Possible New Phase of Thermal QCD

Using lattice simulations, we show that there is a phase of thermal QCD, where the spectral density $\rho(\lambda)$ of Dirac operator changes as $1/\lambda$ for the infrared eigenvalues $\lambda<T$. This behavior persists over the entire low energy band we can resolve accurately, over three orders of magnitude on our largest volumes. We propose that in this"IR phase", the well-known non-interacting scale invariance at very short distances (UV, $\lambda \rightarrow \infty$, asymptotic freedom), coexists with very different interacting type of scale invariance at long distances (IR, $\lambda<T$). Such dynamics may be responsible for the unusual fluidity properties of the medium observed at RHIC and LHC. We point out its connection to the physics of Banks-Zaks fixed point, leading to the possibility of massless glueballs in the fluid. Our results lead to the classification of thermal QCD phases in terms of IR scale invariance. The ensuing picture naturally subsumes the standard chiral crossover feature at $"\!T_c\!"\,\approx 155$ MeV. Its crucial new aspect is the existence of temperature $T_{IR}$ (200 MeV $<T_{IR}<$ 250 MeV) marking the onset of IR phase and possibly a true phase transition.

FIG. 1. Common thermal phase structure of pure glue QCD (pgQCD) and QCD in terms of scale invariance. Since pgQCD is but a model of QCD glue, setting its physical scales involves a small arbitrariness. Temperatures in black appeared in literature without reference to scale invariance.
invariance being broken at long distances (IR), but present at asymptotically short distances (UV) in the trivial non-interacting form.
Here we propose and support the following behavior of thermal pgQCD. Turning the temperature gradually on, the scale (non)invariance properties of a thermal state remain similar to that of a zero-temperature vacuum, until the scale of thermal agitation becomes comparable to the lowest scale of broken scale invariance ("gluon condensate"). This is characterized by the crossover temperature T A past which the properties of thermal medium change rapidly toward the restoration of scale invariance in IR. The latter then occurs at a well-defined temperature T IR > T A . In the ensuing range T IR < T < T UV (IR phase), gauge fields characteristic of a thermal state are scale invariant at distances larger than ≈ 1/T . Unlike asymptotic scale invariance in UV, present at all temperatures, IR invariance emerges due to the interaction that is still strong at long distances. For T > T UV (UV phase), the field fluctuations in IR regime (λ < T ) effectively disappear, and the notion of IR scale invariance becomes trivial. The system can then be described as a weakly interacting gluon plasma.
This scenario is schematically shown in Fig. 1 (top, middle). Note that the low-temperature region T < T IR (B phase for "broken") is split into two regimes B 0 and B A by T A . The relation to transition temperatures discussed previously in literature without invoking scale invariance is also indicated. Temperature T IR coincides with the well-known T c of Polyakov line first order transition in pgQCD [11]. In addition, we identify T UV with T ch of chiral polarization transition [12][13][14]. Analog of T A has not appeared in the context of pgQCD. 1 1 Transition with analogous physical meaning was in fact discussed in Ref. [14] but, rather than being attributed to a distinct dynamical effect, it was mistakenly identified with T c in pgQCD.
Next, we present evidence that the above T -pattern of scale invariance in gauge field is retained by QCD of nature (Fig. 1, bottom). In other words, scaling properties of QCD glue, which again enters as nominally scale-free entity, are driven by quantum nature of the theory like in pgQCD, rather than quark mass effects. Thus, there is a crossover temperature T A ≈ 150 MeV which we qualitatively associate with chiral "T c " of the standard scenario.
However, here it is simply a characteristic temperature of B phase, marking the onset of changes toward IR scale invariance. Note that the quark condensate now also plays a role in determining the value of T A . The IR phase then emerges at 200 < T IR < 250 MeV.
Before proceeding to lattice evidence, we address several immediate questions.
(i) Since lattice offers good quantitative control over QCD at µ = 0, how did the IR phase escape the detection? The answer is insufficient volumes. Indeed, the usual expectation is that IR scales Λ < T contribute little to physics for T > "T c ". Our proposal not only contradicts this but implies that, for T IR < T < T UV , it is the deep infrared scales Λ T IR that drive a significant IR contribution. Hence, we predict the existence of a "crossover size" L IR (T ) 1/T IR of the system past which the deep infrared physics becomes readily reflected in thermal observables. The systems of sufficiently large spatial sizes L > L IR (T ) are not commonly studied at present. This is expanded upon in Appendix A.
(ii) Given (i), how is the existence of IR phase inferred from lattice simulations? At T T IR , we detect the onset of scale invariant 1/λ behavior of Dirac spectral density ρ(λ) (number of eigenmodes per unit volume and spectral interval) for λ T . We propose that this arises due to the onset of effective IR scale invariance of glue fields dominating the thermal state. 2 While the two notions are not equivalent apriori, they are consistent (Appendix B). Moreover, in theories with IR scale invariant gauge fields, such as those governed by Banks-Zaks fixed point, the pure power law behavior of ρ(λ) is expected due to its proposed connection to mass anomalous dimension [15,16]. 3 This argument also suggests that, up to small quark mass deformations, IR scale invariance of glue extends to quark sector in QCD, which was implicitly assumed already.
(iii) Scale invariance in field theory is normally addressed via the energy-momentum tensor.
Its precise determination in lattice simulations is challenging in part because the minimal system size needed to detect the IR phase grows with temperature (Appendix A).
2. Lattice Evidence. Technical details of our simulations are summarized in Appendix C.
To discuss the results, we start with pgQCD where needed volumes are more readily accessible.
In Ref. [19], a peak at the infrared end of Euclidean Dirac spectral density has been observed in pgQCD above T c . Only recently it was shown [14] that this feature is not a regularization artifact. Here we present evidence that ρ(λ) ∝ λ −1 in IR which, together with ρ(λ) ∝ λ 3 in UV, generates a bimodal structure facilitating scale invariance at both ends of the spectrum.
To that end, we study the spectrum of the overlap Dirac operator on equilibrium backgrounds. A useful quantifier is the volume density of eigenmodes in spectral range [λ, T ], If ρ(λ) ∝ λ −1 for λ < T , a straight line passing through the origin is obtained in variable x = ln T /λ ≥ 0. Note that λ = T corresponds to x = 0 and IR is approached by increasing x.
In (1). In Appendix D, we discuss a more direct approach to exposing the 1/λ dependence of ρ(λ) over wide range of scales.
To assess the relationship of Polyakov line phase transition in pgQCD to its IR phase, we simulate the system at T = 0.98 T c in the otherwise identical setup with large volume.
The resulting ρ(λ) is shown in Fig. 3 (top left). Apart from saturation at the IR edge of the spectrum, we find no linear segment in the corresponding σ(x) (top middle), in direct contrast to 1/λ behavior at T = 1.12 T c (top right). Thus, barely below T c , the system is in the B phase. Note also the characteristic difference in spectral densities between B and IR phases (top left). Given the above and the corroborating spectral evidence of Ref. [14] at T = 1.02 T c , we conclude that T IR coincides with T c . One consequence of this is that B and IR phase of pgQCD are separated by a first order phase transition.
Important feature of the Dirac spectrum at T = 0.98 T c is that ρ(λ) exhibits the IR peak even at T < T IR. Indeed, there is a minimum of ρ(λ) at λ m ≈ 120 MeV (Fig. 3, top left).
Such minimum at λ m > 0 may exist even at zero temperature due to the possible logarithmic divergence at λ → 0 and/or the presence of positive power with negative prefactor [20].
However, this has not yet been confirmed in pgQCD simulations, implying that λ m (T ) is very small or zero at low T . This leads us to propose that it is meaningful to distinguish the T 0 and T T IR regimes by a crossover characterized by temperature 0 < T A < T IR . While the crossover point is a non-unique concept, here we have in mind a commonly used approach based on the rate of change. In other words, we define T A as the position of maximum (peak) in dλ m /dT . In physics terms, T A relates to the point at which gluon condensate becomes significantly affected by thermal agitation. It splits the B phase into regimes B 0 and B A ( Fig. 1) with the latter referred to as anomalous, conforming to terminology of Ref. [14].
Standard expectations suggest that the IR phase, commencing at T IR , ends at temperature T UV (Fig. 1) above which scales λ ≈ T become amenable to perturbative treatment. Since IR peak is not featured in weakly coupled regime, we define T UV as a temperature at which ρ(λ) becomes a nondecreasing function on λ ≥ 0 with ρ(0) = 0. The associated disappearance of IR peak has been observed on moderate volumes in Refs. [12,13], accompanied by the simultaneous loss of chiral polarization in low-lying Dirac modes. Since the latter effect is characterized by temperature T ch , we propose that T UV = T ch as indicated in Fig. 1. We now turn to overlap spectral densities in QCD. More specifically, we study SU (3) gauge theory with N f = 2 + 1 quark flavors at physical masses (see Appendix C), which is a very precise representation of real-world strong interactions. To support the existence of T IR , we show in Fig. 3 (bottom) the analog of IR transition we described in pgQCD. The clearly noninvariant behavior at T = 175 MeV is contrasted with that at T = 250 MeV. The latter exhibits characteristic features of the IR phase, both in terms of ρ(λ) and σ(x). Regarding the latter, note also the similarity to the pattern displayed by volume sequence in Fig. 2.
In Appendix D we present additional results at T = 200 MeV, featuring the behavior more marginally on the B A side. This leads us to the following initial estimates where "T c " is the temperature of chiral crossover. The estimate of T A follows from our analysis in Ref. [14] ( 3. IR-UV Separation and Banks-Zaks Fixed Point. The signature aspect of transition at T IR is a clean separation of IR and UV scales in the gauge field, reflected by almost perfectly bimodal ρ(λ). Additional data illustrating the latter is presented in Appendix D. The analysis of Refs. [13,14,21,22] revealed that, apart from increasing the temperature, such IR-UV separation is also inducible by decreasing the quark mass or increasing the number of flavors (3) gauge theories with fundamental quarks. Our aim is to integrate the new element of IR scale invariance into these findings, which promises a valuable insight into the nature of IR phase in thermal QCD.
We start in the corner of SU(3) theory space which is native to IR scale invariance, namely the vicinity of conformal window [23] (N f massless flavors, N c f < N f < 16.5, T = 0). In Refs. [13,14,21,22] it was found that small mass dynamics at N f = 12, believed by most researchers to be near-conformal, generates the pattern of IR-UV separation closely mimicking that of QCD in the IR phase. While originally interpreted as indicating an unexpectedly large N c f , the revelation that ρ(λ) may be a pure power in IR begs this to be reconsidered since ρ(λ) ∝ λ p is exactly what one expects near conformality. A consistent inference is the and a weakly coupled part N UV f < N f ≤ 16.5 with p > 0. The parametric trends in IR-UV separation then lead us to propose that strongly coupled regimes T IR < T < T UV of QCD and of the conformal window belong to a single contiguous IR phase in SU (3) theory space, defined by p < 0. In this sense, the observed elements of IR scale invariance in thermal QCD descend from conformality of a strongly coupled Banks-Zaks fixed point.
The above argument introduces an unconventional scenario for dynamics in a strongly coupled conformal window which requires more detail. Consider the T = 0 system at In more detail, the sequence produces B and IR phases as in the thermal case. With N f in a strongly coupled regime, the UV phase does not materialize. Thus, the mass vicinity of a theory in strongly coupled conformal window (its IR phase) is characterized by ρ(λ) ∝ λ p where 4 To see the relevance of this connection, consider the m → 0 limit in SU (3)  The IR scale invariance of glue, inherent to IR phase, is expected to keep correlators of glue operators long-range, and the associated glueball-like excitations massless. 4 Note that conformality constraints on unitarity [24] and the conjectured method [15] of extractingψψ anomalous dimension then raise interesting questions on details of m → 0 limit and its relation to m ≡ 0. 5 We refer to theories defined as lim m→0 lim L→∞ of those with N f mass-degenerate flavors. 6 Due to the above monotonicity properties, the IR phase in this restriction remains contiguous.

Synthesis and Main Points.
We proposed the existence of a new phase in thermal QCD, the IR phase T IR < T < T UV , featuring aspects of scale invariance at distances larger than 1/Λ IR , where Λ IR (T ) T . In particular, our way of probing the system suggests that glue fields dominating the thermal state in the IR phase are statistically self-similar upon rescalings involving such distances (Appendix B).
In the standard scenario, QCD matter enters the near-perfect fluid regime above the chiral crossover temperature "T c " ≈ 155 MeV. However, given that scale invariance underlies model descriptions able to mimic the observed fluidity properties [8], we propose that this transition actually occurs at T IR (200 < T IR < 250 MeV). In other words, the strongly interacting near-perfect fluid is realized by the IR phase. If glue fields continue to follow the described patterns arbitrarily deep into IR, then T IR marks a phase transition where the leading IR power in ρ(λ) changes from p = 0 to p −1. This transition could be consequential for the analysis of heavy ion experiments and for modeling the thermal history of the universe.
The observed elements of IR scale invariance can be understood by viewing thermal QCD in the larger context of asymptotically free SU(3) gauge theories with fundamental quarks.
To that end, we proposed the phase structure in this space that can be summarized by with transitions occurring accordingly. For example, increasing the temperature past T IR in QCD is expected to eventually generate a transition from p −1 to p > 1, identifying T UV . 7 The connection to scale invariance stems from the proposed existence of a contiguous IR phase in SU (3)  The conjecture that IR phase of QCD realizes the near-perfect fluid is expected to have phenomenological consequences. For example, using the above connection to the physics 7 The value of p in thermal UV phase could be infinite if the depletion of modes in the infrared proceeds faster than arbitrary positive power, e.g. if gap develops in the Dirac spectrum. Note also that p = 0 (B phase) includes the case of logarithmically diverging density. We thank the Wuppertal-Budapest collaboration for sharing their lattice ensembles. role is played by the "crossover size" L IR (Sec. 1). 8 Given that the IR contribution is driven by deep infrared ( Λ IR ) rather than the vicinity of Λ IR , it is clear that L IR 1/Λ IR for standard observables. Moreover, the density of Dirac eigenmodes in the IR regime drops quickly with temperature (see e.g. [13]), causing L IR to increase. Sensitivity to scale invariant behavior of glue is then expected on systems of size L satisfying where the last inequality is due to Λ IR (T ) T . Hence, L(T ) 1/T IR applies to all standard observables and all temperatures T IR < T < T UV . Since T IR is comparable to Λ QCD , the relevant sizes are larger than typically considered sufficient for thermal QCD studies.
Lattice introduces a slight complication in that the Dirac operator, serving as the detector of IR scale invariance, is not a unique object: different discretizations capture aspects of continuum behavior to varying degree. Chirality plays a relevant role here. Indeed, the bimodality in ρ(λ) was first observed with overlap operator that fully respects chirality, while it was not seen by the staggered operator on identical backgrounds [19]. However, the IR peak has recently been identified by staggered-type operator in pgQCD on larger volumes [26], confirming that the presence of this feature is discretization independent. This is also consistent with bimodality of the overlap operator persisting into the continuum limit, shown in Ref. [14]. In addition, since L IR has physical origin (see above), we expect that i.e. that L IR (T ) is universal for fixed definition of the crossover point.
In lattice QCD, Dirac operator defining the quark part of the action obviously plays a special role. While the existence of IR peak in this "native" Dirac spectrum appears more difficult to ascertain numerically, the studies focusing on the U A (1) problem [27,28] already suggest that the feature is present at physical light-quark masses, albeit the studied volumes are small. Its absence would in fact be very surprising. Indeed, the ensuing singularity in the space of lattice Dirac operators with respect to (A2), as well as the associated possibility of non-universality in topological susceptibility, make such scenario unlikely. 8 L IR can be viewed as a size at which the associated finite volume correction assumes its asymptotic form.
Our aim here is to illustrate how scale invariance of gauge field A constrains the form of . This is easiest to do in R 4 , the setup relevant for theories in conformal window, but the arguments can be modified to finite temperature. We implicitly assume that A µ (x) ∈ su(N ) although this is not important in the present context. Thus, we are dealing with eigenvalue problem on fixed "classical" background, defined by (A µ (x) is anti-Hermitian) where λ ∈ R and ψ is an eigenmode. Let A (s) be the gauge field obtained from A by the canonical scale transformation. The following are the simultaneous eigensystem triples where the correspondence is one-to-one. Envisioning the potentials singular at origin or infinity, we consider the regularized eigenvalue problem on [ , L] 4 with the ultraviolet and L the infrared regulator. 9 The relation (B2) is then modified as (A, ψ, λ, , L) ←→ (A (s) , ψ (s) , sλ, /s, L/s) The standard (anti)periodic boundary conditions on A, ψ are respected by the correspondence.
With the usual assumption that the spectrum on finite volume is discrete, (B3) implies that the number of eigenmodes in interval [λ 1 , λ 2 ] for setup on the left is the same as that in [sλ 1 , sλ 2 ] for setup on the right. Focusing on A(x) with no singularity at x → ∞ allows us to remove infrared cutoff (L → ∞) and to account for number of eigenmodes in terms of smooth spectral density. This then leads to for all λ 1 and λ 2 . Consequently, where f (x) is an arbitrary non-negative function. Thus, for scale invariant free field (A(x) ≡ 0), with no singularity at the origin, the density is -independent and ρ(λ) ∝ λ 3 . However, no leading infrared power, such as the behavior 1/(λ 4 ), is excluded a priori. In quantum theory, the diverging UV cutoff length scale is replaced by the dynamically generated 1/Λ IR , and we thus have ρ(λ) ∝ Λ 4 IR /λ. These considerations can be generalized to self-similar (rather than strictly scale invariant) gauge backgrounds, providing additional freedom to accommodate the 1/λ dependence. For QCD Dirac spectra, we utilized the gauge ensembles of Wuppertal-Budapest group described in Ref. [30]. More precisely, they were generated in N f = 2 + 1 theory at physical light quark mass of (m u + m d )/2, and the physical "heavy" quark mass of m s . In terms of lattice setup, the simulations used tree-level Symanzik-improved gauge action and stoutimproved staggered fermions. The physical point (thus scale setting) was defined by fixing m π , m K and f K to their physical values at zero temperature. Our analysis is based on 100 gauge configurations in each case.
The main object of our interest is the 4-volume density σ(λ 1 , λ 2 ) of Dirac eigenmodes from spectral interval [λ 1 , λ 2 ] (convention set by Eq. (B1)). This quantity is commonly expressed in terms of the corresponding spectral density ρ(λ), namely Unless stated otherwise, exact zero modes are excluded from counting. On finite 4-volume L 3 /T , the ensemble average is implicitly assumed in (C1), although expressing ρ(λ) in terms of δ-functions makes it meaningful even for a single configuration.
In a numerical study, it is necessary to work with coarse-grained version of ρ(λ). This is achieved by introducing the parameter δ > 0 and defining Only |λ| > δ/2 + with suitably chosen > 0 to avoid finite volume effects is shown or quoted in any given ρ(λ, δ). A Wilson-Dirac based overlap operator [31] with parameters ρ = 26/19 and r = 1 was used in all Dirac spectrum calculations.
Implicitly restarted Arnoldi method [32,33] was used to compute the eigenvalues and eigenvectors of the overlap operator. For all but one ensemble used in this study, it is efficient to first compute the eigenvalues of D † D in a chiral sector, and then reconstruct the eigenvalues of D using standard techniques [34]. For the N = 64 pgQCD lattice at T = 1.12T c , it becomes problematic to distinguish the eigenvalues of near-zero eigenmodes from those of exact zero modes. To ensure the reliability of numerical results in this case, we solved the eigenvalue problem for D directly, utilizing a suitable polynomial spectral transformation to accelerate the convergence.

Appendix D: Additional Data
In this Appendix we present additional lattice data to further support our conclusions.
A key to the proposed picture of thermal phases is the emergence of a remarkable separation of IR and UV physics at T IR . This signature aspect of IR phase is reflected in the sharp bimodality of Dirac spectral density and the resulting clear separation of scales (Sec. 3). To convey this feature explicitly, we show in Fig. 4 spectral densities for both pgQCD and QCD in the IR phase. The data suggests the presence of dynamics in which IR and UV regimes act as separate independent "components" of the theory.
The 1/λ behavior of ρ(λ) over wide IR range of scales can also be checked in a more direct manner, namely by the process of zooming in toward the infrared. In Fig. 5 we show this for pgQCD at T = 1.12 T c on our largest lattice. With the lower spectral edge fixed at 0.1 MeV to avoid finite volume effects, we plot ρ(λ) up to 240, 24 and 2.4 MeV respectively, with bin sizes correspondingly scaled. In each case we fit the data to 1/λ with bin size taken into account in the procedure (solid line) to avoid the finite bin distortion.
In addition to QCD data shown in Fig. 3 (bottom), we also computed the Dirac spectra at T = 200 MeV. The relevant comparison to spectral behavior in the IR phase (analog of which vanishes in L → ∞ limit for all p > −1, where λ p is the leading IR behavior of ρ(λ).
Its 1/L behavior in our pgQCD ensembles is shown in Fig. 7 (left). Since a turn toward zero for 1/L → 0 is not observed, the available data doesn't suggest the existence of Λ min IR .
Since the proportionality constant c of 1/λ is stable with changing L (see Fig. 2 which holds for arbitrary k > 0 under the assumptions of (D2). The result for k = 3, suitable for our range of 1/L and the statistics, is shown in Fig. 7 (right), confirming the trend toward small positive value of in L → ∞ limit.