Finite-Temperature QCD Sum-Rule Framework

• The finite-temperature QCD sum-rule framework is a nonperturbative method that extends SVZ QCD sum rules to analyze hot QCD matter and extract thermal hadron properties.
• It combines the operator product expansion with finite-temperature field theory to relate thermal QCD condensates to hadronic spectral functions via dispersion relations.
• The approach reveals significant temperature-induced modifications such as sharp drops in tensor meson masses near deconfinement, indicating chiral symmetry restoration.

The finite-temperature QCD sum-rule framework is a nonperturbative analytic method that extends the SVZ (Shifman–Vainshtein–Zakharov) QCD sum rules to hot QCD matter, enabling the extraction of hadronic properties and order parameters from QCD correlators in thermal equilibrium. The approach combines operator product expansion (OPE) with finite-temperature field theory, equating the OPE-side (in terms of temperature-dependent QCD condensates) to a hadronic spectral representation via dispersion relations or Hilbert moments. The application of this framework to light and heavy mesons, and the interaction vertices at finite temperature, provides insight into phase transitions such as chiral symmetry restoration and deconfinement, as well as quantitative evolution of hadron masses, decay constants, and form factors.

1. Foundations of the Finite-Temperature QCD Sum-Rule Framework

The foundation of thermal QCD sum rules is the assumed validity of the OPE and quark-hadron duality at finite temperature, with the vacuum expectation values of local QCD operators replaced by their thermal Gibbs averages,

$$\langle A \rangle_T = \frac{\operatorname{Tr}(e^{-\beta H} A)}{\operatorname{Tr}(e^{-\beta H})}, \quad \beta = 1/T.$$

The basic object is the thermal two-point correlation function for an interpolating current $J_{\mu\nu}$,

$$\Pi_{\mu\nu,\alpha\beta}(q, T) = i \int d^4x \; e^{iqx} \langle \mathcal{T}[J_{\mu\nu}(x) \bar{J}_{\alpha\beta}(0)] \rangle_T.$$

The OPE is extended to finite temperature by including all operators permitted by the reduced symmetry $O(3) \times SO(1)$ (thermal rest frame), notably thermal condensates:

• $\langle \bar{q}q\rangle_T$, $\langle G^2 \rangle_T$ (replacing vacuum condensates),
• additional operators arising from Lorentz symmetry breaking, such as $u^\mu u^\nu \langle \Theta^f_{\mu\nu} \rangle$ (fermionic part of the energy-momentum tensor).

The spectral representation encodes the hadronic content, with the spectral density $\rho(s)$ modeled for the relevant channel. For practical extraction, Borel or Hilbert moments are employed.

2. Implementation: Operator Product Expansion and Thermal Modifications

At finite temperature, the OPE's Wilson expansion is generalized to account for the thermal bath. The condensate contributions include:

• Replacement of $$\langle \bar{q} q \rangle \to \langle \bar{q} q \rangle_T$$ and analogously for gluon condensates,
• New tensor operators involving the heat-bath four-velocity $u^\mu$,
• Medium-induced mixing and modification of Wilson coefficients.

A concrete example is the appearance of operators like $\langle u \Theta^f u \rangle$, which account for the directional structure imposed by the heat bath, introducing anisotropies into the OPE.

The resulting sum rule for mass $m_{f_2(a_2)}(T)$ and decay constant $f_{f_2(a_2)}(T)$ matches the Borel-transformed OPE to the hadronic representation,

$$f_{f_2(a_2)}^2(T) m_{f_2(a_2)}^6(T) e^{-m_{f_2(a_2)}^2(T)/M^2} = \int ds\, \rho_{f_2(a_2)}(s) e^{-s/M^2} + \hat{B}\Pi^{\text{non-pert}}_{f_2(a_2)}.$$

The continuum threshold $s_0(T)$ is rendered temperature-dependent via the thermal quark condensate,

$$s_0(T) = s_0 \frac{\langle \bar{q}q\rangle_T}{\langle 0|\bar{q}q|0\rangle } \left(1 - \frac{(m_q + m_d)^2}{s_0}\right) + (m_q + m_d)^2.$$

3. Extraction and Characterization of Thermal Hadron Parameters

By equating the OPE and hadronic sides, and implementing Borel transformation (to suppress excited-state and continuum contributions), the temperature dependence of hadronic parameters can be extracted numerically. The approach yields:

• Masses and decay constants that are temperature-dependent functions,
• Fit formulas to the numerically extracted observables, for example (with $T$ in GeV),

$$m_{f_2(a_2)}(T) = A\, e^{\alpha T} + B, \qquad f_{f_2(a_2)}(T) = C\, e^{\beta T} + D,$$

with fit parameters $(A, B, C, D, \alpha, \beta)$ determined for each tensor meson.

Quantitatively, the pseudoscalar and vector channels can be treated analogously, with appropriate modifications to the interpolating currents and OPE structure.

4. Main Numerical Results for Light Tensor Mesons

For the $f_2(1270)$ and $a_2(1320)$ tensor mesons, the finite-temperature QCD sum-rule analysis yields the following behavior:

• Up to $T \approx 0.1$ GeV, both mass and decay constant are nearly $T$-independent.
• Approaching the deconfinement temperature $T_c \approx 0.175$ GeV, sharp drops are observed:
  • The tensor meson mass decreases by $\sim96\%$ of its vacuum value.
  • The decay constant decreases by $\sim6\%$.
  • The vacuum values for $f_2$: $m_{f_2} = (1.28 \pm 0.08)$ GeV, $f_{f_2} = 0.041 \pm 0.002$; and for $a_2$: $m_{a_2} = (1.33 \pm 0.10)$ GeV, $f_{a_2} = 0.042 \pm 0.002$.

These results are consistent with previous vacuum sum-rule predictions and empirical data.

5. Physical Implications: Deconfinement, Chiral Restoration, and Spectral Evolution

The observed rapid decrease of tensor meson masses near $T_c$ signals significant in-medium modifications of the hadronic spectrum, interpreted as a manifestation of deconfinement and chiral symmetry restoration. The moderate reduction of decay constants indicates diminished overlap between the mesonic state and the surrounding thermal medium, relating to a suppression of the "wavefunction at the origin." The dramatic drop in the mass (relative to the modest change in decay constant) underscores that hadron masses in a hot medium are strongly affected by the transition to a deconfined phase, while the decay constant is a less sensitive, yet non-negligible, probe.

These findings have direct relevance for the interpretation of heavy-ion collision data, particularly the thermal evolution and possible melting or suppression of tensor mesons as observed in experimental spectroscopy.

6. Formal Summary and Key Equations

Principal Equations for the Tensor Channel

Quantity	Expression
Borel sum rule	$$f_{f_2(a_2)}^2(T) m_{f_2(a_2)}^6(T) e^{-m_{f_2(a_2)}^2(T)/M^2} = \int ds\, \rho_{f_2(a_2)}(s) e^{-s/M^2} + \hat{B}\Pi^{\text{non-pert}}_{f_2(a_2)}$$
Continuum threshold	$$s_0(T) = s_0 \frac{ \langle\bar{q}q\rangle_T / \langle 0|\bar{q}q|0\rangle }{1 - \frac{(m_q + m_d)^2}{s_0}} + (m_q + m_d)^2$$
Thermal fit (mass)	$m_{f_2(a_2)}(T) = A\, e^{\alpha T} + B$
Thermal fit (decay constant)	$f_{f_2(a_2)}(T) = C\, e^{\beta T} + D$

These equations typify the general methodology for extracting in-medium hadronic parameters from finite-temperature QCD sum rule analysis. The same formalism applies, with necessary modifications, to other mesonic channels and can be adapted to paper three-point functions (form factors), exotic states, or finite-density effects.

7. Significance and Relation to Broader QCD Sum-Rule Applications

This methodology illustrates the robustness of the finite-temperature QCD sum-rule framework for quantitative studies of in-medium hadron properties. The inclusion of thermal OPE modifications—particularly new operators reflecting the broken Lorentz invariance—and the careful extraction of temperature-dependent thresholds and condensates are pivotal for capturing qualitative and quantitative aspects of deconfinement and chiral symmetry restoration. The resulting predictions serve as theoretical guidance for interpreting experimental observables related to hadron spectroscopy in hot QCD matter and complement lattice QCD, especially in regimes where real-time properties or particular channels remain challenging for lattice techniques.