Phase Diagram and Finite Temperature Properties of Negative Coupling Scalar Field Theory

Phase Diagram and Finite T emp erature Prop erties of Negativ e Coupling Scalar Field Theory P aul Romatsc hke 1 1 Institut f¨ ur The or etische Physik, TU Wien, Wie dner Hauptstr aße 8-10, 1040 Wien, Austria In this w ork, I consider scalar field theory with negative quartic self-interaction, corresponding to an upside-do wn classical p oten tial. Despite not p ossessing a classically stable ground state, such p oten tials are known to b eha ve prop erly when treated quantum mechanically , leading to stable and unitary time ev olution. Using tw o different saddle-p oint expansions for the same theory , I discuss the phase diagram in terms of bare parameters in Euclidean dimensions one to four, as w ell as the generalization to finite temp erature. Comparing to other metho ds where a v ailable, I find that negative coupling field theory is a promising candidate for an interacting scalar field theory in the contin uum. In particular, in four dimensions it exploits a lo ophole in mathematical pro ofs of quan tum trivialit y , suggesting that negative coupling scalar field theory could offer a UV-complete and interacting description of the Higgs. I. INTR ODUCTION In classical ph ysics, upside-down p otentials do not hav e stable ground states, as those who choose to pla y golf on a moun tain will readily ackno wledge. In quantum field theory , the notion that all “goo d” potentials m ust hav e stable classical ground states is enshrined as an “ob vious” condition in almost all textbo oks, and early ideas to build quan tum field theories with negativ e coupling [1] w ere derided as “nonsense” b y some of the most distinguished mem bers of the comm unit y [2]. The same “ob vious” condition could be applied to the electron in the hydrogen atom, b ecause a c harged classical particle on a circular orbit contin ually emits electromagnetic radiation, leading to orbital decay , th us preven ting a stable configuration. Y et the electronic ground state of the h ydrogen atom is quan tum mechanically stable, despite p ossessing no classically stable ground state. F urther information surfaced in the twiligh t of the last millennium [3], where it was found that even non-Hermitian p oten tials can lead to real and b ounded eigensp ectra of quantum mechanical Hamiltonians if they p ossess certain c haracteristics [4 – 7]. Using the similarities b etw een Schr¨ odinger’s w a ve equation and w av e optics and op en quan tum systems [8], exp erimen tal verification of these “exotic” theory predictions was possible (cf. Ref. [9]), creating a new subfield of material physics, see for instance Ref. [10]. Unlik e negativ e coupling quan tum mec hanics, very little is known ab out negative coupling quantum field theory . Studies of non-classical quan tum field theory w ere first conducted for iϕ 3 p oten tials in Refs. [11 – 14], and more recen tly for a broader class of complex p oten tials in Refs. [15 – 17]. Subsequen tly , non-Hermitian quantum field theory has been considered for fermionic theories (cf. Refs. [18 – 22]), quantum gravit y (cf. Refs. [23, 24]) and lattice systems (cf. Refs. [25, 26]). Studies of negative coupling quan tum field theory are even more recen t, see e.g. Refs. [27 – 33], but p oten tially important as descriptions of the Higgs field, cf. Refs. [34 – 36]. One of the issues that studies of negativ e coup ling field theory ha v e faced is th at established tec hniques suc h as w eak coupling p erturbation theory and lattice field theory using Monte Carlo imp ortance sampling fail, simply b ecause the classical probabilistic interpretation underlying these established techniques is absen t. By contrast, non-p erturbative metho ds suc h as large N expansions pioneered in the 1970s [37, 38] were found to work w ell [28, 30, 39 – 41] whereas con tour-deformation metho ds for lattice studies [25, 29, 42] work in principle, but are n umerically to o exp ensive for extracting con tin uum ph ysics in space-time dimensions d > 2. In this w ork, the focus is on uncov ering the broad qualitativ e features of the phase diagram of negativ e coupling scalar field theory , b oth at zero and finite temperature. The main tool for this study is the use of tw o distinct saddle-p oint expansions of the theory , which were presented and cross-c heck ed against other metho ds for the case of p ositiv e- coupling field theory in Ref. [43]. It should be emphasized that while the expansions are systematically impro v able, there is no small expansion parameter, and therefore truncations corresp ond to uncontrolled approximations of the theory . How ev er, the truncations made in this study hav e the adv an tage that results are analytically tractable, and can b e compared to other non-perturbative methods whenev er a v ailable (to date basically only quantum mec hanics, cf. Ref. [44]) to chec k on the quality of the approximation. In this sense, the present w ork is in tended as a survey of the phase diagram of negativ e coupling field theory both at zero and finite temp erature, with the aim of iden tifying the approximate location of features such as transition lines that ma y subsequen tly be prob ed b y n umerical methods, suc h as lattice field theory [25]. 2 I I. CALCULA TION I consider a scalar field ϕ with Euclidean action S = Z dx  1 2 ∂ µ ϕ∂ µ ϕ + 1 2 m 2 B ϕ 2 − g B ϕ 4  , (1) where the time-like Euclidean direction is a circle with radius of in v erse temp erature, β = 1 T . F or further simplicity , it will b e useful to state the results for the propagator and pressure of a free massiv e b oson in dimension d at finite temp erature: ∆ free ( M , T ) = T X n Z d d − 1 k (2 π ) d 1 ω 2 n + k 2 + M 2 , p free ( M , T ) = − T 2 X n Z d d − 1 k (2 π ) d ln  ω 2 n + k 2 + M 2  , (2) where ω n = 2 π nT are the b osonic Matsubara frequencies. Some integrations may b e p erformed to give the results ∆ free ( M , T ) = Γ  1 − d 2  (4 π ) d 2 M d − 2 + Z d d − 1 k (2 π ) d − 1 n B  √ k 2 + M 2  √ k 2 + M 2 , (3) p free ( M , T ) = Γ  − d 2  2(4 π ) d 2 M d − T Z d d − 1 k (2 π ) d − 1 ln  1 − e − β √ k 2 + M 2  , (4) where n B ( x ) = 1 e β x − 1 is the Bose-Einstein distribution function. Y et another form can b e given in terms of the high-temp erature expansion [45, Eq. (2.90)]: ∆ free ( M , T ) = T M d − 3 Γ  3 − d 2  (4 π ) d − 1 2 + 2 T (4 π ) d − 1 2 (2 π T ) 3 − d ∞ X l =0  − M 2 (2 π T ) 2  l Γ  l + 3 − d 2  Γ ( l + 1) ζ (2 l + 3 − d ) , (5) p free ( M , T ) = T d ζ ( d )Γ  d 2  π d 2 − T M d − 1 Γ  3 − d 2  ( d − 1)(4 π ) d − 1 2 + T (4 π ) d − 1 2 (2 π T ) 1 − d ∞ X l =0  − M 2 (2 π T ) 2  l +1 Γ  l + 3 − d 2  Γ ( l + 2) ζ (2 l + 3 − d ) , where p S B ( T ) = T d ζ ( d )Γ ( d 2 ) π d 2 can be recognized as the Stefan-Boltzmann pressure for a free massless boson in d dimension. A. Symmetric Saddle Expansion Using the mathematical identit y e g ϕ 4 = Z dζ √ 16 g π e − ζ 2 16 g + 1 2 ζ ϕ 2 , (6) I rewrite the action of the theory in to a form that only contains ϕ quadratically . Using the R1 resummation [46], I find for the partition function of the theory Z = Z dζ 0 e β V  p free ( M ,T ) − ζ 2 0 16 g B − 2 g B ∆ 2 free ( M ,T )  , (7) where β V is the volume of space-time, and M 2 ≡ m 2 B + ζ 0 + ν 2 , ν 2 = − 8 g B ∆ free ( M , T ) , (8) is the pole-mass of the field ϕ . In the large v olume limit, the partition function may b e calculated using the saddle p oin t method, finding ln Z β V ≡ p ( M , T ) = p free ( M , T ) − ζ 2 0 16 g B − 2 g B ∆ 2 free ( M , T ) , (9) 3 with the saddle p oint condition 0 = ∂ p ( M , T ) ∂ M 2 = − 1 2 ∆ free ( M , T ) − ζ 0 8 g B , (10) This implies ζ 0 = − 4 g B ∆ free ( M , T ), such that ν 2 = 2 ζ 0 and the saddle p oin t condition can b e expressed in terms of the pole mass [46]. One finds for the symmetric phase M 2 = m 2 B − 12 g B ∆ free ( M , T ) , p ( M , T ) = p free ( M , T ) − ( M 2 − m 2 B ) 2 48 g B . (11) B. Brok en Phase Saddle Expansion In con trast to the symmetric phase expansion, one can expand the action using ϕ ( x ) = ϕ 0 + ξ ( x ) , (12) corresp onding to an expansion around a phase with ⟨ ϕ ⟩  = 0. Using the R1 resummation [43], one obtains the pressure in the brok en phase as ˜ p ( ˜ M , T ) = − m 2 B ϕ 2 0 2 + g B ϕ 4 0 + p free ( ˜ M , T ) − 3 g B ∆ 2 free ( ˜ M , T ) , (13) where the pole mass ˜ M of the fluctuation field ξ fulfills ˜ M 2 = m 2 B − 12 g B ϕ 2 0 + ˜ ν 2 , ˜ ν 2 = − 12 g B ∆ free ( ˜ M , T ) . (14) In the large volume limit, the v alue of ϕ 0 is fixed through the saddle p oin t condition 0 = ∂ ˜ p ( ˜ M , T ) ∂ ϕ 2 0 = − m 2 B 2 + 2 g B ϕ 2 0 + 6 g B ∆ free ( ˜ M , T ) . (15) Using this condition, the p ole mass can b e written as ˜ M 2 = − 2 m 2 B + 24 g B ∆ free ( ˜ M , T ) , (16) so that the saddle p oin t condition and brok en phase pressure simplify to ϕ 2 0 = − ˜ M 2 8 g B , ˜ p ( ˜ M , T ) = p free ( ˜ M , T ) + ˜ M 4 96 g B + ˜ M 2 m 2 B 24 g B − m 4 B 48 g B . (17) I II. D=1: P T -SYMMETRIC QUANTUM MECHANICS F or the case of quantum mechanics d=1, the symmetric phase and brok en phase results simplify to p ( M , T ) = − M 2 − T ln  1 − e − β M  − ( M 2 − m 2 B ) 2 48 g B , M 2 = m 2 B − 6 g B M − 12 g B n B ( M ) M , (18) ˜ p ( ˜ M , T ) = − ˜ M 2 − T ln  1 − e − β ˜ M  + ˜ M 4 96 g B + ˜ M 2 m 2 B 24 g B − m 4 B 48 g B , ˜ M 2 = − 2 m 2 B + 12 g B ˜ M + 24 g B n B ( ˜ M ) ˜ M . Let me first discuss the case of zero temp erature T = 0, where one finds solutions M ∈ R + for m 2 B ≥ 243 1 3 g 2 3 B ≃ 6 . 24 g 2 3 B for the symmetric phase, whereas solutions ˜ M ∈ R + exist for all m 2 B . Using the criterion outlined in [43], these solutions corresp ond to differen t phases of the theory , with the phase with the highest pressure (lo w est free energy) b eing thermo dynamically fav ored. Comparing p ( M , 0) − ˜ p ( ˜ M , 0) for the solutions outlined ab ov e, one finds that there is a transition from a brok en phase for small m 2 B g 2 3 B to a symmetric phase at m 2 B ≃ 6 . 66 g 2 3 B . (19) 4 0 1 2 3 4 5 6 7 8 -10 -5 0 5 10 E 0 E 1 -p(M,0) broken -p(M,0) symmetric branch 1 -p(M,0) symmetric branch 2 E n /g B 1/3 m B 2 /g B 2/3 Energy Levels for Negative Coupling Quantum Mechanics 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 2 4 6 8 10 Broken Saddle Symmetric Saddle T c T/g B 1/3 m B 2 /g B 2/3 Phase Diagram for Negative Coupling Quantum Mechanics FIG. 1. Left: Comparison of lo west lying eigenv alues E 0 , E 1 of the Hamiltonian (20) to minus the pressure p ( M , T = 0) , ˜ p ( ˜ M , T = 0) from the symmetric and broken saddles obtained in the R1-level resummation. Right: Phase Diagram at finite temp erature, indicating whic h saddle is thermo dynamically preferred. Note that there is no actual phase transition here, just a transition from one saddle to another saddle. See text for details. As discussed in the supplemental material of Ref. [43], a transition from brok en to symmetric phase is not taking place in the quan tum mechanical theory , even though the pressure in the symmetric phase does giv e an accurate n umerical description of the ground state energy of the system. This is verified in Fig. 1 through the n umerical diagonalization of the Hamiltonian H = p 2 2 + 4 g B x 4 + p 2 g B x − x 2 m 2 B + m 4 B 16 g B , (20) whic h is isospectral to the negativ e coupling ( P T -symmetric) Hamiltonian H = p 2 2 + 1 2 m 2 B x 2 − g B x 4 [6, 44]. A t finite temp erature, real-v alued solutions for the symmetric pole mass M exist only below T < T c ( m 2 B ), whereas real-v alued solutions exist for ˜ M for all m 2 B , T . It is p ossible to track the solution with the highest pressure for all m 2 B , T , and one obtains the phase diagram for the preferred thermo dynamic saddle sho wn in Fig. 1. In the high temp erature limit T ≫ m B , results (18) simplify to p ( M , T ) = − M 2 − T ln ( β M ) − M 4 48 g B , M 2 = − 12 g B T M 2 , (21) ˜ p ( ˜ M , T ) = − ˜ M 2 − T ln  β ˜ M  + ˜ M 4 96 g B , ˜ M 2 = 24 g B T ˜ M 2 . In the case of the symmetric saddle, there are no real-v alued solutions M on the principal Riemann sheet, but instead one has M 2 = ± i (12 g B T ) 1 4 , (22) and complex-v alued pressure at high temperature. This matc hes the well-documented cases of complex-v alued pressure at high temperature for symmetric saddle expansions in the literature in v arious dimensions, cf. the discussions in Ref. [27 – 29, 32, 47, 48]. Ref. [32] prop osed a resolution of the problem b y using solutions to the saddle-p oin t condition on non-principal Riemann sheets. In this w ork, I p oint out a different wa y from Ref. [32] that also resolv es the issue of complex v alued pressures at high temperature. In particular, the brok en saddle p oint condition has real-v alued solutions ˜ M = (24 g B T ) 1 4 , (23) on the principal Riemann sheet, and the resulting pressure in the high temperature limit is ˜ p ( ˜ M ≪ T ) = − T 4 ln  24 g B e 1 T 3  + . . . , (24) whic h is real-v alued and p ositiv e for high-temp eratures, as adv ertised. 5 IV. D=2: NEGA TIVE COUPLING FIELD THEOR Y F or the case of d = 2 − 2 ε , the symmetric phase and brok en phase results lead to p ( M , T ) = − M 2 8 π  1 ε + ln ¯ µ 2 e 1 M 2  − ( M 2 − m 2 B ) 2 48 g B + T M π ∞ X n =1 K 1 ( nβ M ) n , (25) M 2 = m 2 B − 3 g B π  1 ε + ln ¯ µ 2 M 2  − 12 g B π ∞ X n =1 K 0 ( nβ M ) , ˜ p ( ˜ M , T ) = − ˜ M 2 8 π  1 ε + ln ¯ µ 2 e 1 ˜ M 2  + ˜ M 4 96 g B + ˜ M 2 m 2 B 24 g B − m 4 B 48 g B + T ˜ M π ∞ X n =1 K 1  nβ ˜ M  n , (26) ˜ M 2 = − 2 m 2 B + 6 g B π  1 ε + ln ¯ µ 2 ˜ M 2  + 24 g B π ∞ X n =1 K 0  nβ ˜ M  , where ¯ µ 2 = 4 π µ 2 e − γ E is the MS renormalization scale and I used Z dk 2 π ln  1 − e − β √ k 2 + M 2  = − M π ∞ X n =1 K 1 ( nβ M ) n . (27) The div ergencies can be non-p erturbativ ely renormalized using m 2 B = m 2 R ( ¯ µ ) + 3 g B π ε , (28) whic h leads to the running renormalized mass in the MS scheme m 2 R ( ¯ µ ) = 3 g B π ln ¯ µ 2 Λ 2 MS , (29) where Λ MS is the MS parameter. Plugging this result for m 2 B in to the abov e form ulas leads to p ( M , T ) = − M 2 8 π ln Λ 2 MS e 1 M 2 − M 4 + m 4 B 48 g B + ∞ X n =1 T M K 1 ( nβ M ) π n , − π M 2 3 g B = ln Λ 2 MS M 2 + 4 ∞ X n =1 K 0 ( nβ M ) , ˜ p ( ˜ M , T ) = − ˜ M 2 8 π ln Λ 2 MS e 1 ˜ M 2 + ˜ M 4 − 2 m 4 B 96 g B + ∞ X n =1 T ˜ M K 1  nβ ˜ M  π n , π ˜ M 2 6 g B = ln Λ 2 MS ˜ M 2 + 4 ∞ X n =1 K 0  nβ ˜ M  . A t zero temperature T = 0, a pair of real-v alued solutions for the symmetric phase exist for g B ≥ π e 1 3 Λ 2 MS and are giv en b y M 2 = − 3 g B π W − 1 , 0 − π Λ 2 MS 3 g B ! , (30) where W k denotes the Lam b ert W-function of branc h k . F or the brok en phase, real-v alued solutions exist for all v alues of g B ≥ 0 and are giv en b y ˜ M 2 = 6 g B π W 0 π Λ 2 MS 6 g B ! . (31) While p, ˜ p are div ergent, the difference ∆ p ( M , ˜ M , T ) ≡ p ( M , T ) − ˜ p ( ˜ M , T ) , (32) is finite, and can b e calculated to decide which saddle offers the thermo dynamically preferred (higher pressure) phase. A t zero temp erature, one finds that starting at small g B , the dominant phase is given b y the brok en phase un til a critical v alue of g c ≡ g B Λ 2 MS ≃ 3 . 2894 , (33) 6 -0.02 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -p(M,0) broken -p(M,0) symmetric branch 1 -p(M,0) symmetric branch 2 -p/  MS 2 g B /  MS 2 Free Energy for Negative Coupling QFT in d=2 0 0.5 1 1.5 2 0 2 4 6 8 10 Broken Saddle Symmetric Saddle T c high T approx T/  MS g B /  2 MS Phase Diagram for Negative Coupling QFT in d=2 FIG. 2. Left: F ree Energy (negative pressure) from the symmetric and broken phase saddle expansions, as a function of the coupling. Right: Phase diagram at finite temperature, indicating which saddle is thermodynamically preferred. Line lab eled ’high T approx’ is Eq. (42). See text for details. ab o v e whic h the dominant phase is given b y the symmetric phase with solution branch k = − 1 in (30). I note that at the critical coupling, the symmetric pole mass fulfills g B M 2 = − π 3 W − 1  − π 3 g c  ≃ 0 . 64 , (34) whic h lends itself to comparison to lattice-based approaches (see e.g. Ref. [49] for the positive coupling situation). A plot of the free energies at T = 0 for the differen t phases is shown in Fig. 2. A. High temp erature and dimensional reduction As temperature T is increased, the critical coupling v alue where ∆ p ( M , ˜ M , T ) = 0 increases as w ell (see Fig.2). In the high temperature limit T ≫ Λ MS , one can use the expansions (5) to express (30) after renormalization as p ( M , T ) = T 2 π 6 − T M 2 − M 2 8 π ln Λ 2 MS e 2 γ E (4 π T ) 2 − M 4 + m 4 B 48 g B , − π M 2 3 g B = 2 π T M + ln Λ 2 MS e 2 γ E (4 π T ) 2 , (35) ˜ p ( ˜ M , T ) = T 2 π 6 − T ˜ M 2 − ˜ M 2 8 π ln Λ 2 MS e 2 γ E (4 π T ) 2 + ˜ M 4 − 2 m 4 B 96 g B , π ˜ M 2 6 g B = 2 π T ˜ M + ln Λ 2 MS e 2 γ E (4 π T ) 2 . (36) F or high temp eratures, the action (1) dimensionally reduces to S red = Z d d − 1 x  1 2 ∂ i ϕ∂ i ϕ + 1 2 m 2 B , ( d − 1) ϕ 2 − T g B , ( d ) ϕ 4  , (37) where i = 1 , . . . d − 1 and I ha v e rescaled ϕ → ϕ √ T and I hav e indicated the original dimension for the coupling g B b y a subscript. Note that the mass term in (37) need not coincide with m 2 B of the original theory b ecause additional con tributions are typically generated when in tegrating out the “static” mo des (cf. the discussion in Ref. [45]). In this form S red corresp onds to (1) at zero temp erature in one dimension low er, and one can immediately relate the coupling constan ts g B , ( d − 1) = T g B , ( d ) . (38) In order to related the mass parameter, one matches observ ables in the original d-dimensional theory at high temp er- ature, and the dimensionally reduced theory . F or instance, the let us consider the p ole mass as an observ able, whic h is determined by the solution to the saddle p oint condition. Matching the symmetric saddle p oint condition at high temp erature (35) in d = 2 to the saddle p oin t condition at zero temp erature (18) in d = 1 with mass m B , (1) and coupling g B , (1) one finds g B , (1) = T g B , (2) , m 2 B , (1) = − 3 g B , (2) π ln Λ 2 MS , (2) e 2 γ E (4 π T ) 2 , (39) 7 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 1 10 100 g B =5  MS 2 p(M, T) symmetric branch 2 p(M, T) broken Stefan-Boltzmann limit p/  MS 2 p/T 2 T/  MS Pressur e of Negative Coupling QFT in d=2 FIG. 3. Pressure for the dominan t phase as a function of temp erature for g B = 5Λ 2 MS . Left part of the plot shows p ( M ,T ) Λ 2 MS whereas right part shows p ( M ,T ) T 2 to indicate approach to the Stefan-Boltzmann limit p S B ( T ) = πT 2 6 (grey line). where I stress that T here denotes the temp erature in the d = 2 dimensional theory , whereas the effective d = 1 description is at zero temperature. Also, it is worth p oin ting out that the d = 2 renormalization scale Λ MS , (2) in this matc hing fulfills Λ MS , (2) = 4 π T e − γ E − πm 2 B, (1) 6 g B, (2) , (40) e.g. is temperature-dep endent for fixed m 2 B , (1) . A completely analogous construction can b e p erformed for the brok en phase pole mass ˜ M , where one finds exactly the same matc hing conditions (39) as for the symmetric saddle. With the parameters for the effective d = 1 action fixed, one can directly reuse the results found in the previous section. One finds that for quantum mechanics (d=1) at zero temperature, the system is in the brok en phase as long as m 2 B , (1) g B , (1) 2 3 = − 3 g 1 3 B , (2) π T 2 3 ln Λ 2 MS , (2) e 2 γ E (4 π T ) 2 < ∼ 6 . 66 , (41) whic h is alwa ys true in the high temperature limit. F rom this requirement, one can immediately obtain an estimate for the critical temp erature T c in the d = 2 theory as g B , (2) Λ 2 MS , (2) ≃ 6 . 66 3 π 3 T 2 c 27 ln (4 π T c ) 2 e 2 γ E Λ 2 MS , (2) , (42) whic h is also sho wn in Fig.2. The result at high temp erature is also interesting b ecause the solution to the symmetric phase saddle (35) b ecomes M = ( − 3 g B , (2) T ) 1 3 , (43) whic h is not real-v alued for the principal branc h of the ro ot. This leads to the well-documented case of complex-v alued pressure functions at high temperature [27 – 29, 32, 47]. In the previous article [32], I argued that the non-principal branc h of the root function leads to a w ell-b eha ved high-temp erature limit. In this work, the existence of a real-v alued p ole mass and pressure from the principal ro ot of the brok en saddle expansion, sp ecifically ˜ M = (6 g B , (2) T ) 1 3 , (44) 8 mak e appeals to higher Riemann sheets unnecessary , and in my opinion provide a b etter resolution of the problems discussed in Refs. [27 – 29, 32, 47]. A representativ e plot for the pressure for the thermodynamically dominan t phase as a function of temperature is sho wn in Fig. 3 for g B = 5ΛMS. As can b e seen from this figure, the symmetric branch is preferred at lo w temp eratures, but there is a phase transition at T c Λ MS ≃ 1 . F or T > T c , the broken saddle expansion provides a real-v alued pressure that approac hes the Stefan-Boltzmann limit in the high temperature limit. V. D=3: NEGA TIVE COUPLING FIELD THEOR Y F or the case of d = 3, the symmetric phase and brok en phase results lead to p ( M , T ) = M 3 12 π − ( M 2 − m 2 B ) 2 48 g B + T 3 2 π  Li 3  e − β M  + β M Li 2  e − β M  , (45) M 2 = m 2 B + 3 g B M π + 6 g B T π ln  1 − e − β M  , ˜ p ( ˜ M , T ) = ˜ M 3 12 π + ˜ M 4 96 g B + ˜ M 2 m 2 B 24 g B − m 4 B 48 g B + T 3 2 π  Li 3  e − β ˜ M  + β ˜ M Li 2  e − β ˜ M  , (46) − ˜ M 2 2 = m 2 B + 3 g B ˜ M π + 6 g B T π ln  1 − e − β ˜ M  , where I note that because of the absence of logarithmic div ergencies no non-trivial renormalization is necessary , and I used Z d 2 k (2 π ) 2 ln  1 − e − β √ k 2 + M 2  = − T 3 (2 π ) Li 3  e − β M  − M T 2 (2 π ) Li 2  e − β M  , (47) in terms of the p olylogarithm functions Li n . In the zero temp erature limit T = 0, the saddle point equations admit solutions M = 3 g B 2 π 1 ± s 1 + 4 π 2 m 2 B 9 g 2 B ! , ˜ M = − 3 g B π 1 ± s 1 − 2 m 2 B π 2 9 g 2 B ! , (48) where M ∈ R + for at least one branc h for m 2 B ≥ − 9 g 2 B 4 π 2 and ˜ M ∈ R + for the min us branc h and m 2 B < 0. One finds that ∆ p ( M , ˜ M , 0) = 0 for a critical v alue of m 2 B g 2 B ≃ − 0 . 21 , g 2 B M 2 ≃ 2 . 68 , (49) whic h ma y lend itself to comparison to lattice-based approac hes. A. Nonzero temp erature and high temp erature limit One can trac k the transition b et w een the symmetric and brok en saddle by following the line defined by ∆ p ( M , ˜ M , T ) = 0, whic h is sho wn in Fig. 4. In the high temp erature limit, one can use the expansions (5) or more straightforw ardly directly expand (45) to find p ( M , T ) = ζ (3) T 3 2 π − M 2 T 8 π (1 − 2 ln( β M )) , M 2 = 6 g B T π ln ( β M ) , ˜ p ( ˜ M , T ) = ζ (3) T 3 2 π − ˜ M 2 T 8 π  1 − 2 ln( β ˜ M )  , − ˜ M 2 2 = 6 g B T π ln  β ˜ M  , whic h ha ve solutions M 2 = − 3 g B T π W 0 , − 1  − π T 3 g B  , ˜ M 2 = 6 g B T π W 0  π T 6 g B  , (50) 9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 -3 -2 -1 0 1 2 3 Broken Saddle Symmetric Saddle T c T/g B m B 2 /g B 2 Phase Diagram for Negative Coupling QFT in d=3 0 0.05 0.1 0.15 0.2 0.25 0.1 1 10 100 m B 2 =0.5g B 2 p(M, T) symmetric p(M, T) broken Stefan-Boltzmann limit p/g B 3 p/T 3 T/g B Pressure of Negative Coupling QFT in d=3 FIG. 4. Left: Phase diagram at finite temp erature, indicating which saddle is thermo dynamically preferred. Right: pressure as a function of temperature for m 2 B = 0 . 5 g 2 B . Left part of the plot sho ws p ( M ,T ) g 3 B whereas right part sho ws p ( M ,T ) T 3 to indicate approac h to the Stefan-Boltzmann limit p S B ( T ) = ζ (3) T 3 2 π (grey line). F or m 2 B = 0 . 5 g 2 B , the brok en saddle pressure exceeds the symmetric saddle branch ab o ve T c ≃ 0 . 71 g B . See text for details. for the pole masses of the symmetric and broken saddle expansions. Using the matc hing condition (38) and comparing the pole masses to the zero-temperature expressions (30,31) one finds g B , (2) = T g B , (3) , Λ MS , (2) = T . (51) Note that in particular the renormalization scale Λ MS for the t wo-dimensionally reduced theory takes the v alue of the temperature of the three-dimensional theory . As was found b efore, the matc hing conditions are independent of matc hing the symmetric or broken saddle expansion the dimensionally reduced theory . Ho wev er, in the high temp erature limit the symmetric saddle solution M as well as the corresp onding pressure b ecomes complex-v alued, see the disc ussion for d = 2 around Eq. (22). The solution is indicated in Fig. 4: rather than trac king the symmetric saddle, at high temp eratures a real-v alued solution is provided by the broken saddle expansion. A represen tative example of the pressure as a function of temperature is sho wn in Fig. 4 for the case of m 2 B = 0 . 5 g 2 B . F or low temp eratures, the dominan t phase is given by the symmetric saddle expansion until a critical temp erature of T c ≃ 0 . 71 g B , ab ov e which the brok en phase saddle provides higher pressure. At high temperatures, the broken phase saddle pressure approac hes the Stefan-Boltzmann pressure asymptotically from below. VI. D=4: NON-TRIVIAL INTERA CTING SCALAR FIELD THEOR Y Let me now consider the case d = 4. Before presen ting the results, let me preface the discussion by p oin ting out that there is a mathematical pro of that scalar field theory in four dimensions is trivial in the con tin uum, ruling out in teracting field theory in four dimensions [50]. The mathematical pro of is v ery sp ecific, and in particular is only v alid for Euclidean actions that are within the Griffiths-Simon class. Notably , the case of negativ e coupling λ = − g with g ∈ R + is not in the Griffiths-Simon class, providing a loophole for which the mathematical proof in Ref. [50] does not apply [51], whic h has been pointed out b efore in Ref. [52]. Similarly , lattice-based arguments against an interacting con tinuum theory from the 1980s [53, 54] suffer from the same lo ophole in that positive coupling was implicitly assumed as a sine-qua-non condition. Lattice studies for the negative coupling theory using contour deformations and brute force numerical in tegration ha ve only been p erformed recently [25], but only for tiny lattices, preven ting the study of the contin uum theory using this metho d. Since no kno wn result in the literature prohibits an in teracting con tinuum scalar field theory in d = 4 − 2 ε with ε → 0 and negative coupling [52], I now proceed to ev aluate precisely this case using the saddle p oint expansions from 10 ab o v e. I find p ( M , T ) = M 4 64 π 2 1 ε + ln ¯ µ 2 e 3 2 M 2 ! − ( M 2 − m 2 B ) 2 48 g B + M 2 T 2 2 π 2 ∞ X n =1 K 2 ( nβ M ) n 2 , (52) M 2 = m 2 B + 3 M 2 g B 4 π 2  1 ε + ln ¯ µ 2 e 1 M 2  − 6 g B M T π 2 ∞ X n =1 K 1 ( β M n ) n , ˜ p ( ˜ M , T ) = ˜ M 4 64 π 2 1 ε + ln ¯ µ 2 e 3 2 ˜ M 2 ! + ˜ M 4 96 g B + ˜ M 2 m 2 B 24 g B − m 4 B 48 g B + ˜ M 2 T 2 2 π 2 ∞ X n =1 K 2 ( nβ ˜ M ) n 2 , (53) − ˜ M 2 2 = m 2 B + 3 ˜ M 2 g B 4 π 2  1 ε + ln ¯ µ 2 e 1 ˜ M 2  − 6 g B ˜ M T π 2 ∞ X n =1 K 1 ( β ˜ M n ) n , where I used Z dk 2 π 2 k 2 ln  1 − e − β √ k 2 + M 2  = − M 2 T 2 π 2 ∞ X n =1 K 2 ( nβ M ) n 2 . (54) The symmetric saddle can b e non-perturbatively renormalized as 1 g B = 1 g R ( ¯ µ ) + 3 4 π 2 ε , 1 g R ( ¯ µ ) = 3 4 π 2 ln ¯ µ 2 Λ 2 MS , (55) whereas the brok en saddle can b e non-perturbatively renormalized as 1 g B = 1 g R ( ¯ µ ) − 3 2 π 2 ε , 1 ˜ g R ( ¯ µ ) = 3 2 π 2 ln ˜ Λ 2 MS ¯ µ 2 , (56) see the discussions in Ref. [43]. Note that unlike in low er dimensions, symmetric and brok en saddle cannot be sim ultaneously renormalized using the same renormalization condition, whic h in tro duces the need for t w o scales: Λ 2 MS , ˜ Λ 2 MS . Ho wev er, after renormalization, (52) becomes p ( M , T ) = M 4 64 π 2 ln Λ 2 MS e 3 2 M 2 + M 2 m 2 B 24 g B + M 2 T 2 2 π 2 ∞ X n =1 K 2 ( nβ M ) n 2 , (57) 0 = m 2 B g B + 3 M 2 4 π 2 ln Λ 2 MS e 1 M 2 − 6 M T π 2 ∞ X n =1 K 1 ( β M n ) n , ˜ p ( ˜ M , T ) = ˜ M 4 64 π 2 ln ˜ Λ 2 MS e 3 2 ˜ M 2 + ˜ M 2 m 2 B 24 g B + ˜ M 2 T 2 2 π 2 ∞ X n =1 K 2 ( nβ ˜ M ) n 2 , (58) 0 = m 2 B g B + 3 ˜ M 2 4 π 2 ln ˜ Λ 2 MS e 1 ˜ M 2 − 6 ˜ M T π 2 ∞ X n =1 K 1 ( β ˜ M n ) n , implying that the free energy and p ole masses in the symmetric and brok en saddle expansion would b e exactly iden tical if Λ MS = ˜ Λ MS . Ho wev er, for Λ MS = ˜ Λ MS one has differen t v alues of g B in the brok en and symmetric phase, con tradicting the deriv ation from the same action (1). A partial resolution is indicated by considering the high temperature limit of the theory pro vided by (5) which for the case at hand b ecome after renormalization p ( M , T ) = π 2 T 4 90 − M 2 T 2 24 + M 3 T 12 π + M 4 64 π 2 ln Λ 2 MS e 2 γ E (4 π T ) 2 , 0 = 3 M 2 4 π 2 ln Λ 2 MS e 2 γ E (4 π T ) 2 − T 2 + 3 M T π , (59) ˜ p ( ˜ M , T ) = π 2 T 4 90 − ˜ M 2 T 2 24 + ˜ M 3 T 12 π + ˜ M 4 64 π 2 ln ˜ Λ 2 MS e 2 γ E (4 π T ) 2 , 0 = 3 ˜ M 2 4 π 2 ln ˜ Λ 2 MS e 2 γ E (4 π T ) 2 − T 2 + 3 ˜ M T π , (60) 11 Comparing the saddle-point conditions for M , ˜ M with the effective dimensionally reduced theory (45) leads to g B , (3) = − 4 π 2 T 3 ln Λ 2 MS e 2 γ E (4 π T ) 2 , m 2 B , (3) = − T g B , (3) , (61) ˜ g B , (3) = 2 π 2 T 3 ln ˜ Λ 2 MS e 2 γ E (4 π T ) 2 , m 2 B , (3) = − T ˜ g B , (3) . (62) The effective three dimensional theories for the symmetric and broken saddle expansion b ecome iden tical if g B , (3) = ˜ g B , (3) or ˜ Λ 2 MS = (4 π T ) 3 Λ MS e 3 γ E . (63) Note the similarit y to (40). With the effectiv e parameters for the three-dimensional theory fixed, one can use (49) to determine the appro ximate transition temperature T c from brok en to symmetric saddle as m 2 B , (3) g 2 B , (3) = 3 ln Λ 2 MS e 2 γ E (4 π T c ) 2 4 π 2 ≃ − 0 . 21 , (64) implying T c ≃ 0 . 564Λ MS . The saddle-point conditions (59) in the high temp erature limit ha ve the solutions M = − 2 π T ln Λ 2 MS e 2 γ E (4 π T ) 2   1 ± s 1 + 1 3 ln Λ 2 MS e 2 γ E (4 π T ) 2   , ˜ M = 4 π T ln Λ 2 MS e 2 γ E (4 π T ) 2   1 ± s 1 − 1 6 ln Λ 2 MS e 2 γ E (4 π T ) 2   , (65) where the only solution ∈ R + is ˜ M = 4 π T ln Λ 2 MS e 2 γ E (4 π T ) 2   1 − s 1 − 1 6 ln Λ 2 MS e 2 γ E (4 π T ) 2   . (66) In particular, the solution for the symmetric saddle mass M is complex-v alued, and the resulting pressure is complex- v alued as well, which has created muc h discussion in the recen t literature [27–29, 32, 47, 48]. The existence of the real-v alued brok en-saddle solution (66) implies a real-v alued pressure for the high-temp erature limit of negative coupling field theory , thereby resolving the issue of complex-v alued pressure for the four-dimensional scalar field theory . A t zero temperature, the solution to the symmetric phase saddle-p oin t condition (57) is M 2 = 4 π 2 m 2 B 3 g B W 0  4 m 2 B π 2 3 e 1 g B Λ 2 MS  , (67) whic h implies M ∈ R + for m 2 B g B Λ 2 MS ≥ − 3 4 π 2 . F or the broken phase, using the renormalization prescription (56) implies ˜ M = 0 , ∀ m 2 B g B Λ 2 MS = finite (68) and also ˜ p ( ˜ M , T = 0) = 0. This suggests that at low v alues of m 2 B g B Λ 2 MS , the broken phase is dominant, whereas for sufficien tly high m 2 B g B the symmetric phase is dominant. Increasing m 2 B from m 2 B g B Λ 2 MS = − 3 4 π 2 , one finds indeed that ∆ p ( M , ˜ M , T = 0) = 0 at m 2 B g B Λ 2 MS ≃ − 0 . 06264 , (69) 12 0 0.02 0.04 0.06 0.08 0.1 0.12 0.1 1 10 100 m B 2 =0 p(M, T) symmetric p(M, T) broken Stefan-Boltzmann limit p//  MS 4 p/T 4 T/  MS Pressur e of Negative Coupling QFT in d=4 FIG. 5. Pressure as a function of temp erature for m 2 B = 0. Left part of the plot shows p ( M ,T ) Λ 4 MS whereas right part shows p ( M ,T ) T 4 to indicate approach to the Stefan-Boltzmann limit p S B ( T ) = π 2 T 4 90 (grey line). See text for details. suggesting a phase transition from the brok en to symmetric saddle for this v alue of parameters. A t finite temp erature and m 2 B = 0, using (63) directly in (52) leads to tw o solutions for ˜ M , the smaller one of which repro duces (66) in the high temp erature limit. A plot for the pressure in the symmetric phase and broken phase trac king the smaller solution for ˜ M is sho wn in Fig. 5. As can b e seen in Fig. 5, the pressure is w ell-defined for all temp eratures and approaches the Stefan Boltzmann limit asymptotically from b elow. One finds ∆ p ( M , ˜ M , T c ) = 0 for T c ≃ 0 . 543Λ MS , whic h is w ell b elow the temp erature T ≃ 0 . 616Λ MS where the symmetric p ole mass M b ecomes complex v alued [28]. VI I. SUMMAR Y AND CONCLUSIONS In this work I hav e considered scalar field theory with negativ e quartic coupling using saddle p oin t expansions in v arious dimensions. It w as found that the analytic saddle p oin t expansions predict an interesting phase diagram structure for the contin uum theory , whic h persists at finite temp erature. F or the case of quan tum mechanics at negativ e coupling, which is alternativ ely accessible through its isosp ectral Hermitian equiv alent [44], the saddle point expansions were found to b e quan titativ ely reliable, agreeing with the n umerically diagonalized Hamiltonian on the lev el of 15 percent. In t w o dimensions, the saddle p oin t expansions p oin t to a phase transition at zero temp erature that may b e prob ed by numerical metho ds in the near future, suc h as brute-force numerical integration on the lattice [25]. At finite temperature, the issue of complex-v alued pressure rep orted previously for negativ e coupling field theory [32] is resolv ed through a phase transition to the broken phase saddle, leading to a well-defined phase structure of the theory for all temp eratures. F or three dimensions, the saddle p oin t expansions likewise suggest a phase transition at zero temp erature lo cated near bare parameter v alues m 2 B g B ≃ − 0 . 21, and similarly a w ell-defined phase structure for the theory at finite temper- ature. The case of four dimensions is considerably more complex, but the dimensionally reduced theory at high tem- p eratures p oints to a well-defined real-v alued pressure function described b y the brok en phase saddle, while at zero temp erature and p ositive m 2 B the dominant thermo dynamic configuration is describ ed by the symmetric phase sad- dle. Nev ertheless, the fact that negativ e coupling exploits a kno wn loophole in quan tum trivialit y proofs [50 – 52] and suggests asymptotic freedom for scalars [1, 39] mak es further study of this case particularly interesting for the future. [1] K. Symanzik, A field theory with computable large-momen ta b ehavior, Lett. Nuov o Cim. 6S2 , 77 (1973). 13 [2] S. R. Coleman and D. J. Gross, Price of asymptotic freedom, Ph ys. Rev. Lett. 31 , 851 (1973). [3] C. M. Bender and S. Boettcher, Real spectra in nonHermitian Hamiltonians ha ving PT symmetry, Ph ys. Rev. Lett. 80 , 5243 (1998), arXiv:physics/9712001. [4] C. M. Bender, S. Boettcher, and P . N. Meisinger, Pt-symmetric quantum mechanics, Journal of Mathematical Physics 40 , 2201 (1999). [5] P . Dorey , C. Dunning, and R. T ateo, Spectral equiv alences, Bethe Ansatz equations, and realit y prop erties in PT-symmetric quan tum mechanics, J. Phys. A 34 , 5679 (2001), arXiv:hep-th/0103051. [6] C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Ph ys. 70 , 947 (2007), arXiv:hep-th/0703096. [7] P . Dorey , C. Dunning, and R. T ateo, The ODE/IM Correspondence, J. Ph ys. A 40 , R205 (2007), arXiv:hep-th/0703066. [8] R. El-Ganainy, K. G. Makris, M. Kha javikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-Hermitian ph ysics and PT symmetry, Nature Physics 14 , 11 (2018). [9] C. E. R ¨ uter, K. G. Makris, R. El-Ganainy, D. N. Christo doulides, M. Segev, and D. Kip, Observ ation of parity-time symmetry in optics, Nature Ph ysics 6 , 192 (2010). [10] S. Droulias, I. Katsantonis, M. Kafesaki, C. M. Souk oulis, and E. N. Economou, Chiral Metamaterials with P T Symmetry and Beyond, Ph ys. Rev. Lett. 122 , 213201 (2019), arXiv:1811.05344 [ph ysics.optics]. [11] M. E. Fisher, Y ang-lee edge singularity and ϕ 3 field theory , Phys. Rev. Lett. 40 , 1610 (1978). [12] J. L. Cardy , Conformal inv ariance and the y ang-lee edge singularit y in tw o dimensions, Phys. Rev. Lett. 54 , 1354 (1985). [13] Y.-D. Li and Q. W ang, Isosp ectral lo cal Hermitian theory for the PT-symmetric i ϕ 3 quan tum field theory, Ph ys. Rev. D 111 , 025016 (2025), arXiv:2412.10732 [hep-th]. [14] E. Arguello Cruz, I. R. Klebanov, G. T arnop olsky , and Y. Xin, Y ang-Lee Quantum Criticalit y in V arious Dimensions, Ph ys. Rev. X 16 , 011022 (2026), arXiv:2505.06369 [hep-th]. [15] C. M. Bender, N. Hassanp our, S. P . Klev ansky , and S. Sark ar, P T -symmetric quantum field theory in D dimensions, Ph ys. Rev. D 98 , 125003 (2018), arXiv:1810.12479 [hep-th]. [16] A. F elski, C. M. Bender, S. P . Klev ansky , and S. Sark ar, T ow ards p erturbativ e renormalization of ϕ 2(i ϕ ) ϵ quantum field theory, Phys. Rev. D 104 , 085011 (2021), arXiv:2103.07577 [hep-th]. [17] V. Branc hina, A. Chiav etta, and F. Contino, Study of the non-Hermitian PT-symmetric g ϕ 2(i ϕ ) ε theory: Analysis of all orders in ε and resummations, Phys. Rev. D 104 , 085010 (2021), arXiv:2104.12702 [hep-th]. [18] A. Beygi, S. P . Klev ansky , and C. M. Bender, Relativistic P T -symmetric fermionic theories in 1+1 and 3+1 dimensions, Ph ys. Rev. A 99 , 062117 (2019), arXiv:1904.00878 [math-ph]. [19] A. F elski, A. Beygi, and S. P . Klev ansky , Non-Hermitian extension of the Nam bu–Jona-Lasinio model in 3+1 and 1+1 dimensions, Phys. Rev. D 101 , 116001 (2020), arXiv:2004.04011 [hep-ph]. [20] N. E. Mavromatos and A. Soto, Dynamical Ma jorana neutrino masses and axions II: Inclusion of anomaly terms and axial bac kground, Nucl. Ph ys. B 962 , 115275 (2021), arXiv:2006.13616 [hep-ph]. [21] A. F elski and S. P . Klev ansky , F ermion and meson mass generation in non-Hermitian Nam bu–Jona-Lasinio models, Ph ys. Rev. D 103 , 056007 (2021), arXiv:2102.01491 [hep-ph]. [22] N. E. Mavromatos, S. Sark ar, and A. Soto, PT symmetric fermionic field theories with axions: Renormalization and dynamical mass generation, Ph ys. Rev. D 106 , 015009 (2022), arXiv:2111.05131 [hep-th]. [23] N. E. Mavromatos and S. Sark ar, Chern-Simons gravit y and PT symmetry , Phys. Rev. D 110 , 045004 (2024), arXiv:2402.14513 [hep-th]. [24] J. Kuntz, Unitarity through PT symmetry in quantum quadratic gra vity , Class. Quant. Grav. 42 , 175003 (2025), arXiv:2410.08278 [hep-th]. [25] P . Romatschk e, Negative Coupling ϕ 4 on the Lattice, P oS LA TTICE2023 , 367 (2024), arXiv:2310.03815 [hep-lat]. [26] M. C. Ogilvie, M. A. Sc hindler, and S. T. Sc hindler, Exotic phases in finite-density Z 3 theories, JHEP 03 , 077, arXiv:2411.11773 [hep-ph]. [27] W.-Y. Ai, C. M. Bender, and S. Sark ar, PT-symmetric -g φ 4 theory, Phys. Rev. D 106 , 125016 (2022), [hep-th]. [28] P . Romatsc hk e, A solv able quan tum field theory with asymptotic freedom in (3+1) dimensions, In t. J. Mod. Ph ys. A 38 , 2350157 (2023), arXiv:2211.15683 [hep-th]. [29] S. La wrence, R. W eller, C. P eterson, and P . Romatsc hk e, Instan tons, analytic con tin uation, and PT-symmetric field theory, Ph ys. Rev. D 108 , 085013 (2023), arXiv:2303.01470 [hep-th]. [30] R. D. W eller, Can negative bare couplings make sense? The  ϕ 4 theory at large N , (2023), arXiv:2310.02516 [hep-th]. [31] L. Chen and S. Sark ar, Phases of quartic scalar theories and PT symmetry , Phys. Rev. D 111 , 025001 (2025), arXiv:2409.05439 [quant-ph]. [32] P . Romatsc hke, On the negativ e coupling O(N) model in 2d at high temperature, JHEP 04 , 012, arXiv:2412.10496 [hep-th]. [33] D. Barb erena, Large N v ector mo dels in the Hamiltonian framework, (2025), arXiv:2502.08031 [hep-th]. [34] A. F ring and T. T aira, Goldstone b osons in different PT-regimes of non-Hermitian scalar quan tum field theories, Nucl. Ph ys. B 950 , 114834 (2020), arXiv:1906.05738 [hep-th]. [35] A. F ring and T. T aira, Massive gauge particles versus Goldstone b osons in non-Hermitian non-Ab elian gauge theory, Eur. Ph ys. J. Plus 137 , 716 (2022), arXiv:2004.00723 [hep-th]. [36] P . Romatsc hke, C.-W. Su, and R. W eller, Mass generation in an Abelian gauge theory with m ultiple scalar fields and no tree-lev el dimensionful couplings, Ph ys. Rev. D 110 , 113006 (2024), arXiv:2405.00088 [hep-ph]. [37] L. F. Abb ott, J. S. Kang, and H. J. Sc hnitzer, Bound States, T ac hy ons, and Restoration of Symmetry in the 1/N Expansion, Ph ys. Rev. D 13 , 2212 (1976). 14 [38] A. D. Linde, 1/n-Expansion, V acuum Stability and Quark Confinemen t, Nucl. Phys. B 125 , 369 (1977). [39] P . Romatsc hk e, What if ϕ 4 theory in 4 dimensions is non-trivial in the con tinuum?, Phys. Lett. B 847 , 138270 (2023), arXiv:2305.05678 [hep-th]. [40] P . Romatsc hke, Quantum Field Theory in Large- N W onderland: Three Lectures, Acta Phys. Polon. B 55 , 4 (2024), arXiv:2310.00048 [hep-th]. [41] P . Romatsc hk e, Alternative to p erturbative renormalization in (3+1)-dimensional field theories, Phys. Rev. D 109 , 116020 (2024), arXiv:2401.06847 [hep-th]. [42] S. Lawrence, H. Oh, and Y. Y amauchi, Lattice scalar field theory at complex coupling, Ph ys. Rev. D 106 , 114503 (2022), arXiv:2205.12303 [hep-lat]. [43] P . Romatschk e, On self-dualities for scalar ϕ 4 theory, (2026), arXiv:2602.18286 [hep-th]. [44] H. F. Jones and J. Mateo, An Equiv alent Hermitian Hamiltonian for the non-Hermitian -x**4 p oten tial, Ph ys. Rev. D 73 , 085002 (2006), arXiv:quant-ph/0601188. [45] M. Laine and A. V uorinen, Basics of Thermal Field Theory , V ol. 925 (Springer, 2016) arXiv:1701.01554 [hep-ph]. [46] P . Romatschk e, Simple non-p erturbativ e resummation schemes b eyond mean-field: case study for scalar ϕ 4 theory in 1+1 dimensions, JHEP 03 , 149, arXiv:1901.05483 [hep-th]. [47] S. Kamata, Exact WKB analysis for PT-symmetric quantum mechanics: Study of the Ai-Bender-Sark ar conjecture, Phys. Rev. D 109 , 085023 (2024), arXiv:2401.00574 [hep-th]. [48] S. Kamata, Exact quan tization conditions and full transseries structures for PT symmetric anharmonic oscillators, Phys. Rev. D 110 , 045022 (2024), arXiv:2406.01230 [hep-th]. [49] D. Schaic h and W. Loinaz, An Improv ed lattice measurement of the critical coupling in phi(2)**4 theory, Phys. Rev. D 79 , 056008 (2009), arXiv:0902.0045 [hep-lat]. [50] M. Aizenman and H. Duminil-Copin, Marginal trivialit y of the scaling limits of critical 4D Ising and ϕ 4 4 mo dels, Annals Math. 194 , 163 (2021), arXiv:1912.07973 [math-ph]. [51] P . Romatsc hke, A loophole in the pro ofs of asymptotic freedom and quan tum triviality, P oS EPS-HEP2023 , 500 (2024), arXiv:2310.18414 [hep-th]. [52] M. Aizenman and R. Graham, On the renormalized coupling constan t and the susceptibilit y in φ 4 4 field theory and the Ising mo del in four dimensions, Nuclear Physics B 225 , 261 (1983). [53] J. F r¨ ohlic h, On the T riviality of Lambda (phi**4) in D-Dimensions Theories and the Approach to the Critical Poin t in D ≥ F our-Dimensions, Nucl. Phys. B 200 , 281 (1982). [54] K. G. Wilson, The renormalization group and critical phenomena, Rev. Mod. Ph ys. 55 , 583 (1983).