The 'alternative expansion scheme' (T'-expansion) of Borsányi et al. [Phys. Rev. Lett. 126, 232001 (2021)] extrapolates the QCD equation of state to real baryon chemical potential by encoding its μB\mu_B-dependence in a temperature rescaling T′=T(1+κ2BBμ^B2+κ4BBμ^B4+⋯ )T'=T(1+\kappa_2^{BB}\hat\mu_B^2+\kappa_4^{BB}\hat\mu_B^4+\cdots) with coefficients fitted from imaginary-μB\mu_B lattice simulations, reaching μB/T=3.5\mu_B/T=3.5 where the Taylor expansion fails. The open issues flagged in that Letter and their subsequent history are: the strangeness-neutral extension was completed by the same collaboration (2022, 2025), continuum sixth-order baryon fluctuations became available (2024), and model-based convergence studies appeared with mixed conclusions. The surviving open core is threefold: precise continuum determinations of the fourth- and higher-order coefficients κ4BB\kappa_4^{BB} and κ6BB\kappa_6^{BB} (currently consistent with zero within large errors); the unknown convergence domain of this non-Taylor resummation; and a quantitative physical-quark-mass comparison against canonical-ensemble, complex-Langevin, and resummed-Taylor strategies.

Background

Quantum chromodynamics (QCD) at finite temperature and baryon density describes the strongly interacting matter produced in heavy-ion collisions and present in neutron-star cores. Lattice QCD — a Monte Carlo discretization of the path integral on a spacetime grid — is the only systematically improvable first-principles method for equilibrium QCD thermodynamics, but at nonzero baryon chemical potential μB\mu_B the fermion determinant becomes complex (the 'sign problem'), so direct importance sampling fails. The standard workaround expands the equation of state (EOS) — pressure pp, baryon density nBn_B, entropy density ss, energy density ϵ\epsilon — in a Taylor series in μ^B=μB/T\hat\mu_B=\mu_B/T around μB=0\mu_B=0; the expansion coefficients are the generalized baryon-number susceptibilities χnB=∂n(p/T4)/∂μ^Bn\chi_n^B=\partial^n(p/T^4)/\partial\hat\mu_B^n computed at μB=0\mu_B=0. This Taylor strategy deteriorates for μB/T≳2\mu_B/T\gtrsim2–2.5: only a few coefficients are computable, and higher χnB\chi_n^B suffer a rapidly worsening signal-to-noise ratio. In Phys. Rev. Lett. 126, 232001 (2021), Borsányi, Fodor, Guenther, Kara, Katz, Parotto, Pásztor, Ratti and Szabó introduced an alternative now called the T'-expansion: the entire μB\mu_B-dependence of χ1B/μ^B\chi_1^B/\hat\mu_B is encoded in a temperature rescaling

T′(T,μ^B)=T(1+κ2BB(T)μ^B2+κ4BB(T)μ^B4+⋯ ),T'(T,\hat\mu_B)=T\left(1+\kappa_2^{BB}(T)\hat\mu_B^2+\kappa_4^{BB}(T)\hat\mu_B^4+\cdots\right),

with the κ\kappa coefficients fitted from simulations at imaginary chemical potential (where the sign problem is absent) combined with μB=0\mu_B=0 data, and the EOS then reconstructed from the rescaled temperature. Matching this ansatz order by order against the Taylor expansion yields exact relations between the two coefficient sets, e.g. χ4B=6Tκ2BB dχ2B/dT\chi_4^B=6T\kappa_2^{BB}\,d\chi_2^B/dT and χ6B=60T2(κ2BB)2 d2χ2B/dT2+120Tκ4BB dχ2B/dT\chi_6^B=60T^2(\kappa_2^{BB})^2\,d^2\chi_2^B/dT^2+120T\kappa_4^{BB}\,d\chi_2^B/dT; conversely κ4BB\kappa_4^{BB} is determined by a delicate difference of competing terms involving χ6B\chi_6^B and temperature derivatives of χ2B\chi_2^B. The Letter produced continuum-extrapolated κ2BB\kappa_2^{BB} and κ4BB\kappa_4^{BB} and thermodynamic observables up to μB/T=3.5\mu_B/T=3.5 at zero strangeness and electric-charge chemical potentials (μS=μQ=0\mu_S=\mu_Q=0), free of the unphysical oscillations that plague fixed-order Taylor results. Useful reference points are the hadron resonance gas (HRG) model — a gas of established hadrons that describes lattice thermodynamics below the crossover — and the Stefan–Boltzmann (free-quark-gluon) limit at high temperature. The Letter explicitly flagged unresolved issues: the limited precision of κ2BB\kappa_2^{BB} and especially κ4BB\kappa_4^{BB}; extension to the phenomenologically relevant strangeness-neutral conditions realized in heavy-ion collisions (⟨S⟩=0\langle S\rangle=0 with μS\mu_S and μQ\mu_Q fixed by constraints); the computational cost of order-by-order improvement; and the question of whether the scheme outperforms existing 'canonical' (fixed-baryon-number-ensemble) and other finite-density strategies.

Problem Statement

Complete and validate the T'-expansion program for the finite-baryon-density QCD equation of state, whose single common objective is a first-principles, continuum-extrapolated EOS with controlled and quantified errors at baryon chemical potentials beyond the reach of the Taylor expansion (μB/T≳2.5\mu_B/T\gtrsim2.5, target range up to and beyond μB/T≈3.5\mu_B/T\approx3.5), including the phenomenologically relevant strangeness-neutral and fixed-μQ/μB\mu_Q/\mu_B conditions realized in heavy-ion collisions. The jointly supporting open results, as left by the source Letter and its sequels, are: (i) a precision determination of the fourth-order coefficient κ4BB(T)\kappa_4^{BB}(T) — and of the next coefficient κ6BB(T)\kappa_6^{BB}(T) — in the continuum limit, whether from imaginary-μB\mu_B simulations or through the exact susceptibility relations, with statistical and systematic uncertainties small enough to decide whether these coefficients differ from zero across the crossover region (they are currently consistent with zero within large errors); (ii) the domain of convergence of the scheme, which is not a Taylor series for the pressure, so its analytic structure and ultimate range of validity — e.g. relative to the Lee–Yang edge singularities (the singularities of the free energy at complex chemical potential whose distance from the origin sets the Taylor series' radius of convergence) and to the possible QCD critical point — are unknown; and (iii) a quantitative, like-for-like assessment of the scheme against alternative first-principles finite-density strategies — canonical-ensemble simulations (direct simulation at fixed baryon number rather than fixed chemical potential), complex Langevin (a stochastic simulation method that tolerates the complex fermion determinant), and Padé/resummed Taylor expansions (rational-function resummations of the Taylor coefficients) — at physical quark masses, both in the overlap region near μB=0\mu_B=0 and at μB/T\mu_B/T where the Taylor expansion fails. Items (i)-(iii) each stand as independent open sub-results: resolving any one of them is partial progress toward the objective, not its completion, and a refutation of one leaves the others open. A full solution must deliver all three together — the continuum coefficient set with quantified statistical and systematic error budgets, a convergence statement that determines the domain of validity of the scheme or rigorously bounds it, and the benchmark comparison at physical quark masses — and these must establish, or rigorously delimit, the T'-expansion as a controlled EOS in this density regime; an unsupported or inconclusive finding in any part does not meet this standard. The already-published strangeness-neutral κ2\kappa_2/κ4\kappa_4-level EOS counts as progress toward this objective, not as its completion.

References

  1. S. Borsányi, Z. Fodor, J. N. Guenther, R. Kara, S. D. Katz, P. Parotto, A. Pásztor, C. Ratti, K. K. Szabó, 'Lattice QCD equation of state at finite chemical potential from an alternative expansion scheme', Phys. Rev. Lett. 126, 232001 (2021), DOI: 10.1103/PhysRevLett.126.232001, https://arxiv.org/abs/2102.06660
  2. S. Borsányi, J. N. Guenther, R. Kara, Z. Fodor, P. Parotto, A. Pásztor, C. Ratti, K. K. Szabó, 'Resummed lattice QCD equation of state at finite baryon density: Strangeness neutrality and beyond', Phys. Rev. D 105, 114504 (2022), DOI: 10.1103/PhysRevD.105.114504, https://arxiv.org/abs/2202.05574
  3. S. Borsányi, Z. Fodor, J. N. Guenther, S. D. Katz, P. Parotto, A. Pásztor, D. Pesznyák, K. K. Szabó, C. H. Wong, 'Continuum-extrapolated high-order baryon fluctuations', Phys. Rev. D 110, L011501 (2024), DOI: 10.1103/PhysRevD.110.L011501, https://arxiv.org/abs/2312.07528
  4. M. Kahangirwe, S. A. Bass, E. Bratkovskaya, J. Jahan, P. Moreau, P. Parotto, D. Price, C. Ratti, O. Soloveva, M. Stephanov, 'Finite density QCD equation of state: Critical point and lattice-based T′ expansion', Phys. Rev. D 109, 094046 (2024), DOI: 10.1103/PhysRevD.109.094046, https://arxiv.org/abs/2402.08636
  5. A. Abuali, S. Borsányi, Z. Fodor, J. Jahan, M. Kahangirwe, P. Parotto, A. Pásztor, C. Ratti, H. Shah, S. A. Trabulsi, 'New 4D lattice QCD equation of state: Extended density coverage from a generalized T′ expansion', Phys. Rev. D 112, 054502 (2025), DOI: 10.1103/2dmh-26yh, https://arxiv.org/abs/2504.01881
  6. M. Kahangirwe, C. Ratti, V. Vovchenko, I. Gonzalez, J. A. Muñoz, 'Convergence properties of the T′-expansion scheme: Hadron resonance gas and the cluster expansion model', Phys. Rev. D 111, 094034 (2025), DOI: 10.1103/PhysRevD.111.094034, https://arxiv.org/abs/2408.04588
  7. R. Wen, S. Yin, W.-j. Fu, 'Estimates on the convergence of expansions at finite baryon chemical potentials', Phys. Rev. D 110, 016008 (2024), DOI: 10.1103/PhysRevD.110.016008, https://arxiv.org/abs/2403.06770
  8. A. Adam, S. Borsányi, Z. Fodor, J. N. Guenther, P. Parotto, A. Pásztor, L. Pirelli, C. H. Wong, 'Finite density lattice QCD without extrapolations', EPJ Web Conf. 364, 12004 (2026), DOI: 10.1051/epjconf/202636412004, https://arxiv.org/abs/2512.09415
  9. F. Attanasio, B. Jäger, F. P. G. Ziegler, 'QCD equation of state via the complex Langevin method', arXiv:2203.13144 (2022), https://arxiv.org/abs/2203.13144
  10. S. Borsányi, P. Parotto, 'The QCD phase diagram', arXiv:2512.08843 (2025), https://arxiv.org/abs/2512.08843