Is Rho-Meson Melting Compatible with Chiral Restoration?

Utilizing in-medium vector spectral functions which describe dilepton data in ultra-relativistic heavy-ion collisions, we conduct a comprehensive evaluation of QCD and Weinberg sum rules at finite temperature. The starting point is our recent study in vacuum, where the sum rules have been quantitatively satisfied using phenomenological axial-/vector spectral functions which describe hadronic tau-decay data. In the medium, the temperature dependence of condensates and chiral order parameters is taken from thermal lattice QCD where available, and otherwise estimated from a hadron resonance gas. Since little is known about the in-medium axial-vector spectral function, we model it with a Breit-Wigner ansatz allowing for smooth temperature variations of its width and mass parameters. Our study thus amounts to testing the compatibility of the $\rho$-broadening found in dilepton experiments with (the approach toward) chiral restoration, and thereby searching for viable in-medium axial-vector spectral functions.


INTRODUCTION
The structure of the QCD ground state is reflected in its observable hadron spectrum. In vacuum, the formation of quark and gluon condensates leads to the generation of hadron masses and the spontaneous breaking of chiral symmetry (SBCS). The latter induces mass splittings of ca. 0.5 GeV for chiral partners in the light-hadron spectrum, e.g., between π-σ or ρ-a 1 . In a hot medium, chiral symmetry is restored across a region around a pseudo-critical temperature of T pc ≃160 MeV [1,2]. A long-standing question is how this restoration manifests itself in the hadron spectrum, i.e., what its observable consequences are. Dilepton data from ultra-relativistic heavy-ion collisions (URHICs) [3][4][5] are now providing strong evidence that the ρ resonance "melts" when the system passes through the pseudo-critical region [6], while experimental access to the in-medium a 1 spectral functions (e.g., via a 1 → πγ) remains elusive. Thus, to test whether the ρ melting in the vector channel signals chiral restoration, a theoretical evaluation of the inmedium axialvector spectral function is needed.
A straightforward approach to calculate the in-medium axialvector spectral function, by using a chiral Lagrangian paralleling the treatment of the ρ meson, turns out to be challenging [7]. For example, the widely used scheme of implementing the ρ and a 1 mesons into the pion Lagrangian through a local gauging procedure causes considerable problems in describing the vacuum spectral functions as measured in hadronic τ decays [8,9], which led some groups to abandon the local gauging procedure [10,11]. In the present work, we adopt a more modest approach to this problem, by utilizing in-medium sum rules. Specifically, we adopt the well-known Weinberg sum rules (WSRs) [12][13][14] which relate (moments of) the difference between vector and axialvector spec- * Electronic address: pmhohler@comp.tamu.edu † Electronic address: rapp@comp.tamu.edu tral functions to operators signifying SBCS. Using available calculations of the in-medium ρ spectral function together with temperature dependent order parameters as an input, we ask whether a (not necessarily the) axialvector spectral function can be found to satisfy the in-medium sum rules. To tighten our constraints, we simultaneously employ finite-temperature QCD sum rules (QCDSRs) [15,16] in vector and axialvector channels, which additionally involve chirally invariant condensates. Related works have been carried out, e.g., in the lowtemperature limit [17,18], for heavy-quark channels [19], or focusing on chirally odd condensates in the vector channel only [20].
The present analysis builds on our previous work [21] where QCD and Weinberg sum rules have been tested in vacuum with vector and axialvector spectral functions that accurately fit hadronic τ -decays. The combination of four WSRs turned out be a rather sensitive probe of the spectral functions, allowing, e.g., to deduce the presence of an excited axialvector meson, a ′ 1 . This makes for a promising tool at finite temperature (T ), aided by an experimentally tested in-medium vector spectral function and in-medium condensates from lattice QCD (lQCD). In the absence of reliable microscopic models for the a 1 and the excited states, the price to pay is the a priori unknown in-medium behavior of these states. However, with guidance from model-independent chiral mixing theorems to constrain the T dependence of the higher states, one can still hope for a sensitive test of the in-medium a 1 spectral function, and to gain novel insights into (the approach to) chiral restoration in the IJ P = 11 ± chiral multiplet. This is the main objective of our work.
The Letter is organized as follows. We recall the inmedium QCDSRs and WSRs in Sec. 2 and specify the T dependence of their "right-hand sides" (condensates) in Sec. 3. The finite-T axial-/vector spectral functions ("left-hand sides") are detailed in Sec. 4, followed by quantitative sum rule analyses in Sec. 5. We conclude in Sec. 6.

FINITE TEMPERATURE SUM RULES
The basic quantity figuring into WSRs and QCDSRs is the isovector current-current correlator in the vector (V ) and axialvector (A) channels, In the quark basis with two light flavors, the currents read J µ V =q τ γ µ q and J µ A =q τ γ µ γ 5 q, ( τ : isospin Pauli matrices). From here on, we focus on charge-neutral states (isospin I 3 =0) and drop isospin indices. In vacuum, the currents can be decomposed into 4D transverse and longitudinal components as (2) Vector-current conservation implies Π L V (q 2 )=0, while the pion pole induces the partial conservation of the axialvector current (PCAC), Lorentz symmetry breaking at finite T splits the 4Dtransverse polarization functions into 3D-transverse and 3D-longitudinal parts. From here on, we focus on vanishing 3-momentum ( q=0), for which the 3D components are degenerate. We define pertinent spectral functions as The QCDSRs equate a dispersion integral on the left-hand-side (LHS) to an operator product expansion (OPE) on the right-hand-side (RHS); for the axial-/vector channels they read [22][23][24] where the space-like q 2 is traded for the Borel mass M 2 by a standard Borel transform. On the RHS, we include all operators up to dimension-6, i.e., the common scalar operators already present in the vacuum (quark, gluon, and 4-quark condensates, qq , αs π G 2 µν , and O V,A

4
, respectively), as well as non-scalar operators induced by thermal hadrons (h), organized by dimension (d) and twist (τ ). The T dependencies are detailed in Sec. 3.
The WSRs relate moments of the difference between the vector and axialvector spectral functions to chiral order parameters. Their formulation at finite T was first carried out in Ref. [14]. Subtracting the two channels of the finite-T QCDSRs from one another, Taylorexpanding the Borel exponential, and equating powers of M 2 on each side of the sum rule yields where ∆ρ = ρ V − ρ A . The chiral breaking 4-quark condensate is given by the axial-/vector ones as Since the WSRs only contain chiral order parameters, they are particularly sensitive to chiral symmetry restoration, whereas the QCDSRs are channel specific thus providing independent information.

IN-MEDIUM CONDENSATES
We now turn to the T dependence of each condensate figuring into the QCDSRs. To leading order in the density of a hadron h in the heat bath, the in-medium condensate associated with a given operator O can be approximated by where O 0 is the vacuum value of the operator, h( k)|O|h( k) its hadronic matrix element, E 2 h =m 2 h + k 2 , and d h , m h , and n h are the hadron's spin-isospin degeneracy, mass, and thermal distribution function (Bose (n b ) or Fermi (n f )), respectively. Working at zero baryon chemical potential (µ B =0), we absorb anti-baryons into the degeneracy factor of baryons. Corrections to Eq. (10) figure via multi-hadron matrix elements of the operator.
We approximate the medium by a hadron resonance gas (HRG) including all confirmed states with mass m h ≤ 2 GeV [25]. For the temperatures of interest here, T 170 MeV, the HRG is known to reproduce the equation of state from lQCD quite well [26]. Since the calculation of the in-medium ρ spectral function is also based on HRG degrees of freedom, the OPE and spectral function sides of the sum rules are evaluated in the same basis. For the subsequent discussion, we define the integrals Note that m h I h 1 is the scalar density, ̺ h s .

Quark Condensate
The HRG correction to the quark condensate is [27,28] qq The Goldstone boson contribution can be inferred from current algebra (with decay constants given in Tab. I).
The contributions from baryons (B) and other mesons (M ) can be derived from the HRG partition function via ∂ ln Z/∂m q , which is nothing but the in-medium condensate. They are determined by their σ-terms which to lowest order are given by the (current) quark masses, m q , of the light valence quarks in the hadron [29]. However, important contributions arise from the hadron's pion cloud [30,31]. We write where N q (N s ) is the number of all (strange) valence quarks in h. We adjust the proportionality constant to σ 0 =2.81, to recover the recent value, σ N =59 MeV [32], of the nucleon and assume it to be universal for all hadrons. This leads to fair agreement with estimates of σ h for other ground-state baryons [32]. Note that the decomposition of the σ terms into quark core and pion cloud effects parallels the medium effects of the ρ spectral function [33]. Our HRG results reproduce lQCD "data" [1] for T 140 MeV, see Fig. 1(a). To improve the agreement at higher T without affecting the low-T behavior, we introduced a term αT 10 on the RHS of Eq. (12), with α=1.597 ·10 7 GeV −10 . The quark condensate then vanishes slightly above T =170 MeV, signaling the breakdown of our approach. Choosing a somewhat higher power in T (with accordingly adjusted α) has no significant impact on our results, while a smaller power adversely affects the agreement with lQCD data at low T .

Gluon Condensate
For the gluon condensate, the contributions from pions and nucleons have been evaluated in Refs. [22,24,34]. The HRG effect can be inferred from the trace anomaly, Parameter fπ fK fη f η ′ mq mπ Value (MeV) 92.4 113 124 107 7 139.6  (12). For hadron masses not listed we take averages from the particle data group [25].
The change in light-quark condensate is taken from Eq. (12). For the strange-quark condensate, we assume its suppression from individual resonances to scale with the valence strange-quark content of each hadron h, paralleling the procedure of determining the σ-term for each hadron. One has where ∆ qq h is from Eq. (12). The HRG suppression of the gluon condensate reaches 13% at T =170 MeV.

Four-Quark Condensates
For medium dependence of the vector and axialvector 4-quark condensates induced by Goldstone bosons, we adopt the results from current algebra [22]. For the non-Goldstone bosons and baryons, arguments based on the large-N c limit [28,35] suggest a factorization approximation, i.e., the medium effect linear in their (scalar) density amounts to a factor of 2 times the reduction in the quark condensate, with the same factorization parameter as in vacuum (we have checked that an increase of the in-medium factorization parameter by a factor of 2 has a negligible impact on the OPEs and thus on the resulting spectral functions). The T dependence of the vector and axialvector 4-quark condensates then takes the form As for the quark condensate, we augmented the T dependence by a term β V,A T 10 . Since thermal lQCD data are not available for 4-quark condensates, we adjusted β V,A for each channel to render them vanishing at the same temperature as the quark condensate, resulting in β V =3.05 · 10 7 GeV −10 and β A =1.74 · 10 7 GeV −10 . The T dependence of the chiral breaking 4-quark condensate follows from the axial-/vector ones via Eq. (9); relative to the quark condensate, their initial fall-off is faster but slows down above T ≃140 MeV, cf. Fig. 1(b).

Non-Scalar Condensates
Hadrons in the heat bath also induce non-scalar condensates. For our QCDSR analysis the relevant ones are T , and dimension-6 twist-4, O d=6,τ =4 T . We adopt their T dependence as elaborated in Refs. [22][23][24], given by each hadron as The parameters A 2 and A 4 , which control the twist-2 operators, are related to moments of parton distribution functions for the u and d quarks in the hadron One can think of A 2 as twice the momentum fraction of the up and down quarks in the hadron, with A 4 a higher moment. Their values are reasonably well known for the pion and nucleon, A π 2 = 0.97, A π 4 =0.255, A N 2 =1.12, A N 4 =0.12, while there is substantial uncertainty for other hadrons. For baryons, we assume A 2 and A 4 to be identical to the nucleon values, but weighted by the lightquark fraction; e.g., the A 2 of the Λ is 2 3 A N 2 . The kaons and etas are approximated with the pion's parton distribution functions, reduced by the strange-quark content. For other mesons, Eq. (19) is used with the nucleon parton distributions functions, rescaled by the valence-quark content and also reduced by the strangequark content. This gives A 2 =0.801 and A 4 =0.086 for non-strange mesons. The B 2 's are related to integrals of the twist-4 part of the spin-averaged (longitudinal) structure function, F τ =4 2(L) [23,36]. For the nucleon, it has been extracted as B N 2 =−0.247 GeV 2 . Since there is no empirical information for other hadrons, we assume their B 2 to be the same as for the nucleon (suppressed by the strange-quark content); varying it by a factor of 2 produces no noticeable changes in the final spectral functions. Gluonic contributions are believed to be numerically insignificant [22,23] and have been neglected.

FINITE TEMPERATURE SPECTRAL FUNCTIONS
Our starting point are the vacuum axial-/vector spectral functions of Ref. [21] 1 . They are comprised of contributions from the ground state (ρ and a 1 peaks), a first excited state (ρ ′ and a ′ 1 ), and a chirally invariant (i.e., identical) continuum for both channels. The vacuum ρ is taken from the microscopic model of Ref. [37], while a 1 , ρ ′ and a ′ 1 are parameterized with Breit-Wigner functions. For the present analysis, we have slightly modified the vacuum parameters of the ρ ′ to shift its threshold energy to higher energies. This avoids its low-mass tail to reach well below 1 GeV where the τ -decay data do not exhibit any 4π contributions. The modification to the ρ ′ formfactor is compensated by a small modification of the mass and width of the a ′ 1 as to recover a near-perfect agreement with WSR-1 and WSR-2. The re-evaluation of the vacuum QCDSRs requires numerical values of 4-quark factorization parameter of κ=2.1 in O SB 4 = 16 9 κ qq 2 , and of the gluon condensate of αs π G 2 µν =0.017 GeV 4 . The updated vacuum spectral functions, shown in Fig. 2, are very similar to the ones in Ref. [21].
Finite-temperature effects in the spectral functions are implemented as follows. For the ρ meson, we employ the microscopic calculations using hadronic effective theory [38] at vanishing baryon chemical potential. This is the key input to our analysis, as these spectral functions are consistent with dilepton data in URHICs [6], and thus provide a direct link to experiment. The only amendment we allow is a reduction of the vector-dominance coupling strength (as routinely done in QCDSR analyses [22,24,39,40]). Optimal agreement with the QCDSR requires a reduction of up to 7% at T =170 MeV.
For the a 1 meson, the lack of quantitative calculations at finite T leads us to parameterize the medium modifi- cations of its spectral function. We introduce four parameters which control the a 1 peak's location, width, and strength in-medium. For the a 1 mass, we write M T a1 = M a1 (1 − δM a1 (T )/M a1 ), and for the current coupling C T a1 = C a1 (1 − δC a1 (T )/C a1 ). The width is increased and extended below the vacuum threshold by adding the following term to the vacuum width, Γ a1 (s), where Γ T 1 and Γ T 2 are T -dependent constants, and the last factor is a formfactor with the same scale, Λ a1 , as in vacuum. The resulting ground-state axialvector spectral function in medium takes the form with Γ T a1 (s, T ) = Γ a1 (s) + ∆Γ a1 (s). The temperature dependence of the excited states is even less known. Instead of introducing additional parameters for their in-medium Breit-Wigners (which are hard to control), we rather apply the model independent low-temperature effect known as chiral mixing [41,42] to the ρ ′ and a ′ 1 states. However, in the spirit of the HRG, we go beyond the mixing induced by only thermal pions by including the effect from the virtual pion cloud of the thermal hadrons. This effect has been worked out for the pion cloud of the nucleon in cold nuclear matter [43,44]. To extend it to other hadrons (not including the non-pion Goldstone bosons), we define a mixing parameter The total mixing parameter,ǫ, is the sum of the individualǫ h plus that of the pion,ǫ π = 2̺ π s /(3m π f 2 π ). As with the quark condensates, we introduce an additional T 10 -term to renderǫ = 1/2 at the temperature where qq T = 0. The in-medium spectral functions for the excited axial-/vector states then follow as The a 1 contribution to the excited vector channel admixes only the part which is not included in the microscopic calculation of the ρ, see Ref. [45] for details. Our approximate extension of the mixing beyond the low-T pion gas limit is only carried linear in the (scalar) hadron densities, but in line with the in-medium treatment of the condensates. However, no finite-momentum nor finitemass effects of the (virtual) pions have been accounted for.
The chirally invariant continuum is assumed to be Tindependent (e.g., chiral mixing would not affect it).
Lastly, we need to address the T dependence of the 4D longitudinal part of the axial-vector spectral function, i.e., the pion pole. We approximate the pion mass by the leading-order prediction of chiral perturbation theory, i.e., induced by the pion gas only. This produces a weak T dependence as expected for a Goldstone boson. Assuming the Gell-Mann-Oakes-Renner relation to hold at finite T , allows us to infer f π (T ) from the aboveconstructed T -dependence of the quark condensate.
To summarize this section, we have supplemented a microscopic model for the ρ spectral function with a 4parameter ansatz for the in-medium a 1 , chiral mixing for the excited states, and a weakly T -dependent pion mass from chiral perturbation theory. We now investigate whether this setup can satisfy QCDSRs and WSRs.

FINITE-TEMPERATURE SUM RULE ANALYSIS
Let us start by describing the quantitative criteria which govern the numerical values of the in-medium a 1 parameters introduced in the previous section.
To evaluate the QCDSRs, we adopt the conventional method of Refs. [39,46] to calculate an average deviation between the LHS and RHS over a suitable Borel window, referred to as a d-value. The same procedure and Borel window criteria as for the vacuum analysis in Ref. [21] are adopted. A d-value of below 1% has been argued to reasonably bracket remaining uncertainties in the matching procedure [39]; we adopt this as our figure of merit in both A and V channels below.
To evaluate the WSRs, we define a similar measure of deviation between the two sides as This measure is much simpler than the QCDSR analog because it does not involve any Borel window. However, it also has its subtleties. The integrands of the LHS of each WSR are oscillatory functions with appreciable cancelations to yield the RHS (cf. Fig. 2 in Ref. [21]), especially for the higher moments. Since we only use a finite number of moments (3), this could, in principle, lead to "fine-tuned solutions" to the WSRs where the oscillations are still large, and thus ρ V (s) = ρ A (s) even close to restoration. To probe this behavior (and thus the sen-  sitivity to any "artificial" fine tuning), we introduce an "absolute-value" version of the LHS bỹ Though these moments are not directly related to chiral order parameters, they should diminish toward restoration. We define pertinent ratios r n =w n (T )/w n (T = 0). Our analysis proceeds as follows. We first evaluate the QCDSR for the vector channel. With a small reduction in the vector dominance coupling, we find acceptable d V values ranging from 0.43% to 0.67% for all T =0-170 MeV (cf. Tab II). This is a nontrivial result by itself. For the axialvector channel, the QCDSRs and two WSRs are used simultaneously to search for in-medium a 1 parameters which minimize while requiring a smooth T dependence. The thus obtained finite-T axialvector spectral functions are shown in Fig. 3. For all cases, the percentage deviation of WSR-1 and WSR-2 is below 0.1%, and d A remains below 0.6%. Deviations of WSR-3 are much larger, but comparable to the vacuum up to T ≃150 MeV. At T =160 and especially 170 MeV, the magnitude of the RHSs is small and enters into the denominator of d WSR , thus greatly magnifying residual deviations. The r n measures decrease monotonically with T suggesting acceptable deviations even for WSR-3. We therefore conclude that our spectral functions are compatible with both QCDSRs and WSRs.
To probe the uncertainties in our method, we depict in Fig. 4  spectral functions with d A <1% were included). From this collection of curves, we then select those whose agreement with WSR1 is within 1%, producing a much narrower (shaded) region bordered by dotted lines. The combined constraints of QCDSRs and WSRs are thus shown to noticeably increase the selectivity of the in-medium axialvector spectral function. A visual inspection of the in-medium spectral functions supports the trend toward restoration, cf. Fig. 3: the a 1 peak gradually merges into the ρ while the excited states degenerate somewhat earlier through chiral mixing. The ρ-a 1 merging is largely dictated by the WSRs, but the concrete shape close to chiral restoration is more sensitive to the QCDSRs. Note that our analysis not only complies with a "trivial" degeneracy at the restoration point, but rather provides a systematic temperature evolution, starting from the vacuum, compatible with current best estimates for the T dependent chiral order parameters and condensates (at T =170 MeV, our condensates are close to zero, undershooting the lQCD data for the 2quark condensate; our axialvector spectral function at this temperature is thus more of an illustration of the ex-pected degeneracy at higher T where qq T ≃0). The inmedium a 1 mass shift is consistent with a leading T 4 behavior, in line with model-independent constraints from the chiral Lagrangian. Our analysis also suggests that the approach toward restoration "burns off" the chiral mass splitting between the ρ and a 1 , while "bare" masses of m 0 ≃0.8 GeV essentially persist, similar to Ref. [7].

CONCLUSION
The objective of this work was to test whether inmedium vector spectral functions which describe dilepton data in heavy-ion collisions are compatible with chiral symmetry restoration. Toward this end, we deployed QCD and Weinberg sum rules in a combined analysis of vector and axialvector spectral functions, using lattice-QCD and the hadron resonance gas to estimate the inmedium condensates and chiral order parameters, and chiral mixing to treat the T dependence of excited states. We first found that the QCDSR in the vector channel is satisfied with a small (order 5%) amendment of vector dominance. We then introduced a 4-parameter ansatz for the in-medium a 1 spectral function and found that a smooth reduction of its mass (approaching the ρ mass) and large increase in width (accompanied by a lowmass shoulder) can satisfy the axialvector QCDSR and 3 WSRs over the entire temperature range from T =0-170 MeV, ultimately merging with the vector channel. This establishes a direct connection between dileptons and chiral restoration, and thus the answer to the originally raised question is positive. Our findings remain to be scrutinized by microscopic calculations of the a 1 spectral function. Work in this direction is ongoing.