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<p>The formalism of Dashen, Ma and Bernstein (DMB) expresses the thermal partition function of a system in terms of the S-matrix operator, roughly <i>Z</i>(<i>β</i>) ∝ ∫<i>dEe</i><sup>−<i>βE</i></sup> Tr ln <i>S</i>(<i>E</i>), where <i>S</i> denotes the full scattering operator on the asymptotic Fock space — i.e. including all multi-particle sectors — defined via the Lippmann-Schwinger equation. Recently we have employed this formalism to compute the free energy of flux tubes (essentially a two-dimensional theory of derivatively coupled scalars) and the two-loop <i>O</i>(<i>α</i><sub><i>s</i></sub>) QCD thermal free energy.</p><p>Moving to higher orders, it is well known that at <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi>O</mi> <mfenced close=")" open="("> <msubsup> <mi>α</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mfenced> </math></EquationSource> <EquationSource Format="TEX">\( O\left({\alpha}_s^2\right) \)</EquationSource> </InlineEquation> in QCD, or e.g. at <i>O</i>(<i>λ</i><sup>2</sup>) in <i>λϕ</i><sup>4</sup> theory, the free energy develops IR divergences. These IR divergences are resolved by the screening Debye mass. However, the DMB formalism expresses the free energy in terms of a trace of the S-matrix operator in the vacuum. How, then, does the Debye mass arise in this framework? In this work we address this question, thereby paving the way for higher-order applications of the DMB formalism in relativistic QFT.</p>

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Thermodynamics from the S-matrix reloaded: emergent thermal mass

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  • 发表日期：

    2026年3月24日

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  • Pietro Baratella,
  • Joan Elias Miró

摘要

The formalism of Dashen, Ma and Bernstein (DMB) expresses the thermal partition function of a system in terms of the S-matrix operator, roughly Z(β) ∝ ∫dEe−βE Tr ln S(E), where S denotes the full scattering operator on the asymptotic Fock space — i.e. including all multi-particle sectors — defined via the Lippmann-Schwinger equation. Recently we have employed this formalism to compute the free energy of flux tubes (essentially a two-dimensional theory of derivatively coupled scalars) and the two-loop O(αs) QCD thermal free energy.

Moving to higher orders, it is well known that at O α s 2 \( O\left({\alpha}_s^2\right) \) in QCD, or e.g. at O(λ2) in λϕ4 theory, the free energy develops IR divergences. These IR divergences are resolved by the screening Debye mass. However, the DMB formalism expresses the free energy in terms of a trace of the S-matrix operator in the vacuum. How, then, does the Debye mass arise in this framework? In this work we address this question, thereby paving the way for higher-order applications of the DMB formalism in relativistic QFT.