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Quark Matter at Four Loops: Hardships and How to Overcome Them

Aapeli Kärkkäinen, Pablo Navarrete, Mika Nurmela, Risto Paatelainen, Kaapo Seppänen, and Aleksi Vuorinen

  • Department of Physics and Helsinki Institute of Physics, University of Helsinki, P.O. Box 64, FI-00014, Finland

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Phys. Rev. Lett. 135, 021901 – Published 7 July, 2025

DOI: https://doi.org/10.1103/627n-5g6l

Abstract

Knowledge of the pressure of cold and dense quark matter is known to significantly constrain the equation of state of neutron-star matter, a quantity playing a key role in deciphering the stars’ internal structure. In this Letter, we make important progress toward determining the last unknown contribution to the pressure at next-to-next-to-next-to-leading order in the strong coupling constant αs, available through the sum of all four-loop vacuum diagrams of dense quantum chromodynamics. We demonstrate the cancellation of both the covariant gauge parameter and all infrared divergences in the sum, showcasing the effective-field-theory paradigm in action. For the remaining infrared-finite four-loop integrals, we demonstrate the efficacy of the dense loop tree duality method—a novel numerical framework for multiloop calculations in thermal field theory. Together, our results show that completing the next-to-next-to-next-to-leading order pressure of cold quark matter is no longer merely possible but for the first time within reach.

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References (37)

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  29. Note that in the notation of [12], the regulatory scale parameter in this expression would be the factorization scale Λh instead of Λ¯. Given that we have explicitly witnessed the cancellation of Λh between the full theory and HTL contributions to the pressure, we have identified the two scales here.

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Phys. Rev. Lett. 135, 021901– Published 7 July, 2025
  • Received 15 April 2025
  • Revised 27 May 2025
  • Accepted 16 June 2025
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