Quark Hadron Duality at Finite Temperature Presented by ERA at the LIV Cracow School of Theoretical Physics on QCD meets experiment, June 12-20, 2014.

Quark Hadron Duality at Finite Temperature thanks: Presented by ERA at the LIV Cracow School of Theoretical Physics on QCD meets experiment, June 12-20, 2014.

E. Ruiz Arriola, L.L. Salcedo
Departamento de Física Atómica, Molecular y Nuclear
and Instituto Carlos I de Física Teórica y Computacional,
Universidad de Granada, E-18071 Granada, Spain
   E. Megías
Grup de Física Teòrica and IFAE, Departament de Física,
Universitat Autònoma de Barcelona, Bellaterra E-08193 Barcelona, Spain
Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, D-80805 Munich, Germany
Abstract

At low temperatures we expect that all QCD observables are defined in terms of hadrons. This includes the partition function as well as the Polyakov loop in all representations. We analyze the physics underlying a microscopic derivation of the hadron resonance gas.

PACS numbers: 12.38.Lg, 11.30, 12.38.-t


1 Introduction

The promise of a phase transition at finite temperature in QCD from the confined hadronic phase to a deconfined quark-gluon phase has triggered a lot of activity both theoretically as well as experimentally. This original prediction by early lattice calculations [1, 2, 3] has turned into a smooth crossover after many years of accumulated experience [4]. The determination and understanding of the QCD equation of state (EoS) from the lattice [5] plays a crucial role in the current analysis of ultra-relativistic heavy ion collisions [6].

While in the limit of massless quarks the QCD Lagrangian is scale invariant, implying a vanishing trace of the energy momentum tensor, the symmetry is broken explicitly by the finite quark masses and anomalously by the necessary renormalization which introduces a energy scale and generates a trace anomaly. The most recent up-to-date results for 2+1 flavor lattices have been obtained by the Wuppertal-Budapest(WB) [7] and HotQCD [8] collaborations and after continued discrepancies there is a final consensus that maximal violation of scale invariance occurs at where the trace anomaly reaches its maximum value.

At very high temperatures, the typical momentum scales or are large and finite quark mass effects can be neglected. Due to asymptotic freedom the strong and running coupling constant becomes small and thus interactions can be neglected. Thus, one effectively has a gas of free and massless fermions (quarks and anti-quarks) and bosons (gluons), and scale invariance is restored. At the same time colour is delocalized corresponding to a deconfined phase. This allows in principle to count the number of quark and gluon elementary species by means of the Stefan-Boltzmann law. Current analyses show that this happens for temperatures much larger than .

Yet, there is the firm belief that because of confinement, hadron states, composite, extended and most often unstable bound states made of quarks and gluons build a complete basis at very low temperatures, where they effectively behave as point-like, stable, non-interacting and structureless particles. This quark-hadron duality resembles to a large extent the similar duality between different degrees of freedom as the one found in deep inelastic scattering (for a review see [9]). It is remarkable that in the large limit some of these requirements are indeed fulfilled (for a review see e.g. [10]), except and most notably the point-like character. Of course, as the temperature is raised we expect many excited states to contribute, but also that finite width and finite size effects play a role. How this hadronization happens in detail has not really been understood so far. Lattice calculations suggest that there is a smooth transition or crossover from the purely hadronic phase to the purely quark-gluon phase [4]. Actually, it is not obvious when this hadronic picture fails in practice or what is the main mechanism behind quark and gluon liberation at the lowest possible temperature. The present contribution tries to address this problem guided by our own experience on the field.

2 The hadron resonance gas

In the opposite limit of very low temperatures one expects an interacting gas of the lightest colour neutral particles (for pions and kaons) [11]. Because of spontaneous chiral symmetry breaking, the low-lying pseudoscalar particles are the lightest Goldstone bosons made of quarks. For the typical low momenta in the heat bath Goldstone bosons interact weakly through derivative couplings and hence interactions are strongly suppressed. Thus scale invariance violations of the non-interacting gas are due to the finite pion and kaon masses. Therefore, in the chiral limit of massless pseudoscalars scale invariance is also exact at sufficiently low temperatures. Because of the small signal, current lattice calculations of the trace anomaly are just above the edge of this pion and kaon gas, which is expected to work for .

When the temperature is raised, hadronic interactions among pions start playing a role and two- and more particle states contribute to the thermodynamic properties. The calculation may be organized according to the quantum virial expansion [12] where there are two kinds of contributions. The excluded volume corrections come from repulsive interactions which prevent particles to approach each other below a certain distance. The resonance contributions stem from attractive interactions which generate states living long enough to produce pressure in the system, meaning that the resonance can hit the wall of the container before it decays. A well known example is scattering where one has attractive and resonating states in the isospin corresponding to the and resonances whereas one has a repulsive core in the exotic channel [13, 14] providing a measure of the finite pion size. Once a meson is created, it may interact with another pion and produce a resonance, (which is a state) and so on. For baryons, the situation is similar where is a good example of a resonance contribution. This separation between attractive and repulsive contributions leaves out the residual interaction stemming from the background scattering. The Hadron Resonance Gas (HRG) corresponds to assume that all interactions among the lightest and stable particles can be described by an ideal gas of non-interacting resonances which are effectively pictured as stable, point-like and elementary particles, hence the trace anomaly is given by

(0)

where the sum is over all hadronic states including spin-isospin and anti-particle degeneracies, for mesons and baryons respectively, is the energy and are the hadron masses. This collection of masses is usually called the hadron spectrum and most often identified with the PDG compilation which represents an established consensus among particle physicists [15]. The states classification echoes the non-relativistic quark model, namely mesons are states and baryons states. Those falling outside this category are classified as “further states”. The hadron spectrum obtained from the PDG is depicted in Fig. 1 as well as its separation into mesonic and baryonic spectra.

Figure 1: Left: The full hadron spectrum made of quarks from the PDG [15]. Right: Mesonic and Baryonic spectrum.
Figure 2: Trace anomaly for the HRG using the PDG [15] (dashed) and the RQM [16, 17] (full) compared to the continuum extrapolated results of the WB [7] and HotQCD [8] collaborations. We also show the contributions from states with (dotted) and with (dot-dashed).

The result of using Eq. (2) with the PDG states [15] compared to the continuum extrapolated results of the Wuppertal-Budapest (WB) [7] and HotQCD [8] collaborations is shown in Fig. 2. It is amazing that this exceedingly simple picture works accurately almost below (the maximal scale-violating temperature) at about . Note that the lowest lattice data points at are first saturated when states with masses above the vector mesons are included. Higher temperatures start feeling excited hadronic states, which by their nature embody relativistic dynamics. The quantum statistical bosonic and baryonic character of the states accounts by less than correction to the classical Maxwell-Boltzmann distribution (corresponding to take ) at about , already well beyond the range where lattice and HRG agree and much smaller than the lattice uncertainties. The transition from hadrons to quarks has been analyzed on the light of strangeness [18] and an observable which vanishes for the HRG has been proposed.

Actually, the HRG model has arbitrated the discrepancies between the different collaborations in the past and the lattice community have had a long struggle until agreement between them and with the HRG model has been declared. While this has made the HRG model a sort of holy grail (see e.g. [19] and references therein), it is good to remind that despite of the phenomenological success it is not a theorem, nor a well defined approximation from the original QCD Lagrangian. The most compelling argument is that it is to date unclear how corrections to this simple approach should be implemented nor what the error estimate for the HRG should be. For a comparison, we also show the result of Eq. (2) using instead the Relativized Quark Model (RQM) [16, 17] which essentially combines two basic elements, the static energy among the constituents and a relativistic form of the kinetic energy which does not consider the spin of the particles but does not contain more parameters than QCD itself. The likewise impressive agreement of the RQM trace anomaly with the lattice data not only illustrates our point on the lack of uniqueness of what is actually being checked by these comparisons, but also that the RQM may provide information not listed in the PDG booklet.

For instance, the PDG lists the quantum numbers, decay modes, masses and widths of the resonances building the hadron spectrum, but no information on their size is provided. While for unstable particles this is a problematic issue (see e.g. [20] and references therein), within the heavy-ion literature the assumption of a constant volume is frequently made (see e.g. [21]). This information can be accessed by means of quark models or lattice calculations. On the other hand, as discussed in Ref. [22], the purely resonance character makes the very definition of the mass ambiguous, and this allows to generate an error band on the PDG prediction of the HRG model which gives a spread for the trace anomaly about half the lattice uncertainties.

3 QCD at finite temperature

As a guideline, let us provide some of the main features of QCD at finite temperature emphasizing some relevant aspects for the discussion. Many gaps in this sketchy presentation may be filled by consulting e.g. [5] and references therein. The QCD Lagrangian (in Euclidean spacetime) is given by

(0)

The QCD Lagrangian is invariant under colour gauge transformations

The QCD thermodynamics is obtained from the partition function

with the periodic or anti-periodic boundary conditions for gluons and quarks respectively

where the Matsubara frequencies are for gluons and for quarks. Preservation of the quark antiperiodic boundary conditions implies that only periodic gauge transformations are allowed, namely

(0)

Within the convenient Polyakov gauge ( stationary and everywhere diagonal [23]) the most general remaining transformation is either stationary and diagonal or of the type

(0)

where and . Large Gauge Invariance implies periodicity in with period

(0)

This is the finite temperature version of the Gribov copies, i.e., the fact that there is no complete gauge fixing in a non-abelian gauge theory. A drastic consequence of this periodicity property is that it becomes explicitly Broken in perturbation theory [24, 25]. Thus, we may consider this as a signal of the relevance of non-perturbative finite temperature gluons.

In the limit of massless quarks () the QCD Lagrangian is scale invariant, i.e. . This symmetry is broken by quantum corrections due to the necessary regularization. To see this consider the partition function dependence on the coupling constant after the rescaling of the gluon field (and ignoring renormalization issues)

(0)

Note that the only dependence on is the one shown explicitly. Thus,

(0)

where in the last equation we have assumed a vacuum space time independent configuration, with the volume of the system. On the other hand the free energy and internal energy are given by the thermodynamic relations

(0)

and the trace anomaly becomes

(0)

Generally, a renormalization scale has to be introduced to handle both IR and UV divergences. Thus, on purely dimensional grounds one has

(0)

Physical results should not depend on the renormalization scale, thus using that we get

(0)

Introducing the beta function

(0)

the trace anomaly becomes then

(0)

where

(0)

Here we have implemented, in full harmony with standard lattice practice (see, e.g., [26]), a subtraction to renormalize the vacuum contribution [27, 28]. The vanishing of the trace anomaly at zero temperature is consistent with quark-hadron duality, see Eq. (2).

Also in the massless quark limit, the QCD Lagrangian is invariant under chiral transformations,

(0)

Chiral symmetry is spontaneously broken by the chiral condensate in the vacuum down to

(0)

The Gell-Mann–Oakes–Renner relation, which in the simplest case becomes

(0)

relates the quark condensate and the quark mass to physically measurable quantities such as the pion mass and the pion weak decay constant . This relation exemplifies the quark-hadron duality, namely the fact that, in the confined phase of QCD, we expect quark observables to be representable by hadronic observables.

In the opposite limit that all quarks become infinitely massive, , the quark determinant becomes a constant which can be factored out of the path integral

(0)

and the resulting action corresponds to a pure Yang-Mills theory

(0)

This purely gluonic action exhibits a larger symmetry: ’t Hooft Center Symmetry , namely, invariance under gauge transformations which are periodic modulo a center element of

(0)

An example for and in the Polyakov gauge is given by the choice

(0)

with . The Polyakov loop is an order parameter of the center symmetry which is related to the free energy of a colour charge in the medium. In the Polyakov gauge, the Polyakov loop in the fundamental representation reads

(0)

The vacuum expectation value of the Polyakov loop transforms under the previous gauge transformation as

(0)

Therefore unbroken symmetry ()) implies and hence . The divergence of the free energy of a colour charge in the fundamental representation is interpreted as a signal for confinement [29]. The renormalization of the Polyakov loop is a subtle issue addressed in the lattice in Ref. [30].

In gluodynamics the center symmetry is spontaneously broken above a critical temperature and approaches unity (or any other central element) as the temperature increases. In fact, at high temperatures and one may expand [31],

(0)

Note that while this formula suggests that , the renormalization overshoots this value at a perturbative level in a tiny but visible way [32, 33, 31].

In full QCD the center symmetry is explicitly broken, which results in at all temperatures. In the limit of heavy quarks and low temperature one has or , where is the smallest residual binding energy of doubly heavy meson. Likewise for light dynamical quarks where denotes the ground state mass of a heavy-light meson (the mass of the heavy quark excluded) [34].

One can also define Polyakov loops in higher representations which have been subject of attention only a few times despite its very interesting properties [35, 36, 37, 38].

Scale invariance is also broken in gluodynamics. For instance in perturbation theory to two loops one has [39],

(0)

where at lowest order

(0)

Thus, taking , we expect to have a free gas of gluons and quarks in the high temperature limit. In the simple case of non-interacting particles the partition function is given by

(0)

where for bosons, for fermions and is the number of species. From the partition function we have the thermodynamic identities

For the massless quark and gluon gas the pressure is given by

(0)

which is the Stefan-Boltzmann law. Because in the high temperature limit the particles are effectively massless, scale invariance is restored and hence the (reduced) trace anomaly vanishes

(0)

These expectations have been checked in Ref. [40] by a lattice study in a wide temperature window, .

Thus, the quark condensate and the Polyakov loop in the fundamental representation become true order parameters in quite opposite situations. While signals spontaneous chiral symmetry breaking in the massless quark limit, signals confinement for infinitely heavy quarks. The real situation is somewhat intertwined, and can be summarized as follows.

  • Order parameter of chiral symmetry breaking ()
    Quark condensate

  • Order parameter of deconfinement ()
    Polyakov loop: Center symmetry broken

  • In the real world is finite but inflection points nearly coincide (accidental?)

    at about the same temperature .

The chiral-deconfinement crossover is a unique prediction of lattice QCD. Whether or not this result is accidental, could be answered by computing the (connected) crossed correlator,

(0)

which corresponds to the quark mass dependence of the Polyakov loop.

Finally, the correlation function between Polyakov loops in an arbitrary representation exhibits Casimir scaling (quenched approximation) [41]

(0)

It can be shown that this correlation function, which for the fundamental representation is related to the singlet free energy, , can also be written as a ratio of partition functions between sources placed at a distance and the vacuum [42], hence satisfying a spectral decomposition with integral weights and positive energies ,

(0)

where the singlet free energy has been introduced. One important property is that at large distances (for the unquenched full QCD case),

(0)

4 Relativized Quark-Gluon models

4.1 Relativity and thermodynamics

One of the most troublesome aspects of hadron binding is that it makes relatively heavy particles from massless ones, hence most of the mass comes from the interaction. A prominent example is the Glueball in gluodynamics, where the lightest state [43] has a mass while gluons are massless. Fully relativistic few body equations are not only hard to handle but encounter many difficulties regarding cluster decomposition properties [44, 45, 46]. This feature is particularly interesting as it is related to the compositeness nature of relativistic particles which build the hadrons, and strictly speaking we have to deal with relativistic statistical mechanics of interacting particles, a subject which has a long history [47].

Unfortunately, as we have shown, the physics of finite temperature QCD below the phase transition involves the excited hadronic spectrum, and thus relativity becomes an essential ingredient in the game. Relativized quark models (RQM) combine two basic elements, the static energy among the constituents and a relativistic form of the kinetic energy which does not consider the spin of the particles [16, 17].

4.2 The linear potential

In the Born-Oppenheimer approximation the object to be analyzed is the interaction between heavy sources and . In perturbation theory one has one gluon exchange which yields a Coulomb like interaction,

(0)

where and and are the generators111These are more customarily denoted by , while is twice the generator. of the SU(3) colour group corresponding to the representation of the source. This form of the colour interaction exhibits Casimir scaling, a property that is violated only at three loops in perturbation theory [48] and appears to hold non-perturbatively on the lattice with an additional linear potential contribution [49]. Thus, to a good approximation, the interaction between heavy sources on the lattice reads,

(0)

Thus, for quark-antiquark or gluon-gluon pairs coupled to a singlet state or a quark-quark pair coupled to the antifundamental representation (diquark) the following relations are obtained

(0)
(0)
(0)

In what follows we use to denote the string tension . As a consequence of Casimir scaling the ratio between the fundamental , adjoint and diquark colour sources are

(0)

By making simplifying assumptions, easy relations can be derived from Casimir scaling. For instance, consider the lowest glueball state by analyzing two massless spin-1 particles in the CM system assuming spin independent interactions, and similarly for the meson as composed of two massless quarks. Neglecting the Coulomb term, the respective mass operators appearing in the Salpeter equation read

(0)

Simple dimensional considerations imply that the eigenmasses are proportional to the square root of the string tension, thus

Here, as it is customary, we have matched the scales of gluodynamics and QCD by assuming a common value of in both theories. A rough estimate of the mass follows from using the uncertainty principle for the ground state, namely, by taking . For the glueball this yields

4.3 The cumulative number

The spectrum of the RQM model of Isgur and collaborators for in the case of mesons and for baryons [16, 17] is concisely shown in Fig. 3. A detailed comparison to individual states unveils a rather good description of the data. Of course, we do not expect this or any quark model Hamiltonian to describe accurately the individual levels. This, however, poses an interesting problem on how two different spectra including many excited states can quantitatively be compared, beyond eyesight and subjective impression just based on contemplating Figs. 1 and 3. One way suggested by Hagedorn in the early 60’s is by means of the cumulative number of states

(0)

The question is then to decide to what extent or coincide.

Figure 3: Left: The full hadron spectrum made of quarks from the Relativized Quark Model (RQM) of Isgur and collaborators where mesons are described as bound states and baryons as bound states [16, 17]. Right: Mesonic and Baryonic spectrum.

For our discussion we will consider a simplified version of that model where hyperfine splittings due to spin and three-body forces are ignored and the Hamiltonian for constituents (we restrict to and systems) is taken to be

(0)

We will consider explicitly some cases of interest below, but already at this level some important observations can be made on the growth of the cumulative number of states. To this end, let us adopt a semiclassical approximation, which should be reliable when the number of states is large. The number of states in the CM system and at rest, below a certain mass takes the form

(0)

where takes into account the degeneracy. For the sake of the argument, let us neglect the Coulomb term, thus , as well as the current quark masses. In this case, a dimensional argument, , , gives

(0)

It is not hard to show that lifting any of the above approximations only modifies this result by subleading powers of . Thus, for a finite number of degrees of freedom, the leading contribution to the cumulative number scales as a power of the mass. The qualitative power behaviour can be clearly identified as straight lines in the log-log plot of Fig. 4 for the cumulative number, where we compare the resulting cumulative number both for the PDG and the RQM separating the mesonic and baryonic contributions.

Figure 4: Cumulative numbers for the PDG (dashed line) and the RQM (full line). Left panel: mesonic states (log-log scale). Right panel: Baryonic states (log-log scale).

As we can also see, the PDG and RQM spectra look very much alike below . We note, however, that the spectrum for the RQM saturates sharply at which is about the cut-off mass where Isgur, Godfrey and Capstick stopped to compute states. On the other hand, the PDG states saturates at lower energy values in a softer fashion. Note that the RQM looks like a linear extrapolation (mind the log scale) of the PDG spectrum. The agreement at lower masses is not highly surprising since RQM parameters were tuned to reproduce low lying states on the one side and the listed PDG states fit into the quark model scheme on the other side. On the other hand, there are only and the quark masses as fitting parameters, so we should regard the agreement as a further big success of the RQM from a global point of view.

4.4 Bound states vs resonances

One issue which is problematic from the start is that if really counts the number of bound states, then we expect it to be, for flavors, just pions , nucleons and anti-nucleons , so . More generally we have low lying mesons and baryons, while in the RQM diverges as a power.

The cumulative number is by its own stair-case nature a piecewise discontinuous function, but as we go to higher states the discontinuities smooth out. This becomes particularly visible in the RQM in the range .

In addition, while in the PDG we have an issue regarding completeness of states, in the RQM case this is not the case; besides angular and spin flavor quantum numbers, the radial number can just be checked with the oscillation theorem, in the case, or simply by diagonalizing in a complete basis of normalizable states; no state will be left out in the process. In the PDG however, it is unclear if there are missing or redundant states although the consensus of listing states fitting into the quark model makes this argument into a circular one.

Of course, the meaning of and is different. While in the PDG listing we usually encounter a Breit-Wigner resonance parameterization characterized by a mass and a width, in the RQM we have just the mass of a bound or state. We expect that when we couple these bound states obtained from the RQM to the continuum there will be, besides a width, a mass shift , whence the cumulative number for bound states will generally differ as the one for Breit-Wigner resonances, but the sign of is a priori unknown.

4.5 Hadron sizes

The solution of the multiparton Hamiltonian Eq. (4.3) onto colour singlet states yields the corresponding hadron wave functions. To estimate the hadron size we can make use of the virial theorem based on stationarity of eigenstates under unitary coordinate scaling around the value, . For light quarks we may, for simplicity, take the massless quark limit . Thus, for the various terms in the Hamiltonian , likewise and where is the mean distance between particles and . Due to the virial theorem we have for a multiparton state

(0)

In particular,

(0)

for mesons and baryons respectively. So, in this model the size of a hadron grows linearly with the mass per constituent. For instance, from the relations and we have and . From we would get so we can identify the constituent quark mass as . Note that the above virial relation includes the Coulomb like contribution , since this term scales exactly as the kinetic piece. In the case of hadrons with one heavy quark, the virial theorem yields and .

4.6 String breaking

Confinement is often attributed to this ever-linear growing of the energy with the distance. This is true on the lattice only in the quenched approximation, where quark-antiquark creation is suppressed. In full QCD however, the string breaks, a fact that has been observed by lattice calculations at a distance [50]. This happens when a light pair is created in between the heavy quark and anti-quark sources , thus two colour singlet and mesonic states can be created. On the other hand, charge conjugation implies that the binding energy of the and the is the same and equals the residual energy of a heavy-light meson with total mass . Thus the string breaking distance corresponds to

(0)

Good approximations to these states exist in nature for and . From heavy-quark QCD we expect a universal (independent of the heavy quark spin and flavor) spectrum of hybrid hadron masses (heavy-quark mass subtracted). For the lightest, pseudoscalar, hybrid meson, the following sequence should approach a value of , for increasingly heavier quarks and using the PDG values in the -scheme,

(0)

which gives the estimate for . Another estimate can be made based on a constituent quark model picture where the total mass of the quark is , with the constituent quark mass and the current quark mass. Spontaneous breaking of chiral symmetry implies that in the chiral limit (massless current quark masses, ) the total mass is non vanishing, thus . Actually, for light mesons current masses can be neglected. Then the light meson has a mass and the mass of the light baryon is . The mass of a heavy-light meson would then be hence , which for and yields the estimate for the string breaking distance, a quite reasonable value.

4.7 Avoided crossings

The observation of string breaking for a system requires taking into account the mesonic channels into which the system may decay after pair creation from the vacuum. This coupled channel dynamics spans the Hilbert space and features the avoided crossing phenomenon familiar from molecular physics in the Born-Oppenheimer approximation [51]. For two channels, say and , one can compute the direct correlators yielding the lowest energies

(0)

These two channels are orthogonal for all . For simplicity we have disregarded the Coulomb piece as well as the residual interaction between the two heavy-light mesons and which is of van der Waals type and corresponds to meson exchange. Note that these curves cross when . Thus, the spectrum in the Hilbert space with and components reads,

(0)
(0)

where now the states and are piecewise orthogonal. The point corresponding to degenerate states is singular. A linear combination of and involves a crossed correlator between the channels and representing a variational improvement. The avoided crossing occurs because the finite energy of the non-diagonal interaction lifts the degeneracy, a feature called level repulsion. The adiabatic potential curves and appear as avoided crossings on the lattice [50] with a finite and small energy repulsion and a narrow transition region of about , see Fig. 5, which resemble the simple shape of Eq. (4.7). Therefore, as long as the size of the system remains small, we may ignore the string breaking effect. Otherwise, one has to consider a coupled channel dynamics with and states. Excited states potential curves should follow a similar pattern as Eq. (4.7) but with suitable modifications. Before mixing one has the crossing among the energy levels up to pair creation,

(0)

where the double excitation character of the adiabatic potential curve is displayed explicitly and a universal string tension is assumed. The crossings must happen at . Avoided crossings take place when mixing among different sectors is allowed yielding energy curves sketched in Fig. 5 when using the spectrum from the RQM for -hadrons [16, 17].

Figure 5: Avoided crossings structure of adiabatic energy curves (in MeV) for heavy sources as a function of the distance (in fm) between the heavy sources. Left panel: Lowest states from lattice calculations with a threshold added [50]. Right panel: Complete double heavy-light spectrum of the RQM for -hadrons [16, 17] and included a tiny offset to make clear the structure. We also draw the envelope with .

4.8 Limitations in Counting states

As we have said, when the size of the system is large enough the string breaks and the assumption of a linear potential becomes invalid, see Eq. (4.7). Because of the linear dependence of the size with the mass, Eq. (4.5), this provides a maximum mass value beyond which the RQM becomes inapplicable. A rough estimate for mesons can be made by taking which for gives . For baryons the situation is more complex. The value is actually an upper bound for an equilateral triangular configuration, however, the string may break more economically when just two constituents are sufficiently far apart, . This corresponds to an elongated isosceles triangle (quark-diquark configuration) such that . The RQM departs from the PDG at (see Fig. 4). While this poses the problem on the validity of the RQM for masses beyond the PDG saturation, it also suggests that higher mass states break up into weakly bound molecular systems with a small net contribution to the cumulative number. Actually, as argued in [52], counting hadronic states implicitly averages over some scale, and so states such as the deuteron generate fluctuations in a smaller scale.222The cumulative number in a given channel in the continuum with threshold is which becomes due to Levinson’s theorem. In the NN channel where the appearance of the deuteron changes rapidly at by one unit so that , but when we increase the energy this number decreases slowly to zero at about pion production threshold .

For systems with a heavy quark, we have that for mesons and baryons string breaking occurs when .

5 Thermodynamics of bound states of quarks

5.1 The total cumulative number and equation of state in the confined phase

The relativized quark model describes all states as bound states of for mesons and for baryons [16, 17]. The total cumulative number is then defined as

(0)

This counts the number of bound states below which is depicted in Fig. 8 (note the log-scale) where a clear straight line is observed.

By quark-hadron duality, in the limit of very low temperatures we expect to have a gas of pions (the lightest hadrons) which due to spontaneous breaking of chiral symmetry interact weakly at low energies through derivative couplings. In the chiral limit the pions would become massless resulting in a small trace anomaly in the temperature regime where heavier hadrons are suppressed. For a gas of hadrons the pressure reads

(0)

where the sum is over all hadronic states including spin-isospin and anti-particle degeneracies. From here, and using the cumulative number Eq. (5.1) obtained with the RQM [16, 17], it is straightforward to compute the trace anomaly. The comparison, shown in Fig. 2, with the continuum extrapolated results of the WB [7] and HotQCD [8] collaborations is remarkable. As a side remark let us mention that in QCD of the trace anomaly stems from the gluonic part of the operator (right-hand of Eq. (3)) [8].

5.2 Polyakov loop correlators

A straightforward consequence of Eq. (4.7) is that in the confined phase the correlator between Polyakov loops in the fundamental representation at large distances becomes, according to Eq. (3), with and ,

(0)

where by charge conjugation. Then one has

(0)

where we have replaced . The result is depicted in Fig. 6 for using the RQM in the case of c-quarks for . We see that at small temperatures there is little change in qualitative agreement with lattice calculations [53].

Figure 6: Free energy (in MeV) as a function of the distance for a set of increasing temperatures (from top to bottom), using the RQM in the case of c-quarks. We take .

Corrections to Eq. (5.2) are expected, as it is based on a sharp string breaking transition and the fact that we truncated the spectrum to one light pair creation. The avoided crossing structure shown in Fig. 5 is modified by the finite string breaking transition region, which on the lattice and for the ground state was found to be about