Thermodynamics from the S-matrix reloaded: emergent thermal mass

TL;DR

The paper analyzes how the Dashen–Ma–Bernstein S-matrix framework encodes thermal physics and IR structure, focusing on how Debye-like thermal masses emerge when the free energy is expressed as a trace of the vacuum S-matrix. Using a λφ^4 warm-up, the authors compute IR-finite contributions at LO and NLO, unveiling melon and caterpillar topologies and the necessity of E-dependent T(E) regularization for forward-singular diagrams. They demonstrate the emergence of a thermal mass through daisy-like resummations at NNLO and connect this to standard Th-QFT resummations, providing analytic and combinatorial checks, especially in the large-N limit via superdaisy resummation. The results pave the way for applying the DMB formalism to higher-order calculations in relativistic QFT, including QCD, and for exploring other thermal observables within a forward-S-matrix framework.

Abstract

The formalism of Dashen, Ma and Bernstein (DMB) expresses the thermal partition function of a system in terms of the S-matrix operator, roughly $Z(β) \propto \int dE\, e^{-βE}\,\text{Tr}\,\ln S(E),$ 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 $O(α_s)$ QCD thermal free energy. Moving to higher orders, it is well known that at $O(α_s^2)$ in QCD, or e.g. at $O(λ^2)$ in $λφ^4$ 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.

Figures (12)

• Figure 1: One among the $\binom 82 \,6!$ contractions that give a connected diagram in the $8\to 8$ sector; see Eq. (\ref{['OVnton']}). The colouring of lines is immaterial and is only there to help following the two chains of contractions (described in the main text). The lengths of the two chains are 3:5 in this example.
• Figure 2: $(a),(b)$: $2\to2$ one-loop amplitudes that contribute at $O(\lambda^2)$ to the free energy of the theory. $(c),(d)$: $3\to3$ tree-level amplitudes that contribute at the same order. $(e)$: When initial and final state particles with the same label are joined, all diagrams become topologically equivalent to the melon diagram.
• Figure 3: S-matrix elements that have caterpillar topology $(f)$ and contribute to the temperature-dependent part of Eq. (\ref{['master']}). All diagrams vanish in $d$ dimensions except $(b)$, which is divergent in the forward limit and has to be regulated by working with $E$-dependent $T$-matrix elements.
• Figure 4: Example of a Wick contraction that contributes to the caterpillar topology.
• Figure 5: Upper row: seed diagrams with caterpillar topology. To account for windings one has to multiply by $n_B^p$, where $p$ is the number of external lines. Bottom-left: particles whose lines encounter only one interaction vertex are shown in grey for diagrams $A,D$. Bottom-right: taking diagram $B$ as an example, it is shown that independent windings should be considered for each external line. Upper and lower red dots are identified under the Trace, and similarly for the blue ones.