REVIEW 3 major objections 5 minor 2 cited by
On the negative coupling O(N) model in 2d at high temperature
T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read Saddle points on non-principal Riemann sheets give a consistent solution of the 2d negative-coupling O(N) model at all temperatures.
desk verdict A serious large-N calculation with a plausible fix for complex saddles—non-principal sheet saddles—but the branch choice is matched, not derived, so treat the central claim as conditional. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The carrying object is the analytically continued gap equation (65), obtained by substituting $\sqrt{iz_0}\to m\in\mathbb{C}$ in the finite-temperature saddle-point condition and rewriting the zeta-function series so that all Riemann sheets are accessible. The same continuation applied to the free energy density, Eq. (67), permits comparison of phases that would be invisible on the principal sheet. This machinery converts the problem of complex masses and complex free energies into a search over real saddles on other sheets, and it is what links the two-dimensional field theory to the quantum-mechanics solution.
What would settle it
A direct numerical evaluation of the original 2d negative-coupling O(N) path integral at large N and high temperature, for example by contour-deformed lattice simulation, that yields a free-energy density different from $-\frac{NT^2\pi}{6}-\frac{3N(2g)^{1/3}T^{4/3}}{8}+\cdots$ would show that the non-principal-sheet saddle is not the physical vacuum.
Extended reading notes
Core claim
The central discovery is that the finite-temperature saddle-point condition has physically relevant solutions away from the principal Riemann sheet, and one of them is the true large-N vacuum. Written in terms of $m=\sqrt{iz_0}$ and analytically continued to all sheets, the gap equation (65) has a solution $m_2$ that stays real for every temperature and connects smoothly to the high-temperature limit $m=-(2Tg)^{1/3}$. Its free energy is real and, at high temperature, equals $-\frac{NT^2\pi}{6}-\frac{3N(2g)^{1/3}T^{4/3}}{8}+\cdots$, matching the dimensionally reduced quantum mechanics with a self-adjoint Hamiltonian. The principal-sheet saddles become a complex-conjugate pair above a critical temperature, but the non-principal saddle has the lowest free energy (except for a narrow low-temperature window at $g=g_{\rm crit}$ where another real saddle wins) and no tachyons at next-to-leading order, whereas the other saddles are dynamically unstable.
Load-bearing premise
The argument assumes that the physical theory is defined by taking the saddle on a non-principal Riemann sheet selected by free-energy minimization, with no first-principles derivation from the original path integral.
Editorial extensions
If this is right
- The large-N free energy of the 2d negative-coupling O(N) model is real at all temperatures, not just below the temperature where the principal-sheet saddles turn complex.
- In the high-temperature limit the free-energy density is $-\frac{NT^2\pi}{6}-\frac{3N(2g)^{1/3}T^{4/3}}{8}+\cdots$, matching the dimensionally reduced quantum-mechanics result exactly.
- The thermodynamically preferred saddle $m_2$ is dynamically stable at next-to-leading order in the large-N expansion, while the other saddles have tachyons and are unphysical.
- The model makes a sharp finite-N prediction for the high-temperature free energy, including the unusual case $N=3$ where the leading interaction correction vanishes to the shown order.
Reading between the lines
- If the same sheet-selection principle carries over to the four-dimensional O(N) model, the high-temperature complex-saddle problem found there could resolve in the same way; this paper only demonstrates the mechanism in two dimensions.
- The low-temperature window at $g=g_{\rm crit}$ in which $m_3$ has the lowest free energy was not subjected to the next-to-leading-order stability check, so a tachyon there would alter the phase diagram.
- A contour-deformation calculation of the original path integral could test whether the contributing stationary points actually land on the non-principal sheet, turning the sheet choice into a derived result.
- The near-vanishing ground-state energy at $N=3$ is a curiosity that, if it persists beyond the leading large-N order, would mark a special point where the reduction and large-N expansions need separate treatment.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the large-N limit of the two-dimensional O(N) scalar field theory with a negative quartic coupling, focusing on its finite-temperature behavior. The author derives the finite-temperature saddle-point equations, observes that the principal-sheet saddles become complex at high temperature, and shows that at very high temperature the theory dimensionally reduces to a PT-symmetric quantum-mechanical model. Using the known non-perturbative ground-state energy of that quantum-mechanical theory (computed via a Hermitian-equivalent reformulation and numerical diagonalization), the author argues that the correct high-temperature saddle corresponds to a non-principal Riemann sheet of sqrt(iz0). This choice is then extended to all temperatures through an analytically continued gap equation (65), producing a real saddle m2 whose free energy is generally the lowest (Fig. 1) and which is claimed to be dynamically stable at next-to-leading order. The central claim is that saddle points on non-principal Riemann sheets provide a fully consistent description of the model for all temperatures.
Significance. If the central claim holds, the paper would constitute a significant step toward understanding negative-coupling field theories beyond perturbation theory, offering a concrete mechanism (non-principal Riemann sheet saddles) that resolves the long-standing puzzle of complex saddle-point free energies. The high-temperature matching to an independent PT-symmetric quantum-mechanics result is a strong external consistency check, and the finite-N predictions in Table I and Eq. (60)-(62) are falsifiable by lattice or other non-perturbative methods. The paper is therefore potentially valuable to the large-N and PT-symmetry communities. However, as detailed below, the derivation contains several opaque or incorrect intermediate steps that are load-bearing for the main conclusion; these need to be corrected before the claims can be accepted.
major comments (3)
- [IIIB, Eqs. (51)-(53)] The reduction of the N-component path integral to the single-field partition function is not derived correctly as printed. In Eq. (51), the delta-function argument is written as ε p_i^2 - p_1^2/2 + ε ṗ_i - ε ṗ_1; substituting p_i = p_1 and ṗ_i = ṗ_1 leaves ε p_1^2 - p_1^2/2, which is not zero, so Eq. (52) does not follow. If the intended argument is ε[(p_i^2 - p_1^2)/2 + ṗ_i - ṗ_1], the equations should be corrected accordingly. Furthermore, the claim that the remaining integrations produce 'only one overall non-trivial factor' leading to the -(N-1) ln[ε ∑ p_1(x)] term in Eq. (53) is asserted without showing the Jacobian. Because this reduction underlies the ground-state energy E0 and hence the high-T matching, the derivation must be made explicit and correct.
- [IV, Eqs. (64)-(65)] The analytic continuation sqrt(iz0) = m to a non-principal Riemann sheet is introduced as a formal substitution rather than derived from the original path integral. No contour deformation or Lefschetz-thimble argument is given to demonstrate that the functional integral is dominated by saddles on non-principal sheets. The only justification offered is the high-temperature matching to the quantum-mechanics result (59). Since the all-temperature solution m2 and the central claim rest entirely on this branch choice, this is a load-bearing gap that needs to be addressed, ideally by relating the analytic continuation to a first-principles definition of the path integral, or at least by a clear physical argument for why non-principal saddles contribute.
- [IVA, around Eq. (78)] The inequality governing the presence of tachyonic poles appears to be reversed. For g > gcrit, the numerical values in the zero-temperature limit give π m_n^2/g > 1 for the larger-mass saddles m_{-1}, m_1 and π m_n^2/g < 1 for the smaller-mass saddles m_0, m_2 (e.g., for g = 2gcrit, π m_{-1}^2/g ≈ 2.68 and π m_0^2/g ≈ 0.23). From D^{-1}(0) = N/(8g)(-1 + g/(π m_n^2)), one sees that the condition for D^{-1}(0) < 0, and hence for no zero-crossing of D^{-1}(k), is π m_n^2/g > 1, not < 1. The text's statement that 'there are no poles ... as long as π m_n^2/g < 1' is therefore inconsistent with the equations, and the subsequent assignment of tachyons to m_0, m_2 versus m_{-1}, m_1 should be re-examined. This directly affects the claimed next-to-leading-order stability of the preferred saddle.
minor comments (5)
- [Eq. (51)] The delta-function argument appears to have a typographical error: the factor ε is missing from p_1^2/2. If the intended expression is δ(ε(p_i^2 - p_1^2)/2 + ε(ṗ_i - ṗ_1)), it should be written explicitly to avoid confusion.
- [IVB, Eq. (86)] The sentence 'this relation implies that there are poles of D(k), and hence no tachyons' is self-contradictory; presumably it should read 'there are no poles of D(k), and hence no tachyons'.
- [Table I] The entry for N=3 is listed as 0.0000 with the remark that E0 appears to vanish within numerical precision; the authors should state the numerical uncertainty and confirm that this is not an artifact of the discretization or diagonalization truncation.
- [References] Reference [9] is listed as 'in preparation' and clearly cannot be consulted by the reader; if it is essential, its contents should be summarized or replaced by a published reference.
- [Fig. 1] The figure caption notes that only real-valued free energies are shown; for completeness, the reader would benefit from a statement about the imaginary parts of the omitted curves and their behavior as the temperature varies.
Circularity Check
No circular reduction: the non-principal-sheet saddle is anchored to an independently computed quantum-mechanics result, not fitted from the target 2d prediction.
full rationale
The paper's derivation chain is not circular. The high-temperature limit is obtained by dimensional reduction (Eqs. (31)-(38)), and the quantum-mechanical ground-state energy is computed independently: the large-N minimum of the equivalent Hermitian potential (Eqs. (55)-(58)) gives E0 = -3N(2g)^{1/3}/8, and Table I reports numerical diagonalization for finite N, with N=1 matching the external Bender-Boettcher result [1]. This is an external benchmark, not a fit to the 2d free energy. The non-principal sheet is introduced in Eqs. (64)-(65) by analytically continuing sqrt(iz0) -> m; the high-temperature saddle m = -(2Tg)^{1/3} in Eq. (66) then reproduces the dimensional-reduction result (59) in Eq. (68). This matching is used as a discrete branch selector, not as a fitted parameter. The all-temperature solutions m2 are subsequently obtained by solving Eq. (65) with no further input from the quantum mechanics, and the free-energy comparison (Fig. 1) and NLO stability analysis follow from that branch choice. The main caveats are that the non-principal-sheet choice is an assumption not derived from a contour deformation of the original path integral, and that the boundary-condition reduction around Eqs. (51)-(53) is opaque as printed (the delta-function argument needs careful evaluation). These are correctness/rigor risks, not circular reductions: the 2d all-temperature prediction is not equivalent by construction to the QM input, since it solves a distinct gap equation. Self-citations such as Refs. [24], [27], and [48] appear as technical or motivational support but are not the load-bearing basis for the central claim, which is independently anchored by the numerical QM calculation and the external N=1 benchmark.
Assumptions & free parameters
assumptions (5)
- domain assumption The large-N saddle-point expansion is valid: fluctuations xi(x) around z0 are 1/N suppressed and the leading saddle dominates.
- domain assumption The Bender-Brody-Chen-Jones-Milton-Ogilvie transformation maps the PT-symmetric upside-down quartic oscillator to a Hermitian quartic Hamiltonian with an anomaly.
- ad hoc to paper Periodic boundary conditions and delta-function integrations reduce the N-field partition function to the single-field form Eq. (55), including the (N-1) ln term.
- ad hoc to paper The analytic continuation sqrt(iz0) = m to a non-principal Riemann sheet in Eqs. (64)-(65) captures the physical vacuum of the original field theory.
- domain assumption At high temperature the zero Matsubara mode dominates and the dimensionally reduced theory with m2_1d = 0 reproduces the 2d propagator and free energy.
Cite this review
Pith. "Pith review of On the negative coupling O(N) model in 2d at high temperature." pith.science (2026). https://pith.science/paper/5OA2EOGO
@misc{pith2026241210496,
author = {Pith},
title = {Pith review of: On the negative coupling O(N) model in 2d at high temperature},
year = {2026},
howpublished = {\url{https://pith.science/paper/5OA2EOGO}},
note = {Machine review of arXiv:2412.10496}
}
abstract
In this work, I consider N-component scalar quantum field theory in two dimensions interacting with an upside-down quartic potential. Working in the large N limit, the model can be solved non-perturbatively using the saddle-point method for sufficiently strong negative coupling. At high temperature, the O(N) model dimensionally reduces to ${\cal PT}$-symmetric quantum mechanics, for which powerful non-perturbative solution methods exist. It is found that the solution from quantum mechanics can be matched by the saddle-point method in quantum field theory when allowing for saddles beyond the principal Riemann sheet. I show that saddle points on non-principal Riemann sheets lead to a fully consistent solution of the 2d negative-coupling O(N) model for all temperatures.
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Reference graph
Works this paper leans on
-
[1]
Real spectra in non Hermitian Hamiltonians having PT symmetry,
Carl M. Bender and Stefan Boettcher, “Real spectra in non Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998), arXiv:physics/9712001
arXiv 1998
-
[2]
Carl M. Bender, Patrick E. Dorey, Clare Dunning, Andreas Fring, Daniel W. Hook, Hugh F. Jones, Sergii Kuzhel, G´ eza L´ evai, and Roberto Tateo, PT Symmetry (WSP, 2019)
work page 2019
-
[3]
(23) In analogy to the numerically obtained solutions to (21), (23) are co mplex-valued and cor- respond to a pair of complex-conjugate solutions. From (3), the renormalized free-energy density at finite tempera ture is given by Ω N = T 2 ∑ n ∫ dk 2π ln ( ω2 n +k2 +iz0 ) + (iz0)2 + 2iz0m2 B +m4 B 16g , (24) which after renormalization (10) and subtracting ...
-
[4]
Optical Solitons in PT Periodic Potentials,
Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Ch ristodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100, 030402 (2008)
work page 2008
-
[5]
Observation of Chiral State Transfer Without Encircling an Exceptional Point
Hadiseh Nasari, Gisela Lopez-Galmiche, Helena E. Lopez -Aviles, Alexander Schumer, Ab- sar U. Hassan, Qi Zhong, Stefan Rotter, Patrick LiKamWa, Dem etrios N. Christodoulides, and Mercedeh Khajavikhan, “Observation of Chiral State Tra nsfer Without Encircling an Exceptional Point,” Nature 605, 256–261 (2022), arXiv:2205.15230 [physics.optics]
work page Pith review arXiv 2022
-
[6]
Sidney R. Coleman and David J. Gross, “Price of asymptoti c freedom,” Phys. Rev. Lett. 31, 851–854 (1973)
work page 1973
-
[7]
A field theory with computable large-momen ta behavior,
K. Symanzik, “A field theory with computable large-momen ta behavior,” Lett. Nuovo Cim. 6S2, 77–80 (1973). 20
work page 1973
-
[8]
Marginal tri viality of the scaling limits of crit- ical 4D Ising and φ4 4 models,
Michael Aizenman and Hugo Duminil-Copin, “Marginal tri viality of the scaling limits of crit- ical 4D Ising and φ4 4 models,” Annals Math. 194, 163 (2021), arXiv:1912.07973 [math-ph]
arXiv 2021
Show all 50 references
-
[9]
Instantons, analytic continuation, and PT -symmetric field theory,
Scott Lawrence, Ryan Weller, Christian Peterson, and Pa ul Romatschke, “Instantons, analytic continuation, and PT -symmetric field theory,” Phys. Rev. D 108, 085013 (2023), arXiv:2303.01470 [hep-th]
2023 arXiv
-
[10]
Large N vector models in the Hamiltoni an framework,
Barberena Diego, “Large N vector models in the Hamiltoni an framework,” in preparation (2024)
2024
-
[11]
Negative Coupling φ4 on the Lattice,
Paul Romatschke, “Negative Coupling φ4 on the Lattice,” PoS LA TTICE2023, 367 (2024), arXiv:2310.03815 [hep-lat]
2024 arXiv
-
[12]
Lefschetz thimbles and stochastic quanti zation: Complex actions in the complex plane,
Gert Aarts, “Lefschetz thimbles and stochastic quanti zation: Complex actions in the complex plane,” Phys. Rev. D 88, 094501 (2013), arXiv:1308.4811 [hep-lat]
2013 arXiv
-
[13]
Latt ice scalar field theory at complex coupling,
Scott Lawrence, Hyunwoo Oh, and Yukari Yamauchi, “Latt ice scalar field theory at complex coupling,” Phys. Rev. D 106, 114503 (2022), arXiv:2205.12303 [hep-lat]
2022 arXiv
-
[14]
Symplectic quantization and the Feyn- man propagator: a new real-time numerical approach to latti ce field theory,
Martina Giachello and Giacomo Gradenigo, “Symplectic quantization and the Feyn- man propagator: a new real-time numerical approach to latti ce field theory,” (2024), arXiv:2403.17149 [hep-lat]
2024
-
[15]
Lefschetz thimble-inspired weight reg- ularizations for complex Langevin simulations,
Kirill Boguslavski, Paul Hotzy, and David I. M¨ uller, “ Lefschetz thimble-inspired weight reg- ularizations for complex Langevin simulations,” (2024), a rXiv:2412.02396 [hep-lat]
2024 arXiv
-
[16]
The thermal bootstrap for the critical O(N) model,
Julien Barrat, Enrico Marchetto, Alessio Miscioscia, and Elli Pomoni, “The thermal bootstrap for the critical O(N) model,” (2024), arXiv:2411.00978 [he p-th]
2024 arXiv
-
[17]
Bootstra pping time-evolution in quantum mechanics,
Scott Lawrence, Brian McPeak, and Duff Neill, “Bootstra pping time-evolution in quantum mechanics,” (2024), arXiv:2412.08721 [hep-th]
2024 arXiv
-
[18]
Critical Exponents above Tc to O(1/N),
Shang-keng Ma, “Critical Exponents above Tc to O(1/N),” Physical Review A 7, 2172 (1973)
1973
-
[19]
AdS dual of the critic al O(N) vector model,
I. R. Klebanov and A. M. Polyakov, “AdS dual of the critic al O(N) vector model,” Phys. Lett. B 550, 213–219 (2002), arXiv:hep-th/0210114
2002 arXiv
-
[20]
Quantum field theory in the large N limit: A Review,
Moshe Moshe and Jean Zinn-Justin, “Quantum field theory in the large N limit: A Review,” Phys. Rept. 385, 69–228 (2003), arXiv:hep-th/0306133
2003 arXiv
-
[21]
Large-n expansion for unitary superfluid fermi gases,
Martin Y Veillette, Daniel E Sheehy, and Leo Radzihovsk y, “Large-n expansion for unitary superfluid fermi gases,” Physical Review A 75, 043614 (2007)
2007
-
[22]
Finite-Temperature Conformal Fiel d Theory Results for All Cou- plings: O(N) Model in 2+1 Dimensions,
Paul Romatschke, “Finite-Temperature Conformal Fiel d Theory Results for All Cou- plings: O(N) Model in 2+1 Dimensions,” Phys. Rev. Lett. 122, 231603 (2019), [Erratum: 21 Phys.Rev.Lett. 123, 209901 (2019)], arXiv:1904.09995 [he p-th]
2019 arXiv
-
[23]
Three dimensional Yukawa models and CFTs at strong and weak couplings,
Marcus Benghi Pinto, “Three dimensional Yukawa models and CFTs at strong and weak couplings,” Phys. Rev. D 102, 065005 (2020), arXiv:2007.03784 [hep-th]
2020 arXiv
-
[24]
Interacting CFTs for all couplings: ther mal versus entanglement entropy at large N,
Seth Grable, “Interacting CFTs for all couplings: ther mal versus entanglement entropy at large N,” JHEP 10, 133 (2022), arXiv:2205.15383 [hep-th]
2022 arXiv
-
[25]
Quantum Field Theory in Large- N Wonderland: Three Lectures,
Paul Romatschke, “Quantum Field Theory in Large- N Wonderland: Three Lectures,” Acta Phys. Polon. B 55, 4–A2 (2024), arXiv:2310.00048 [hep-th]
2024 arXiv
-
[26]
Bound States, Tachyons, and Restoration of Symmetry in the 1/N Expansion,
L. F. Abbott, J. S. Kang, and Howard J. Schnitzer, “Bound States, Tachyons, and Restoration of Symmetry in the 1/N Expansion,” Phys. Rev. D 13, 2212 (1976)
1976
-
[27]
1/n-Expansion, Vacuum Stability and Quark Confinement,
Andrei D. Linde, “1/n-Expansion, Vacuum Stability and Quark Confinement,” Nucl. Phys. B 125, 369–380 (1977)
1977
-
[28]
A solvable quantum field theory with a symptotic freedom in (3+1) dimen- sions,
Paul Romatschke, “A solvable quantum field theory with a symptotic freedom in (3+1) dimen- sions,” Int. J. Mod. Phys. A 38, 2350157 (2023), arXiv:2211.15683 [hep-th]
2023 arXiv
-
[29]
A fully solvable model of fe rmionic interaction in 3 + 1d,
Seth Grable and Max Weiner, “A fully solvable model of fe rmionic interaction in 3 + 1d,” JHEP 09, 017 (2023), arXiv:2302.08603 [hep-th]
2023 arXiv
-
[30]
Can negative bare couplings make sense ? The ⃗φ4 theory at large N ,
Ryan D. Weller, “Can negative bare couplings make sense ? The ⃗φ4 theory at large N ,” (2023), arXiv:2310.02516 [hep-th]
2023
-
[31]
Revisiting O(N) σ model at unphysical pion masses and high temperatures. II. The vac uum structure and ther- mal σ pole trajectory with cross-channel improvements,
Yuan-Lin Lyu, Qu-Zhi Li, Zhiguang Xiao, and Han-Qing Zh eng, “Revisiting O(N) σ model at unphysical pion masses and high temperatures. II. The vac uum structure and ther- mal σ pole trajectory with cross-channel improvements,” Phys. R ev. D 110, 094054 (2024), arXiv:2405.11313...
2024 arXiv
-
[32]
Asympt otic freedom in a strongly interacting scalar quantum field theory in four Euc lidean dimensions,
J¨ urgen Berges, Razvan Gurau, and Thimo Preis, “Asympt otic freedom in a strongly interacting scalar quantum field theory in four Euc lidean dimensions,” Phys. Rev. D 108, 016019 (2023), arXiv:2301.09514 [hep-th]
2023 arXiv
-
[33]
Coupling renormalization flow in the strongly interacting regime of an asymptotically free quantum field theory in four dimensions,
J¨ urgen Berges, Razvan Gurau, Hannes Keppler, and Thim o Preis, “Coupling renormalization flow in the strongly interacting regime of an asymptotically free quantum field theory in four dimensions,” Phys. Rev. D 110, 036007 (2024), arXiv:2405.08153 [hep-th]
2024 arXiv
-
[34]
The Theory of Nonrenormalizable Interacti ons. 1. The Large N Expansion,
G. Parisi, “The Theory of Nonrenormalizable Interacti ons. 1. The Large N Expansion,” Nucl. Phys. B 100, 368–388 (1975)
1975
-
[35]
What if φ4 theory in 4 dimensions is non-trivial in the continuum?
Paul Romatschke, “What if φ4 theory in 4 dimensions is non-trivial in the continuum?” Phys. Lett. B 847, 138270 (2023), arXiv:2305.05678 [hep-th]. 22
2023 arXiv
-
[36]
Mass ge neration in an Abelian gauge theory with multiple scalar fields and no tree-level di mensionful couplings,
Paul Romatschke, Chun-Wei Su, and Ryan Weller, “Mass ge neration in an Abelian gauge theory with multiple scalar fields and no tree-level di mensionful couplings,” Phys. Rev. D 110, 113006 (2024), arXiv:2405.00088 [hep-ph]
2024 arXiv
-
[37]
PT -symmetric -g ϕ4 theory,
Wen-Yuan Ai, Carl M. Bender, and Sarben Sarkar, “ PT -symmetric -g ϕ4 theory,” Phys. Rev. D 106, 125016 (2022), arXiv:2209.07897 [hep-th]
2022 arXiv
-
[38]
Exact WKB analysis for PT -symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture,
Syo Kamata, “Exact WKB analysis for PT -symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture,” Phys. Rev. D 109, 085023 (2024), arXiv:2401.00574 [hep-th]
2024 arXiv
-
[39]
Exact quantization conditions and full tr ansseries structures for PT symmetric anharmonic oscillators,
Syo Kamata, “Exact quantization conditions and full tr ansseries structures for PT symmetric anharmonic oscillators,” Phys. Rev. D 110, 045022 (2024), arXiv:2406.01230 [hep-th]
2024 arXiv
-
[40]
925 (Springer, 2016) arXiv:1701.01554 [hep-ph]
Mikko Laine and Aleksi Vuorinen, Basics of Thermal Field Theory , Vol. 925 (Springer, 2016) arXiv:1701.01554 [hep-ph]
2016 arXiv
-
[41]
Phases of quartic scala r theories and PT symmetry,
Leqian Chen and Sarben Sarkar, “Phases of quartic scala r theories and PT symmetry,” (2024), arXiv:2409.05439 [quant-ph]
2024 arXiv
-
[42]
Collective m odes of an anisotropic quark gluon plasma,
Paul Romatschke and Michael Strickland, “Collective m odes of an anisotropic quark gluon plasma,” Phys. Rev. D 68, 036004 (2003), arXiv:hep-ph/0304092
2003 arXiv
-
[43]
QCD plasma instabilities and bottom up thermalization,
Peter Brockway Arnold, Jonathan Lenaghan, and Guy D. Mo ore, “QCD plasma instabilities and bottom up thermalization,” JHEP 08, 002 (2003), arXiv:hep-ph/0307325
2003 arXiv
-
[44]
Collective non -Abelian instabilities in a melting color glass condensate,
Paul Romatschke and Raju Venugopalan, “Collective non -Abelian instabilities in a melting color glass condensate,” Phys. Rev. Lett. 96, 062302 (2006), arXiv:hep-ph/0510121
2006 arXiv
-
[45]
Color instabilities in the quark–gluon plasma,
Stanislaw Mrowczynski, Bjoern Schenke, and Michael St rickland, “Color instabilities in the quark–gluon plasma,” Phys. Rept. 682, 1–97 (2017), arXiv:1603.08946 [hep-ph]
2017 arXiv
-
[46]
An Equivalent Hermitian Hamil tonian for the non-Hermitian −x4 potential,
H. F. Jones and J. Mateo, “An Equivalent Hermitian Hamil tonian for the non-Hermitian −x4 potential,” Phys. Rev. D 73, 085002 (2006), arXiv:quant-ph/0601188
2006 arXiv
-
[47]
Equivalence of a Complex PT -Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly,
Carl M. Bender, Dorje C. Brody, Jun-Hua Chen, Hugh F. Jon es, Kimball A. Milton, and Michael C. Ogilvie, “Equivalence of a Complex PT -Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly,” Phys. Rev. D 74, 025016 (2006), arXiv:hep-th/0605066
2006 arXiv
-
[48]
Simple non-perturbative resummati on schemes beyond mean- field: case study for scalar φ4 theory in 1+1 dimensions,
Paul Romatschke, “Simple non-perturbative resummati on schemes beyond mean- field: case study for scalar φ4 theory in 1+1 dimensions,” JHEP 03, 149 (2019), arXiv:1901.05483 [hep-th]. 23
2019 arXiv
-
[49]
Alternative to perturbative renorm alization in (3+1)-dimensional field theories,
Paul Romatschke, “Alternative to perturbative renorm alization in (3+1)-dimensional field theories,” Phys. Rev. D 109, 116020 (2024), arXiv:2401.06847 [hep-th]
2024 arXiv
-
[50]
Con- tour deformations for nonholomorphic actions,
Scott Lawrence, Semeon Valgushev, Jianan Xiao, and Yuk ari Yamauchi, “Con- tour deformations for nonholomorphic actions,” Phys. Rev. D 110, 074512 (2024), arXiv:2401.16733 [hep-lat]
2024 arXiv
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