Abstract
At high baryon chemical potential μ_{B}, the equation of state of QCD allows a weak-coupling expansion in the QCD coupling α_{s}. The result is currently known up to and including the full next-to-next-to-leading order α_{s}^{2}. Starting at this order, the computations are complicated by the modification of particle propagation in a dense medium, which necessitates nonperturbative treatment of the scale α_{s}^{1/2}μ_{B}. We apply a hard-thermal-loop scheme for capturing the contributions of this scale to the weak-coupling expansion, and we use it to determine the leading-logarithm contribution to next-to-next-to-next-to-leading order: α_{s}^{3}ln^{2}α_{s}. This result is the first improvement to the equation of state of massless cold quark matter in 40years. The new term is negligibly small and thus significantly increases our confidence in the applicability of the weak-coupling expansion.
Highlights
At high baryon chemical potential μB, the equation of state of QCD allows a weak-coupling expansion in the QCD coupling αs
No new perturbative orders have been determined for the equation of state (EOS) since 1977, when Freedman and McLerran derived the full next-to-next-to-leading order (NNLO) result for the pressure as a function of quark chemical potentials in the limit of massless quarks [9,10]
While the naive loop expansion of the EOS leads to a series of terms analytic in αs, this need not be the case for the resummed soft sector: In particular, loop integrals that are sensitive to both the hard and the soft scales can receive contributions from the semisoft region between the two
Summary
QCD matter with massless quarks can be written in the form (see, e.g., Ref. [13]). 0.911 ln μBΛ=3α2s þ c3;2α3s ln2αs þ c3;1ðΛ Þα3s ln αs þ c3;0ðΛ Þα3s þ Oðα4s Þ; ð1Þ where Λis the renormalization scale, and where the c3;i are the as-yet-uncalculated N3LO terms. When the integration momentum in Eq (2) becomes soft, P ∼ α1s=2μB, adding an arbitrary number of (one-loop) self-energy insertions to the gluon line does not change the magnitude of the diagram. To the leading order in the external momenta, this gives rise to the well-known hard-thermal-loop (HTL) power counting [24] and allows for a convenient computation of the resummed diagrams within the framework of the HTL ð2Þ where the (2 þ 1) corresponds to two transverse polariza- FIG. The semisoft contribution to the pressure is simple, as the propagator can be treated as if it were both soft and hard: Because P ≪ μB, instead of all topologies, only the restricted HTL set of diagrams contribute, but because P ≫ α1s=2μB the diagram can be expanded in the number of self-energy insertions. Introducing two semisoft momentum-space cutoffs α1s=2μB ≪ Λ1 ≪ Λ2 ≪ μB, we are left with the integral psIRem;1isoft
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