PT contribution [ P T (T )-3P P T (T )]/T 4 in Eq. ( 49) will produce a rather small numerical correction at high temperatures due to the SB limit (42) for the trace anomaly. The numerical results presented in Table 1 confirm this statement. It shows rather good numerical agreement between our NP analytical trace anomaly, as it enters Eq. ( 49), and its lattice counterpart at hight temperatures starting already from 2T c [START_REF] Boyd | [END_REF].

In conclusion, let us note that in lattice QCD at non-zero temperatures there is a possibility to distinguish between the magnetic (m) and electric (e) parts of the gluon condensate (see, for example Refs. [START_REF] Boyd | [END_REF]59,60] and references therein). In Refs. [59,60] and Ref. [START_REF] Boyd | [END_REF], such a gauge-invariant field-strength correlators in pure YM theory have been calculated at finite temperatures around T c and above T c , respectively. Following Ref. [60], we can formally write < G 2 > T =< G 2 e > T + < G 2 m > T , but at this stage we do not know how to calculate these two parts separately within our approach. There is therefore as yet no possibility to compare with lattice data. In any case, such an analytical investigation at zero and non-zero temperatures is beyond the scope of the present investigation. The NP dynamical structure of the GM within our approach is highly non-trivial. In the whole temperature range we have the two different massive gluonic excitations ω and ω with the effective masses m ef f = 1.17 GeV and meff = 0.585 GeV, respectively. Both effective masses are due to the mass gap ∆ 2 , which is responsible for the largescale structure of the QCD ground state. It is dynamically generated by the nonlinear interaction of massless gluon modes [43,[START_REF] Gogokhia | [END_REF]46]. The first massive excitations can be interpreted as the glueballs, since m ef f is comparable to the masses of scalar glueballs (though it is lighter than the pseudoscalar ones) [61]. The second one meff might be identified with an effective gluon mass of about (500 -800) MeV, which arises in different approaches (see again the above-mentioned review [61] and references therein).

We also have the two different massless gluonic excitations ω: the NP massless gluons, propagating according to the integral P 1 (T ) in Eq. ( 31), and almost SB massless gluons, propagating according to the integral P 2 (T ) in Eqs. [START_REF] Golub | Matrix Computations[END_REF]. We stress that the integral (31) should be multiplied by (6/π 2 )∆ 2 , and all other integrals [START_REF] Golub | Matrix Computations[END_REF] are to be multiplied by (16/π 2 )T , when one speaks about different NP contributions to the NP pressure (30).

It is worth emphasizing once more that all these NP gluonic excitations are of dynamical origin, and only such can be accounted for within our approach. Whether the PT part [START_REF] Rubakov | Classical Gauge Fields[END_REF] of the gluon pressure (29) will introduce a new massless or even massive gluonic excitations (or, equivalently, an effective gluonic degrees of freedom [16,62]) is an open question at this stage. In our opinion, however, they may be really present in the full GM EoS. In addition to the mass gap the PT part [START_REF] Rubakov | Classical Gauge Fields[END_REF] depends on the asymptotic scale parameter of QCD. The combination of these two fundamental scale parameters may lead to the creation of new massive gluonic excitations around T c , for example, pseudoscalar glueballs [60,61,63]. In any case, we should definitely have the SB free massless gluons far away from T c . However, one thing should be made perfectly clear. Because of these and a possible new massive excitations in the GM, the behavior of all the thermodynamic variables in the moderate temperatures region (1 -3.75)T c will be substantially different from the behavior of a gas of free massless gluons. This our qualitative yet conclusion is in agreement with recent lattice data and quasi-particle approaches both cited above. The recent lattice data will be reproduced quantitatively after the evaluation of the NP (30), PT [START_REF] Rubakov | Classical Gauge Fields[END_REF], and free gluon contributions to the full GM pressure.

The existence of the SB-type relations ( 43)- [START_REF] Jaffe | Yang-Mills Existence and Mass Gap[END_REF] indicates that in the transition region near T c a dramatic increase in the number of effective gluonic degrees of freedom will appear (for example, the glueballs will begin rapidly to dissolve). This effect will be reflected in the strong increase of the thermodynamic variables near T c , in agreement with recent lattice results [60,63,64]. This will lead to drastic changes in the structure of the GM. A change in this number is enough to generate pressure gradients, but not enough to affect the pressure itself. It varies slowly and therefore remains continuous in this region. At the same time, the pressure gradients, such as the energy and entropy densities, etc., should undergo sharp changes in their behavior. The penetration of the PT part [START_REF] Rubakov | Classical Gauge Fields[END_REF] of the gluon pressure below T c should not go deeply into the transition region (0.6 -1)T c , since the INP effective charge is logarithmical suppressed. However, it should go sufficiently deep in order to eliminate "non-physical" maxima and other effects, created by the derivatives of P NP (T ) in the transition region. Instead of "nonphysical" maxima, the full thermodynamic quantities (apart from the pressure) should have discontinuities at T c , as follows from our approach. This has also been established by the thermal QCD lattice cited above.

Of course, not all the glueballs will be dissolved in the transition region. Some of them will remain above T c , along with other massive and massless gluonic excitations, forming thus a possible mixed and extended mixed phases around T c [53]. After the evaluation of the PT part [START_REF] Rubakov | Classical Gauge Fields[END_REF], which is also "contaminated" by the NP contributions, we hope that the NP physics of the mixed phases will be well understood in the GM. At very high temperatures, starting at 3.75T c the NP effects become very small, and the structure of the GM will be mainly determined by the SB relations ( 40)-( 42) between all the important thermodynamic quantities.

Conclusions

The effective potential approach for composite operators [26] has been generalized to non-zero temperatures in order to derive analytical EoS for pure SU(3) YM fields, shown in Eqs. ( 29)- [START_REF] Rubakov | Classical Gauge Fields[END_REF]. In its NP part (30) there is no dependence on the coupling constant, only the dependence on the mass gap, which is responsible for the large-scale structure of the QCD ground state. A key element of this work is the generalization of the expression for the bag constant at zero temperature [27] to non-zero temperatures. The NP part (30) of the gluon pressure (29) has been exactly evaluated, while its PT part [START_REF] Rubakov | Classical Gauge Fields[END_REF] has been left undetermined at this stage, since it requires a separate investigation.

Our main quantitative and qualitative results in this investigation, which will not be changed (or only slightly changed) after the inclusion of the PT part [START_REF] Rubakov | Classical Gauge Fields[END_REF], are:

(i). In Eq. ( 29) the confining dynamics at non-zero temperatures ( 15) is taken into account through the T -dependent bag constant (22) and the mass gap ∆ 2 .

(ii). Other NP effects are also taken into account via the YM part (27).

(iii). The mass gap ∆ 2 or, equivalently, ω ef f is the only one independent input scale parameter needed to calculate the NP thermodynamic quantities.

(iv). The presence of the four different types of massive and massless gluonic excitations of the NP origin.

(v). The characteristic temperature T c = 266.5 MeV is a temperature at which the maximum of the NP part of the gluon pressure is achieved.

(vi). The low-temperatures region up to 0.6T c is under control. The exponential fall off in the T → 0 limit of all the main NP thermodynamic quantities (see Fig. 6) is expected to be preserved for their full counterparts as well.

(vii).

The behavior of all the thermodynamic quantities in the transition region (0.6 -1)T c depends on how deeply the PT part [START_REF] Rubakov | Classical Gauge Fields[END_REF] penetrates this region.

(viii). The existence of the SB-type relations ( 43)-( 44) at T c shows that the structure of the GM below and above T c may be really rather different.

(ix). Since the NP entropy and energy densities satisfy them, we expect for their full counterparts to have jump discontinuities at T c , while the full pressure remains continuous.

(x). Since the NP heat capacity does not satisfy them, we expect for its full counterpart to have an essential discontinuity at T c , while the full speed of sound squared is to be zero at T c .

(xi). In the moderate temperatures region (1 -3.75)T c the NP vacuum effects are still significant.

(xii). Because of this, our qualitative prediction is that in this region the behavior of all the thermodynamic variables will be substantially different from the behavior of a gas of free massless gluons.

(xiii). All the full thermodynamic quantities, therefore, approach their SB limits rather slowly.

(xiv). A possible understanding of new forms of the GM below and above 3.75T c as strong and weak coupling GM, respectively, has to be pointed out.

(xv). The existence of the mass gap term ∆ 2 T 2 in the NP pressure (46), which remains in the gluon pressure (29) as well.

(xvi). Because of the mass gap term ∆ 2 T 2 the full trace anomaly and the gluon condensate will go down as 1/T 2 at high temperatures, and not as 1/T 4 .

In the subsequent paper (part II) we shall evaluate the PT part [START_REF] Rubakov | Classical Gauge Fields[END_REF] of the gluon pressure (29), as well as include the free gluons contribution. It will make it possible to establish the order of the phase transition at T c . We will be able to compare our numerical results with thermal QCD lattice calculations [START_REF] Boyd | [END_REF]52] at high temperatures above T c and in the transition region [52,60,63] as well. Only after completion of this programme, we will include the quark degrees of freedom in order to derive the NP QGP EoS within our formalism. For example, this will allow one to confirm a possible existence of Quarkyonic Matter (QM) [65,66] and of a triple point in the QCD phase diagram (see Ref. [67] and references therein). Finally, let us note that recently [68] the pressure and the trace anomaly relation both scaled by T 4 have been calculated in 3d "electrostatic QCD" in apparent contradiction with the 4d lattice data [START_REF] Boyd | [END_REF]52]. It is worth noting once more that our calculations are in a good agreement with them (see Table 1 and discussion above). Let us note that ω ef f , which appears first in the integral (21), is the only free parameter of our approach. In frequency-momentum space it is

ω ef f = q 2 ef f -ω 2 c , (B.1)
where we introduce the "constant" Matsubara frequency ω c , which is always positive.

Hence ω ef f is always less or equal to the q ef f of four-dimensional QCD, i.e., ω ef f ≤ q ef f . One can then conclude that q ef f is a very good upper limit for possible values of ω ef f . In this connection, let us recall that the bag constant B Y M at zero temperature has been successfully calculated at a scale q 2 ef f = 1 GeV 2 , in fair agreement with other phenomenological quantities such as gluon condensate [27]. So ω ef f is fixed as follows:

ω ef f = q 2 ef f = 1 GeV. (B.2)
The mass gap squared ∆ 2 , also calculated at this scale, has a value The above-mentioned gluon condensate at zero temperature < G 2 > 0 calculated at the scale (B.2) and at the same confining effective charge [START_REF] Karsch | 2007 PoS CPOD07 026[END_REF] in Ref. [27] is

< G 2 > 0 ≡ 0| 1 4
G a µν G a µν |0 = 0.1052 GeV 4 . (B.5)

We need this value for the calculation of the temperature-dependent gluon condensate < G 2 > T via Eq. (39). Multiplying the right-hand-side of the relation (B.5) by (4α s /π), where α s = 0.1187 [69], it numerically becomes

0| α s π G a µν G a µν |0 ≈ 0.016 GeV 4 , (B.6)
It is in a good agreement with its phenomenological value, < G 2 > ph ≡ 0|(α s /π)G a µν G a µν |0 ≈ 0.012 GeV 4 , which can be changed by a factor of ∼ 2, as mentioned in Ref. [41] (see also Ref. [70] and references therein). In this connection, we recall [27] that the quark contribution to the bag constant, and hence to the gluon condensate, is approximately an order of magnitude less than the pure YM one. So the above-mentioned agreement of our YM value (B.6) with the phenomenological value is impressive.

Figure 1 .

 1 Figure 1. Infinite series for the gluon part of the VED to log-loop level.

3 Figure 3 .

 33 Figure 3. The NP pressure, the entropy and energy densities, the heat capacity as functions of T/T c . The NP pressure has a maximum at T c = 266.5 MeV. Here the horizontal dashed line is the general SB number (40), while the vertical dashed line at 0.6T c separates the low-temperatures region from the transition region (0.6 -1)T c .

4 Figure 4 . 2 Figure 5 .

 4425 Figure 4. The NP velocity of sound, conformity and the trace anomaly relation as functions of T/T c .

3 Figure 6 .

 36 Figure 6. The NP gluon pressure, the entropy and energy densities, the heat capacity as functions of T/T c in the low-temperatures region shown up to 0.6T c .

4 Figure 7 . 2 Figure 8 .

 4728 Figure 7. The NP velocity of sound, conformity and the trace anomaly relation as functions of T/T c . They are shown in the low-temperatures region up to 0.6T c .

4 Figure 9 . 4 Figure 10 .

 49410 Figure 9. The zero temperature gluon condensate and the NP trace anomaly relation with minus sign, as it enters Eq. (49). Both scaled by T 4 and shown as functions of T/T c .

From

  -ω) + ln 1 -e -βω 1 -e -βω .(A.10)Appendix B. The scale-setting scheme

[ 27 ]∆ 2

 272 = 0.4564 GeV 2 , ∆ = 0.6756 GeV. (B.3)The effective gluon masses, defined in the relations(20) and(26), then become m

Table 1 .

 1 The NP analytical and lattice trace anomalies at high temperatures.

		Numbers
	T /T c Ours, Fig. 9 Lattice, Ref. [49]
	2	0.762	0.860
	3	0.453	0.475
	4	0.252	0.255
	5	0.166	0.166

Acknowledgments

This paper is dedicated to the memory of the late Prof. J. Zimanyi, who has initiated this investigation. Support by HAS-JINR Scientific Agreement (P. Levai) is to be acknowledged. We would like to thank L. Csernai, T. Biró, T. Csorgö, P. Van, G. Barnaföldi, V. Skokov and especially C. Wilkin and S. Pochybova for useful discussions, comments, remarks and help.

Appendix A. The summation of the thermal logarithms

In the second terms of Eqs. (19) and (25) the summation over the Matsubara frequencies can be done explicitly [START_REF] Kapusta | Finite-Temperature Field Theory[END_REF], as follows:

In terms of the above-introduced parameters, the sums in Eq. ( 19) containing the corresponding logarithms look like:

It is convenient to introduce the notations:

and similarly, letting ω → ω

In these expressions we introduced the following notations:


Fl Stancu
email: fstancu@ulg.ac.be

Can Y(4140) be a ccss tetraquark ?

published or not. The documents may come

Can Y(4140) be a tetraquark?

I. INTRODUCTION

The CDF Collaboration [1] has recently observed a narrow structure in the J/ψφ mass spectrum of B + → J/ψφK + decays, which has been named Y(4140). Its mass and decay width are M = 4143.0 ± 2.9(stat) ± 1.2(syst) MeV/c 2 and Γ = 11.7 +8. 3 -5.0 (stat) ± 3.7(syst) MeV/c 2 respectively, which suggest that its structure does not fit conventional expectations for charmonium states. The CDF Collaboration expects that the J/ψφ final state, with positive C-parity and two J P C = 1 --vector mesons (VV), is a good candidate for an exotic meson search. This resonance is well above the threshold for open charm decay D + s D - s at 3936.68 MeV and a charmonium cc with this mass would decay into an open charm pair predominantly and have a small branching fraction into J/ψφ [2]. The mass of Y(4140) is below the threshold of the decay channel D * + s D * - s at 4224.6 MeV, and not far above the J/ψφ threshold at 4116. [START_REF] Liu | [END_REF] MeV.

More recently the Belle Collaboration reported preliminary results on Y(4140) [START_REF] Yuan | for the BES and the Belle Collaborations[END_REF]. No significant signal was found but their efficiency is low for the mass of Y(4140). The upper limit on the production rate B(B + → Y (4140)K + , Y (4140) → J/φ) is 6 × 10 -6 at 90% C.L.

This upper limit is lower than the central value of the CDF measurement (9±3.4±2.9)×10 -6

[1] which is thus considered not to contradict the CDF measurement.

The Belle Collaboration also searched for Y(4140) in the J/ψφ mass spectrum of the twophoton process γγ → J/ψφ [START_REF] Yuan | for the BES and the Belle Collaborations[END_REF]. Again, the efficiency was low and no signal was reported. In exchange, evidence was found for a new narrow structure at 4.35 MeV and width 13. [START_REF] Yuan | for the BES and the Belle Collaborations[END_REF] MeV, with a statistical significance of about ∼ 3.5σ in the J/ψφ mass spectrum. This resonance was named X(4350).

As such, the present situation allows a new opportunity to look for exotics. The fashionable option of a D * s D * s molecule has been considered in Refs. [START_REF] Liu | [END_REF][5][6][7] and the QCD sum rules in Ref. [8][9][10]. where states with J P C = 0 ++ or 2 ++ are favoured. Let us note however that the Belle Collaboration measurement of a two-photon partial width difavours the scenario of Y(4140) to be a D * s D * s molecule with J P C = 0 ++ or 2 ++ [START_REF] Yuan | for the BES and the Belle Collaborations[END_REF]. Prior to the observation of Y(4140) by the CDF Collaboration, predictions for tetraquarks ccss seen as diquark-antidiquark systems with various J P C were made in a simple nonrelativistic model including = 0 and 1 partial waves in Ref. [11] and in a relativistic framework based on the quasipotential approach in Ref. [12]. In the latter, states with 0 ++ and 1 +± acquired masses in the range 4.1 -4.2 MeV.

We should also mention that the resonance Y(4140) was studied as the second radial excitation of the P-wave charmonium χ cJ (J = 0 and 1), looking at the hidden charm decay mode. The conclusion was that such a description is problematic [13].

Deciphering the nature of Y(4140), if confirmed in the future, (presently the B-factories have a poor acceptance for B → KJ/ψφ in the desired range [14]), is a new challenge.

Thus it is legitimate to consider the tetraquark interpretation without correlated quarks or antiquarks, and try to find out if the Y(4140) fits into the spectrum of the ccss system.

Most important, we search for the decay pattern given by this possible structure. For simplicity, we use the model of Ref. [15] which successfully describes the X(3872) as a ccq q tetraquark. In Ref. [15] it was shown that X(3872) can be interpreted as an eigenstate of the chromomagnetic interaction, where the lowest 1 ++ has a dominant octet-octet component (0.9997) and a very small singlet-singlet component (0.026) which explains why this state decays with a very small width into J/ψ + ρ or J/ψ + ω, in agreement with the experimental value for the total width Γ < 2.3 MeV of X(3872) [16], and that the J/ψ + pseudoscalar channel is absent. As Y(4140) is seen to be narrow and decays into two vector mesons we wonder whether or not the same mechanism can give an explanation of its small width, about 5 times larger than that of X(3872), and similar to that of X(4350), but considerably narrower than the decay width of every other X,Y or Z resonances.

The paper is organized as follows. In Sec. II we introduce the quark model used in this study. In Sec. III we recall the basis states in the direct meson-meson channel with emphasis on the charge conjugation quantum number. In Sec. IV we present the matrix elements of the Hamiltonian [15] for J P C = 0 ++ , 1 ++ , 1 +-and 2 ++ states. In Sec. V we show the calculated spectrum and discuss its features. The last section is devoted to conclusions. In Appendix A we derive the orthogonal transformation from the direct meson-meson channel to the exchange meson-meson channel for states 0 ++ , in Appendix B for states with 1 ++ and 1 +-and in Appendix C for states with 2 ++ . Appendix D is devoted to an attempt to dynamically derive effective masses in a standard constituent quark model in order to justify the simplicity of the present study and enlighten the choice of effective masses.

II. THE MODEL

This is an exploratory study, based on the simple model of Ref. [15] which can reveal the basic features of the ccss tetraquark, especially the structure of the wave functions. In the next section we introduce the relevant basis states in the color-spin space, including both the singlet-singlet channels and the octet-octet, simply called hidden color channels. There are no correlated quarks or diquarks, as in Ref. [11], for example.

Accordingly, the mass of a tetraquark is given by the expectation value of the effective Hamiltonian [15]

H = i m i + H CM , (1)
where

H CM = - i,j C ij λ c i • λ c j σ i • σ j . (2)
The first term in Eq. ( 1) contains the effective masses m i as parameters. The constants C ij represent integrals in the orbital space of some unspecified radial forms of the chromomagnetic part of the one gluon-exchange interaction potential and of the wave functions.

A warning should be given to the way of determining the effective masses m i to be used for multiquark systems. Besides the kinetic energy contribution, they incorporate the effect of a Coulomb-like term and of the confinement, the latter still being an open problem [17]. Thus, in principle, they cannot be directly extracted from meson or baryon spectra as discussed in Appendix D. Lack of better knowledge we however use the compromise proposed in Ref.

[15]

m c = 1550 MeV, m s = 590 MeV, (3)
but due to the arbitrariness in the choice of effective masses of quarks, precise estimates of the absolute values of tetraquark masses is difficult to make. One can have an approximate idea about the range where the spectrum should be located. But a shift of the whole spectrum is justified and sometimes even performed, like in the popular work of Maiani et al. [18], which deals with diquarks, where the arbitrariness in mass is even larger.

However, the relative distances between the eigenstates obtained from the chromomagnetic Hamiltonian (2) and the structure of its eigenstates do not depend on the effective masses, which is important for exploring the strong decay properties. The parameters C ij have been taken from Ref. [19] where a more complete list, containing also parameters needed in this work, is given. The required values are

C cs = 5.0 MeV, C cc = 5.5 MeV, C cs = 6.7 MeV, C ss = 8.6 MeV. (4)
We should mention that the above parameters were extracted from a global fit to meson and baryon ground states. For some mesons into which Y(4140) can decay in Table I we compare the experimental masses of PDG [20] with the theoretical values obtained from the two-body version of ( 1) and (2) in the parametrization (4) which is

m q q = m q + m q -λ c 1 • λ c 2 σ 1 • σ 2 C q q (5)
where q stands here for any light or heavy quark. From Table I one can see that the Due to this discrepancy it is meaningless to compare the tetraquark states with the theoretical threshold. This work questions the practice of using identical effective masses in both ordinary and exotic multiquarks. In such a case we would return us to the schematic treatment of the never observed "stable" H-dibaryon [21] predicted to be strongly bound by the chromomagnetic interaction. We do not intend to make a fine tuning of the effective masses. We are mostly interested in the structure of the tetraquark wave functions which essentially depends on the hyperfine interaction. We shall compare the calculated spectrum to the experimental thresholds. In Appendix D we give a simple proof that one cannot use the same effective masses both in mesons and tetraquarks.

In the following, an important parameter in this study is the difference between the values of C cs and C cs . In fact we shall see that the replacement of the light quarks q = u, d with the strange quark s does not much modify the structure of the ccss with respect of that of ccq q.

III. THE BASIS STATES

Here we use a basis vectors relevant for understanding the decay properties of tetraquarks.

The total wave function of a tetraquark is a linear combination of these vectors. We suppose that particles 1 and 2 are quarks and particles 3 and 4 antiquarks, see Fig. 1. In principle the basis vectors should contain the orbital, color, flavor and spin degrees of freedom such as to account for the Pauli principle. But, as we consider = 0 states the orbital part is symmetric and anyhow irrelevant for the effective Hamiltonian described in the previous section. Moreover, as the flavor operators do not explicitly appear in the Hamiltonian, the flavor part does not need to be specified. A detailed description of the three distinct bases corresponding to the three choices of internal coordinates shown in Fig. 1 is presented in Refs. [22,23]. It was found that the inclusion of meson-meson channels accelerate the convergence, for example in ccq q tetraquarks [24].

We remind that in the color space there are three distinct bases: a) |3 12 3 34 , |6 12 6 34 , b) |1 13 1 24 , |8 13 8 24 , and c) |1 14 1 23 , |8 14 8 23 , associated to the three distinct internal coordinate systems shown in Fig. 1. The 3 and 3 are antisymmetric and 6 and 6 are symmetric under interchange of quarks and antiquarks respectively. This basis is convenient for diquark-antidiquark models, where usually the color space is truncated to contain only |3 12 3 34 states [18]. This reduces each J P C spectrum to twice less states than allowed by the Pauli principle [START_REF] Stancu | 11th International Conference on Nuclear Reaction Mechanisms[END_REF] and influences the tetraquark properties. The sets b) and c) contain a singlet-singlet color and an octet-octet color state. The amplitude of the latter vanishes asymptotically, when the mesons, into which a tetraquark decays, separate. These are called hidden color states by analogy to states which appear in the nucleon-nucleon problem, defined as a six-quark system [START_REF] Harvey | [END_REF]. The contribution of hidden color states to the binding energy of light tetraquarks has been calculated explicitly in Ref. [22]. Below we shall point out their role in the description of the of ccss tetraquarks. The situation is similar to the interpretation of the X(3872) resonance as a ccq q tetraquark in Ref. [15], where its small width has been explained as due to a tiny J/ψ + ρ or J/ψ + ω component in the wave function of the 1 ++ tetraquark state.

As the quarks and antiquarks are spin 1/2 particles the total spin of a tetraquark can be S = 0 , S = 1 or S = 2.

For S = 0 there are two independent basis states (two Young tableaux) for each channel.

The spin states associated to the three distinct internal coordinates depicted in Fig. 1 are:

a) |S 12 S 34 , | A 12 • A 34 , b) |P 13 P 24 , |(V 13 V 24 ) 0 , c) |P 14 P 23 , |(V 14 V 23 ) 0 , respectively,
where S stands for scalar, A for axial and P and V for pseudoscalar and vector subsystems and the lower index 0 indicates the total spin. The relation between the three different bases can be found in Ref. [23].

For S = 1 there are three independent spin states, corresponding to three distinct Young tableaux. Presently we are interested into those corresponding to Fig. 1b, named the direct meson-meson channel. In this channel we remind that the basis vectors are [23]

|(P 13 V 24 ) 1 , |(V 13 P 24 ) 1 , |(V 13 V 24 ) 1 . (6)
As above, the lower index indicates the total spin 1.

In this case the charge conjugation operator is related to permutation properties of the basis vectors in a simple way. Under the transposition (13) manifestly one has (13)|P 13 = -|P 13 , (13)|V 13 = +|V 13 ,

and similarly for the transposition (24) (24)|P 24 = -|P 24 , (24)|V 24 = +|V 24 .

The case S = 2 is trivial. There is a single basis state

χ S = |(V 13 V 24 ) 2 , (9)
which is symmetric under any permutation of quarks.

From Ref. [START_REF] Stancu | Group theory in subnuclear physics[END_REF] Ch. 10, one can see that the permutation (13)(24) leaves invariant the color basis vectors |1 13 1 24 and |8 13 8 24 . Then, with the identification 1 = c, 2 = s, 3 = c and 4 = s the permutation (13)( 24) is equivalent to the charge conjugation operator [START_REF] Stancu | AIP Conf. Proc[END_REF].

Thus all basis states introduced below have a definite charge conjugation, which is easy to identify.

IV. MATRIX ELEMENTS

For a ground state tetraquark the possible states are J P C = 0 ++ , 1 ++ , 1 +-and 2 ++ . In The matrix elements introduced below appeared in the Proceedings [START_REF] Stancu | 11th International Conference on Nuclear Reaction Mechanisms[END_REF]. For the reader's convenience we present them here again. They correspond to the scalar, axial and tensor tetraquarks introduced above. Later on, the authors of Ref. [19] calculated the matrix elements of the chromomagnetic interaction (2) in a basis corresponding to Fig. 1a. Although the spectrum is the same, one cannot distinguish between charge conjugation C = 1 and C = -1 because in that basis J P = 1 + states do not have a definite charge conjugation. To identify C one must return to our basis. Therefore we found it convenient to use our basis which can give direct information to experimentalists.

For J P C = 0 ++ the basis constructed from products of color and spin states associated to Fig. 1b

The chromomagnetic interaction Hamiltonian with minus sign, -H CM , acting on this basis leads to the following symmetric matrix

                        16(C 13 + C 24 ) 0 0 8 2 3 (C 12 + C 23 ) - 16
3 (C 13 + C 24 ) -8 2 3 (C 12 + C 23 ) 16 √ 2 3 (C 23 -C 12 ) -2(C 13 + C 24 ) 4 √ 3 (2C 12 -7C 23 ) 16 3 C 12 + 56 3 C 23 + 2 3 (C 13 + C 24 )                         (11)
For J P = 1 ++ there are two linearly independent basis vectors built as products of color and the third spin state of Eq. ( 6) .

ψ 1 1 ++ = |1 13 1 24 (V 13 V 24 ) 1 , ψ 2 1 ++ = |8 13 8 24 (V 13 V 24 ) 1 . (12)
The matrix associated to the chromomagnetic interaction -H CM is

          - 16
3 (C 13 + C 24 ) 8 √ 2 3 (C 23 -C 12 ) 2 3 (4C 12 + 14C 23 + C 13 + C 24 )           (13)
which has been previously related to X(3872). Its lowest state gave a mass of 3910 MeV to X(3872) [15,[START_REF] Stancu | 11th International Conference on Nuclear Reaction Mechanisms[END_REF], quite close to the experimental value [16].

For J P = 1 +-there are four linearly independent basis vectors built as products of color states b) and the first and second spin states of Eq. ( 6)

ψ 1 1 +-= |1 13 1 24 (P 13 V 24 ) 1 , ψ 2 1 +-= |1 13 1 24 (V 13 P 24 ) 1 , ψ 3 1 +-= |8 13 8 24 (P 13 V 24 ) 1 , ψ 4 1 +-= |8 13 8 24 (V 13 P 24 ) 1 . (14)
The matrix associated to the chromomagnetic interaction -H CM is

                        16(C 13 - 1 3 C 24 ) 0 0 8 2 3 (C 12 + C 23 ) 0 - 16
3 (C 13 -C 24 ) 8 2 3 (C 12 + C 23 ) 0 -2(C 13 - 1 3 C 24 ) - 4 3 (2C 12 -7C 23 ) 2 3 (C 13 -3C 24 )                         (15)
For J P C = 2 ++ the basis vectors are

ψ 1 2 ++ = |1 13 1 24 χ S , ψ 2 2 ++ = |8 13 8 24 χ S , (16)
where χ S is the S = 2 spin state (9). The corresponding -H CM 2 × 2 matrix is

          - 16
3 (C 13 + C 24 ) - 8 √ 2 3 (C 23 -C 12 ) - 2 3 (4C 12 + 14C 23 -C 13 -C 24 )           (17)
In the calculation of the matrix elements we have used the equalities

C 14 = C 23 , C 12 = C 34 , (18)
due to charge conjugation.

The above matrices have been first used to calculate the full spectrum of ccq q with q = u, d [START_REF] Stancu | 11th International Conference on Nuclear Reaction Mechanisms[END_REF]. They can be used in any quark model containing a chromomagnetic interaction. In that case the coefficients C ij should be replaced by integrals containing the chosen form factor of the chromomagnetic interaction and the orbital wave functions of the model.

Note that the matrices ( 11), ( 13) and ( 17

V. THE SPECTRUM OF ccss

The calculated spectrum is exhibited in Fig. 2. There are several states in the range 4.1 -4.2 MeV, consistent with predictions of more realistic models [12]. This implies that the choice of the effective masses ( 3) is quite adequate for ccss tetraquarks. Here we are mostly interested in those states with a small amplitude in the VV channel in the present parametrization.

A. J P C = 0 ++

In the order indicated by the basis (10)

decaying substantially into PP channels and much less into the VV channel J/ψφ. The last two amplitudes correspond to hidden color channels which do not decay strongly.

The tetraquark states mentioned above can also decay into the D + s D - s , the threshold being 3936.68 MeV. The corresponding amplitudes can be obtained from the orthogonal transformation going from the direct meson-meson channel, Fig. 1b, to the exchange mesonmeson channel, Fig. 1c. In Appendix A we present the exchange basis vectors (A2) in terms of the direct basis vectors (10) given by this transformation. Using the expressions(A3)-(A6) and the amplitudes (19) obtained in the direct channel we can write the lowest state 0 ++ in the exchange channel basis as

ψ 0 ++ (3995) = -0.7244 |1
Looking at Fig. 1c, again with 1 = c, 2 = s, 3 = c and 4 = s, we can identify the color singlet-singlet channels in (A2) with the asymptotic meson-meson channels. Thus we have

ψ 1ex 0 ++ = |1 14 1 23 |P 14 P 23 = D s D s (22)
ψ 2ex 0 ++ = |1 14 1 23 |(V 14 V 23 ) 0 = D * s D * s (23)
From the wave function (21)

C. J P C = 1 ++

For the lowest 1 ++ state at 4194 MeV, which is quite close to the experimental range, the amplitudes of its components in the basis (12) are shown in Table II for two values of C cs . The singlet-singlet channnel 1 13 1 24 (V 13 V 24 ) 1 has a very small amplitude for both values of C cs . The hidden color state 8 13 8 24 (V 13 V 24 ) 1 is by far the dominant component. The situation is entirely analogous to that of the resonance X(3872) in the same model [15,[START_REF] Stancu | 11th International Conference on Nuclear Reaction Mechanisms[END_REF].

The clue was to have a nonvanishing, but small, value for C 23 -C 12 ≡ C cq -C cq in Eq. (13). For X(3872) one had 1.5 MeV, here we have C cs -C cs = 1.7 MeV imposed by the parametrization (4). As seen from Table II a decrease of C cs will make the hidden color state even more dominant but it will somewhat deteriorate the value of the M D * s -M Ds splitting, the experimental value of which is 143.8 MeV. Combined with the phase space of the decay Y(4140) → J/ψφ obtained from the experimental threshold, the lowest state 1 ++ of the tetraquark ccss would acquire a rather small width, as required by experiment, and could be the best candidate for Y(4140) in a tetraquark interpretation.

It is useful to write the wave function of the lowest state also in the exchange channel basis, as for the scalar tetraquaks above. For this purpose we use the transformation between the direct and exchange channel basis vectors derived in Appendix B, namely the Eqs. (B14) and (B15). Together with the amplitudes from Table II

From Fig 1c and Eq. (B1) one has

ψ 1ex 1 + = D s D * s , ψ 2ex 1 + = D * s D s (25)
According to (B8) a molecular-type component with C = + is obtained in the exchange channel as

ψ 1ex 1 ++ = 1 √ 2 (D s D * s -D * s D s ) ( 2 6 )
having a very large probability of 91.3 % in the 1 ++ ground state. The phase space is larger than for the J/Ψφ channel, so that a large width is expected in the D s D * s channel. The second term in ( 24) is a hidden color component, which does not decay, but vanishes asymptotically.

D. J P C = 2 ++

The spectrum is formed of two, nearly degenerate states, both too high for Y(4140), by about 200 MeV. In the parametrization (4) the wave function of the lowest state has the amplitudes (-0.4675, 0.8840) [START_REF] Stancu | Group theory in subnuclear physics[END_REF] in the order of the basis (16). One can see that the color singlet-singlet state ψ 1 2 ++ = |1 13 1 24 χ S has a small amplitude and the hidden color state ψ 2 2 ++ = |8 13 8 24 χ S is dominant, again due to the smallness of the off-diagonal matrix element C cs -C cs . With C cs = 6.0 MeV the amplitudes become ( -0.2274, 0.9738). This would give rise to a even smaller decay width into J/ψφ. The calculated mass fits better into the newly found narrow structure X(4350) reported by the Belle Collaboration [START_REF] Yuan | for the BES and the Belle Collaborations[END_REF]. According to Appendix C the wave function of the lowest state obtained from the latter amplitudes becomes

ψ 2 ++ (4343) = 0.8442 D * s D * s -0.5390 ψ 2ex 2 ++ (28)
where we have replaced ψ 1ex 2 ++ by its physical content. This state has a dominant moleculartype structure plus a hidden color component (C2) which would vanish asymptotically, but is important at short range. In a standard hadronic molecule interpretation [START_REF] Liu | [END_REF][5][6][7] the second component is absent because the emitted mesons do not have a structure.

VI. CONCLUSIONS

Prior to the CDF experiment [1], among other multiquark systems, the tetraquark ccss has been studied with a different parametrization from the one considered here and with a different basis using an SU(6) classification [START_REF] Cui | High Energy Phys[END_REF]. In that basis it is difficult to identify the VV component. Moreover a distinction between charge conjugation C = 1 and C = -1 has not been made.

Our study favours mostly the 1 ++ sector for the ccss tetraquark interpretation of the recently observed narrow structure Y(4140) [1]. If correct, Y(4140) would be the strange analogue of X(3872), when interpreted as a ccq q tetraquark. This observation follows from the fact that in the schematic model of Ref. [15] the chromomagnetic interaction leads to a similar composition of the wave function in the basis (12) for tetraquarks containing either u and/or d, like X(3872), or s quarks, like Y(4140).

Note however that one should consider the effective masses (3) with caution. They have been obtained from fitting baryon and meson spectra. A natural question raises whether or not these masses are adequate for tetraquarks. This study questions their use in tetraquarks, inasmuch as they contain the effect of the kinetic energy and of the confinement. The confinement has been thoroughly studied in lattice calculations. A Y-shape confinement potential is almost confirmed by lattice results (see i. e. [29]). Information from lattice calculations on tetraquark (see i. e. [30]) may lead to a better understanding of the effective masses to be used in simple models. Thus, with the present parametrization it is meaningless to look at the tetraquark spectrum relative to the theoretical threshold. A detailed discussion based on a simple example is presented in Appendix D. A correct interpretation of the narrow structure Y(4140) observed by CDF [1] would be possible if its existence was confirmed and its quantum numbers J P C were found experimentally, in order to remove the doubt cast by some theoretical interpretations [31]. Also,

Next, using Appendix C of Ref. [23] for the spin states we obtain the spin-color exchange channel basis in terms of the spin-color direct channel basis (10).

For J P C = 0 ++ the exchange channel basis vectors are defined by In terms of the direct channel basis vectors (10) the orthogonal transformation ex1is given Although we rely on the same PDG data [20] these masses are different from those of Eq.

ψ 1ex 0 ++ =
(3) proposed in Ref. [15]. The difference is however very small for the c quark and this can be explain by the cancellation of the kinetic and potential energies, as one can see from Table III. Such a cancellation does not take place for the quark s. Thus in a dynamical approach based on a Hamiltonian like (D1) there is a cancellation of various parts of the Hamiltonian. The cancellation is more subtle in a tetraquark which has 6 distinct quarkquark or quark-antiquark pairs, while in a meson there is only one pair. It follows then that the effective masses needed for a tetraquak can be different from those of Eq. (D6). Indeed, using Table III, we obtain

m ef f c + m ef f s = 1 2 H 0 ccss = 2238.5 MeV (D7)
which is different from the sum of masses in (D6). This proves that one cannot use the same effective masses in mesons and tetraquarks. In this light we can consider the choice (3) acceptable and understand why the agreement with the experiment in Table I is unsatisfactory for mesons. A better knowledge of the confinement and more precise calculations

  two-body Hamiltonian (5) with the masses effective (3) systematically overestimates the meson masses. Therefore the threshold energies of the channels J/Ψφ, D s D s , D * s D * s , and D * s D * s , are considerably overestimated.

FIG. 1 :

 1 FIG. 1: Three independent relative coordinate systems. Solid and open circles represent quarks and antiquarks respectively: (a) diquark-antidiquark channel, (b) direct meson-meson channel, (c) exchange meson-meson channel.

  the direct meson-meson channel, in each case a basis can be built with the quark-antiquark pairs(1,[START_REF] Yuan | for the BES and the Belle Collaborations[END_REF] and(2,[START_REF] Liu | [END_REF] as subsystems, where each subsystem has a well defined color state, a singlet-singlet or an octet-octet. This arrangement is convenient to describe hidden charm J/ψ + light meson or η c + light meson channels, the light meson quantum numbers being consistent with J P C . The other quark-antiquark pairs,(1,[START_REF] Liu | [END_REF] and(2,[START_REF] Yuan | for the BES and the Belle Collaborations[END_REF] describe open charm meson channels, here called exchange channels (see Fig.1c) as e. g.D s D s , D s D * s or D * s D * s .One can fix a basis in terms of the problem one looks at, but for convenience, in the calculations one can pass from one basis to another by an orthogonal transformation. In this study the adequate basis is that related to the direct meson-meson channel, depicted in Fig.1b. The orthogonal transformations from the direct to the exchange meson-meson channel for J P C = 0 ++ and 1 ++ are given in Appendices A and B respectively.

FIG. 2 :

 2 FIG. 2: The spectrum of the ccss tetraquark with the Hamiltonian introduced in Sec. II and the color-spin bases of Sec. III

Finally, we should

  mention that the present study does not exclude the 0 ++ sector. In fact, in the molecular D * s D * s , Y(4140) can have the quantum numbers 0 ++ or 2 ++ . As mentioned in the introduction, here we stress again that the Belle Collaboration measurement of a two-photon partial width disfavors the scenario of Y(4140) to be a D * s D * s molecule with J P C = 0 ++ or 2 ++ [3].

  the measurement of the decay widths of other open channels such as η c + η or η c + η is important. If such decays are observed, the 0 ++ sector is favored, if not, the sector 1 ++ is favoured in a tetraquark interpretation. Complementary information can also be obtained from the decays to D s D s , D s D * s , D * s D * s etc. In conclusion, as a start, we have presented results in a tetraquark schematic model to get a hint on the interpretation of Y(4140), which remains an open problem. Perhaps a more realistic view, if Y(4140) was confirmed, would be to have a compact tetraquark structure at short range and a molecular structure at medium or large range. Anyhow, a more elaborate study of the ccss tetraquark system is worth by itself.VII. ACKNOWLEDGMENTSI am most grateful to S. L. Olsen and S. Eidelman for valuable correspondence on the Belle Collaboration preliminary measurements.

14

 14 APPENDIX A: DIRECT TO EXCHANGE CHANNEL BASIS FORJ P C = 0 ++In the following we need to express the color exchange basis in terms of the color direct basis vectors. The well known relations are |1

TABLE I :

 I Theoretical and experimental meson masses in MeV

	Meson	J P C	Theory	Exp
	J/ψ	1 --	3121	3096.9
	φ	1 --	1225.9	1019.5
	D s	0 -?	2032.0	1968.5
	D * s	1 -?	2175.7	2112.3

  are ψ 1 0 ++ = |1 13 1 24 P 13 P 24 , ψ 2 0 ++ = |1 13 1 24 (V 13 V 24 ) 0 , ψ 3 0 ++ = |8 13 8 24 P 13 P 24 , ψ 4 0 ++ = |8 13 8 24 (V 13 V 24 ) 0 .

  the lowest state, 3995 MeV, has the amplitudes

	(-0.7737, 0.0594, 0.1789, 0.6049)	(19)
	The first number in the bracket implies that this state can decay substantially into a PP
	channel, i. e. η c + η (threshold 3528 MeV) or η c + η (threshold 3938 MeV). The second
	number indicates a very weak coupling to the VV channel. The mass is too low for the
	decay into J/ψφ.
	A better candidate would be the first excited state at 4135 MeV with the amplitudes
	(-0.6172, -0.1774, 0.4006, -0.6536)

  14 1 23 |P 14 P 23 + 0.0743 |1 14 1 23 |(V 14 V 23 ) 0 -0.2088 |8 14 8 23 |P 14 P 23 + 0.6529 |8 14 8 23 |(V 14 V 23 ) 0

  we can see that the open D s D s channel acquires a large ampli-

	tude in the ground state (3995 MeV), corresponding to a probability of about 50%, which
	will imply a large decay width in this channel and a negligible amplitude 0.5% to the closed
	channel D * s D	*

s .

TABLE II :

 II The D * s -D s splitting (MeV) and the amplitudes of the basis vectors (12) of the 1 ++ state at 4194 MeV as a function of C cs (MeV). D s 1 13 1 24 (V 13 V 24 ) 1 8 13 8 24 (V 13 V 24 ) 1 In this sector the lowest state has an appropriate mass, but not the convenient charge conjugation. It would decay exclusively into a PV channel for = 0 tetraquarks. For general interest the exchange meson-meson basis is also given in Appendix B.

	C cs s -6.0 D * 128.0	0.0245	0.9997
	6.7	143.7	0.0399	0.9992
	B. J P C = 1 +-

  associated to C cs = 6.7 Mev we obtain for the lowest 1 ++ state

	ψ 1 ++ (4194) = -0.9554 ψ 1ex 1 ++ + 0.2954ψ 2ex 1 ++ .

  |1 14 1 23 P 14 P 23 , ψ 2ex 0 ++ = |1 14 1 23 (V 14 V 23 ) 0 , ψ 3ex 0 ++ = |8 14 8 23 P 14 P 23 , ψ 4ex 0 ++ = |8 14 8 23 (V 14 V 23 ) 0 .

	(A2)

TABLE III :

 III Expectation values of H 0 , Eq. (D1) and of its kinetic and potential parts obtained from a trial wave function with the parameter a (see text).

	System	a	Kinetic	Potential	H 0
		(fm -1 )	( M e V )	( M e V )	( M e V )
	cc	2.5	2 2 9 . 6	-3 1 7 . 8	3 0 9 2
	ss	1.4	3 3 6 . 7	3 . 2 7	1 0 2 0
	ccss	2.1	6 5 6 . 8	-3 9 . 8	4 4 7 7

by the following relations

These relations are used to derive Eq. ( 21).

APPENDIX B: DIRECT TO EXCHANGE CHANNEL BASIS FOR

In the exchange channel corresponding to Fig. 1c the basis states can be defined as above. Note however that in this case they do not all have a definite charge conjugation.

By analogy to the direct channel, let us first introduce the J P C = 1 + the exchange channel basis vectors as Using Appendix C of Ref. [23] which gives the transformations in the spin space, the exchange channel basis vectors (B1) can be written as linear combinations of the direct channel basis vectors (12) and (14). The orthogonal transformation gives the equations

From these relations one can see that only ψ 3ex 1 + and ψ 6ex 1 + have a definite charge conjugation C = -. But in the exchange channel one can further introduce definite charge conjugation from the basis vectors in the following way. For C = + the normalized states are

Then for C =the normalized states are

Lastly, replacing the expressions of ψ 1ex 1 + , ψ 2ex 1 + , ψ 4ex 1 + and ψ 5ex 1 + in Eqs. (B8) and (B9) we get the orthogonal transformation relating the exchange channels with the direct channel wave functions (12) for C = +

This transformation will be used in the subsection C of Sec. V.

APPENDIX C: DIRECT TO EXCHANGE CHANNEL BASIS FOR

The relations between the exchange and direct basis are in this case a direct consequence of the definitions (A1) inasmuch as the spin state χ S is symmetric under any permutation of S 4 . One obtains

This transformation will be used in the subsection D of Sec. V.

APPENDIX D: EFFECTIVE MASSES

First we establish the relation between effective quark masses used in these calculations and masses m 0 i of a constituent quark model. For this purpose we start from the spinindepenent part of a simple model of the commonly used type [START_REF] Bhaduri | [END_REF]

with a kinetic part from which the center of mass energy has been removed and a potential part containing a two-body linear confinement V (r ij ) and a Coulomb-like term

Together with a spin-spin part identical to that of Ref. [START_REF] Bhaduri | [END_REF] (not necessary to be specified here, also used in other studies as e. g. Ref. [23]), we have fitted the parameters of (D1) to reproduce resonably well the mass of J/ψ and φ mesons by choosing a trial wave function of the form φ 0 ∝ exp(-a 2 r 2 ij /2). These calculations are aimed at understanding the basic reason behind the difference between effective masses and bare masses m 0 i . The fitted parameters are Below they are used to estimate the expectation value of H 0 corresponding to a ccss system described by a trial wave function of the form R ∝ exp[-a(σ 2 + σ 2 + λ 2 )], a being a variational parameter, as above. Here, for convenience, we use the internal coordinates From these results we can define effective masses in a similar way as in Ref. [15]. We have m ef f q = 1 2 H 0 q q (D5)