{"id":"64852595-8046-47d2-ad1b-ed42b6c9a2c7","arxiv_id":"2511.22711","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Terminal bubble-wall velocities in local equilibrium can be found by requiring degenerate minima of a field-only pseudopotential, avoiding scalar equations of motion and profile or equation-of-state assumptions.","lead":"In a first-order cosmological phase transition, the terminal speed of an expanding bubble wall is found by requiring two minima of a newly defined \"pseudopotential\" to be degenerate, so the net pressure on the wall vanishes. The paper tests this in a singlet-extended Standard Model and finds sub-percent agreement with full numerical solutions of the scalar field equation.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Load-bearing approximation T(φ,φ')≈T(φ,0) is validated in only one weakly supercooled model; the advertised general method needs a stronger-transition check.","rationale":"The reader's weakest assumption—the neglect of gradient dependence in T(φ,φ')—is indeed the load-bearing point, and I agree with that identification. The paper is honest about this assumption and provides a meaningful numerical check: solving the full static EOM and comparing terminal velocities. The sub-percent agreement in one model with very small ΔT/T is real evidence that the method works in that regime. My concern is not that the derivation is wrong, but that the advertised generality of the method is broader than the tested regime. The T(h,h') expansion in Eq. (35) shows the leading local correction is quadratic in h', but the force error involves derivatives of that correction, so a small ΔT/T does not automatically guarantee a small error in the terminal velocity. A concrete integrated-force check on the existing exact solutions would settle whether the approximation is the actual reason for the agreement, and a stronger-transition benchmark would test the method's claimed scope. The abstract's statement about confirming a hybrid dip is also difficult to square with Sec. 6, where no stationary hybrid solutions are found and hybrid numerics are called uncertain; this is secondary to the core method but supports keeping the verdict conditional rather than accepting the abstract at face value. Overall, the mathematical structure is internally consistent, no ad hominem issues arise, and the concern is a limitation of evidence, not a demonstrated failure.","tokens_in":21437,"tokens_out":10609,"duration_ms":99880,"concrete_test":"Using the exact EOM solutions already computed in Sec. 5, evaluate the integrated omitted gradient-force term δP = ∫ dz h'(z) [ (∂V/∂T)(∂T/∂h(h,h')−∂T/∂h(h,0)) ]_{h=h(z),h'=h'(z)} and compare |δP| with |d(ΔVhat)/dvw|·Δvw for each λ_HS. If δP is negligible relative to the pressure scale, the 0.5% agreement is explained and the approximation is quantitatively confirmed in the tested regime; if δP is comparable or larger, the agreement is partly coincidental and the method should be re-run for a stronger first-order transition (e.g., λ_HS=0.6 or N=8) before claiming general validity.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central criterion ΔVhat=0 is derived only after replacing T(φ,φ') by T(φ,0) in the pseudopotential (Eq. 18). The paper's own caveat immediately after Eq. (18) and in Sec. 7 states this explicitly. If gradient corrections to T are non-negligible, Eq. (21) no longer follows from Eq. (10), the energy E of Eq. (22) is not conserved, and degenerate pseudopotential extrema are not the terminal-velocity condition. The numerical support in Sec. 5 and Fig. 4 is real but limited: for N=4 and λ_HS∈[0.70,0.95], ΔT/T<1e-5 and wall velocities agree with full EOM solutions at ~0.5%. This validates the approximation in that benchmark. What is not established is the advertised general method. The argument that T has no linear term in φ' (Eq. 35) only controls the local temperature correction; the force error entering the EOM is δF=(∂V/∂T)(∂T/∂h(h,h')−∂T/∂h(h,0)), a derivative of the quadratic correction that can be amplified in thin-wall or strongly supercooled transitions. Thus the claim 'this allows to compute bubble velocities without solving the EOM' is conditional on a regime tested only in one weakly supercooled model. This is a scope limitation, not an internal inconsistency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new method for computing terminal bubble-wall velocities in first-order phase transitions under local thermal equilibrium (LTE). The authors define a 'pseudopotential' V̂(φ) by integrating ∂V/∂φ along the field profile using the hydrodynamic temperature T(φ,0), and argue that the difference ΔV̂ = V̂(φ₊) − V̂(φ₋) equals the net outward pressure on the wall. They derive that for stationary bubbles the two relevant extrema must be degenerate, ΔV̂ = 0, so that the wall velocity can be obtained without solving the scalar equation of motion, without a tanh profile ansatz, and without a bag-like equation of state. The method is illustrated in a Standard Model plus N complex singlet model. For N = 4 and λ_HS ∈ [0.70, 0.95], the pseudopotential predictions agree with full solutions of the scalar equation of motion at the ~0.5% level (Fig. 4). The authors also compute the net pressure as a function of v_w, finding stable deflagrations, unstable detonations, and no stationary hybrids in the tested parameter space.","tokens_in":21899,"tokens_out":6084,"duration_ms":56653,"significance":"If the method holds beyond the single tested benchmark, it offers a computationally efficient and profile-independent route to wall velocities in LTE, avoiding both the scalar-field shooting and the bag-model approximation. The analytic derivation leading to Eq. (24) is clean, and the numerical validation against independent exact solutions is a genuine strength: no parameters are fitted to produce ΔV̂ = 0, and the comparison with the full EOM is a nontrivial check. The stability interpretation of the pressure curve is physically appealing and connects to known results in the literature. However, the advertised generality is conditioned on the approximation T(φ,φ') ≈ T(φ,0), whose domain of validity is tested only in weakly supercooled transitions with very small temperature corrections. The paper is honest about this caveat, but the central claim of a general method would be materially strengthened by a second, stronger benchmark or a quantitative error estimate.","major_comments":[{"comment":"The central identity ΔV̂ = 0 relies on replacing T(φ,φ') by T(φ,0). The paper's own caveat after Eq. (18) and in Sec. 7 acknowledges this. The numerical validation, however, covers only one weakly supercooled model (N = 4, λ_HS ∈ [0.70, 0.95]) with ΔT/T < 10⁻⁵. The argument that T has no linear term in φ' (Eq. 35) controls only the local temperature correction; the force error entering the EOM (33) is δF = (∂V/∂T)[∂T/∂h(h,h') − ∂T/∂h(h,0)], a derivative of the quadratic correction that can become amplified in thin-wall or strongly supercooled transitions. Thus the claim 'this allows to compute bubble velocities without solving the EOM' is currently established only in a narrow regime. I recommend adding a benchmark with a stronger transition (e.g., larger λ_HS, smaller N, or a model with stronger supercooling) or providing a quantitative bound on the gradient correction in terms of the w","section":"Sec. 3, Eqs. (18)–(24) and Sec. 5, Fig. 4"},{"comment":"The abstract states that the paper confirms 'the dip in outward pressure found in the literature for hybrid bubbles', but Sec. 6 finds no stationary hybrid configurations and the maximum backreaction pressure occurs near the speed of sound in the deflagration branch, not in the hybrid/Jouguet regime. The abstract therefore overstates the hybrid result. Either soften the abstract to say that the pressure curve shows a peak near c_s and that no stationary hybrids were found, or clarify what exactly is meant by 'dip'. This is a presentation issue, but it affects the paper's advertised conclusions.","section":"Abstract and Sec. 6, Fig. 5"},{"comment":"The absence of stationary hybrids is not established as a robust result. The authors state that 'we could only probe negative values of |v₋|−cₛ⁻, never truly reaching zero', so the hybrid constraint |v₋| = cₛ⁻ could not be satisfied within numerical accuracy. The conclusion 'there is no configuration ... with ΔV̂ = 0' is therefore a numerical limitation rather than a definitive no-go. The paper should present this as an inconclusive finding, possibly with a discussion of the numerical uncertainty, rather than as a confirmed property of the model.","section":"Sec. 6, 'Hybrids' paragraph"},{"comment":"The derivation shows that if a static solution of the approximate EOM (21) exists, then the energy E of Eq. (22) is conserved and ΔV̂ = 0 follows. The converse — that ΔV̂ = 0 with a barrier between the two extrema guarantees a finite-energy kink solution — is assumed but not proved. In practice the numerical checks validate the sufficiency in the tested cases, but a brief statement of the conditions under which degeneracy implies existence (e.g., monotone profile, barrier height) would make the method more rigorous.","section":"Sec. 3, Eqs. (21)–(24)"}],"minor_comments":[{"comment":"The term 'dip in outward pressure' is inconsistent with the body text, which describes a peak in the backreaction force. Please align the wording.","section":"Abstract"},{"comment":"The constants c₁ and c₂ are not explicitly related to the asymptotic values at z → ±∞; a reader may initially confuse them with the pressure parameters in the bag model. A one-sentence clarification would help.","section":"Sec. 2, after Eq. (11)"},{"comment":"The left panel shows Δv_w/v_exact; please state explicitly the reference value (deflagration vs. detonation) and whether the deviation is signed or absolute. The caption says 'relative deviation' but the text uses Δv_w = |v_pseudo − v_exact|.","section":"Sec. 5, Fig. 4"},{"comment":"The boundary condition h''(z) → 0 at z → −∞ is mentioned but not derived; it follows from the requirement that h− minimizes V(h,T(h,h')). This could be stated explicitly.","section":"Sec. 5, Eq. (33)"},{"comment":"The phrase 'corrections ... are thus expected to be small' is an expectation, not a proof; given that the approximation is the main assumption, this sentence could be strengthened by referencing the quantitative bound requested above.","section":"Sec. 7"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid contribution with a clean central idea and a fair numerical benchmark, but its advertised generality needs a stronger-transition test or a quantitative error estimate. The hybrid-related claims in the abstract are overreaching relative to the results. With those additions or revisions, it would likely be publishable in this journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my honest take on arXiv:2511.22711.\n\nWhat is new: the pseudopotential construct is genuinely new. Casting the net pressure as a difference of a field-dependent free energy, with stationary walls corresponding to degenerate minima, is an elegant reformulation of the LTE force balance. It avoids solving the scalar EOM, avoids the bag EOS, and avoids a tanh profile ansatz. The derivation in Sec. 3 is clean; Eqs. (18)–(24) are straightforward once you accept the T(φ,φ')≈T(φ,0) replacement. The numerical validation in Sec. 5 is real: sub-percent agreement with the full EOM solutions for the SM+singlet benchmark, with temperature differences below 10^-5. They also explicitly flag the assumption themselves after Eq. (18) and again in Sec. 7 — good practice.\n\nSoft spots, in proportion.\n\n1. The abstract claims confirmation of the 'dip' for hybrid bubbles and states that deflagrations are stable while detonations are unstable. But Sec. 6 explicitly says no stationary hybrid solutions were found; the hybrid branch could only be probed with negative |v-|-c_s, never reaching zero. The peak they see is near the speed of sound, not at the Jouguet velocity as in Laurent–Cline. So the abstract overstates the hybrid confirmation. This is a face-value error, not a deep flaw.\n\n2. The stability argument has a sign typo in the deflagration paragraph: perturbations to the left (lower vw) are said to give ΔV̂<0, but the text then says this accelerates the wall, which requires ΔV̂>0. Minor, but confusing.\n\n3. The load-bearing approximation T(φ,φ')≈T(φ,0) is justified by the absence of a linear term in φ' and validated numerically for one model with weak supercooling. The stress-test concern is legitimate: the force error is δF=(∂V/∂T)(∂T/∂h(h,h')-∂T/∂h(h,0)), a derivative of the quadratic correction that could be amplified in thin-wall or strongly supercooled transitions. The paper does not test that regime. The method may well hold there, but the advertised generality rests on a single benchmark. This is a scope limitation, not an internal inconsistency.\n\nOverall: a solid methodological contribution. The math is coherent, the validation is honest, and the limitations are acknowledged. Who benefits: anyone computing bubble velocities for phase transitions in LTE, for gravitational-wave or baryogenesis studies. A serious referee should be assigned. With moderate revision — fixing the abstract, fixing the sign typo, and either adding a stronger-transition test or explicitly tempering the generality claim — it will be a useful paper.\n\nMy verdict: conditional, but clearly worth peer review. I'd cite it and bring it to the study group.","headline":"Elegant new pseudopotential method for LTE bubble velocities, cleanly derived and sub-percent validated in one model, but the abstract oversells the hybrid claim and the key T(φ,φ')≈T(φ,0) approximation is tested only in a weakly supercooled regime.","tokens_in":22294,"tokens_out":2629,"would_cite":true,"duration_ms":22309,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The terminal speed of a bubble wall in local equilibrium is the speed at which the pseudopotential's extrema become degenerate.","keywords":["bubble wall velocity","first-order phase transition","local thermal equilibrium","pseudopotential","hydrodynamic backreaction","scalar field equation of motion","electroweak phase transition","cosmological phase transitions"],"falsifier":"Compute, for a local-equilibrium model with a first-order transition and strong coupling, the dimensionless quantity ΔT = (½ d²T/d(φ')²|φ'=0 φ'²)/T(φ,0) along the wall profile; if this quantity is not small (say, above a few percent), the terminal velocity from ΔV=0 should deviate from the full solution of the scalar equation of motion by more than 0.5%, contradicting the paper's central claim.","tokens_in":21370,"feed_emoji":"🫧","tokens_out":4074,"duration_ms":35753,"temperature":0.7,"pith_summary":"This paper proposes a new method to compute the terminal speed of expanding vacuum bubbles in a first-order cosmological phase transition when the surrounding plasma is in local thermal equilibrium. The central idea is a 'pseudopotential' for the scalar field whose shape changes with the wall velocity; the difference between its two extrema equals the net outward pressure on the wall. Stationary bubbles correspond exactly to degenerate pseudopotential extrema, giving a simple algebraic condition for the terminal velocity. This bypasses solving the scalar equation of motion, assuming a tanh wall profile, or using a simplified bag equation of state. In tests on a Standard Model plus singlet model, the method matches full numerical solutions to about 0.5%.","feed_headline":"Pseudopotential degeneracy fixes bubble-wall terminal velocity","feed_subtitle":"New method finds bubble speeds without solving the scalar equation of motion or assuming a wall profile.","key_machinery":"The pseudopotential is defined by V(φ) = ∫₀^φ dφ' (∂V(φ',T(φ'))/∂φ'), where T(φ) is obtained from the hydrodynamic conservation equations by setting field gradients to zero. Its derivative equals the ordinary finite-temperature force, so its extrema coincide with the minima of the thermal potential. The crucial property is that, when the temperature's dependence on field gradients is neglected, the energy E = ½(φ')² − V(φ) is conserved along a stationary wall profile, implying that stationary walls have degenerate pseudopotential extrema. The method replaces the second-order scalar differential equation with a single algebraic condition on the wall velocity and upstream temperature.","core_discovery":"The paper establishes that in local thermal equilibrium, the terminal velocity v_w of a planar bubble wall is the value for which the pseudopotential V(φ), defined as an integral of the finite-temperature force along the field, has degenerate extrema: V(φ₊) − V(φ₋) = 0, where φ₊ and φ₋ are the field values at the extrema, coinciding with the minima of the ordinary thermal potential. This condition directly expresses the balance between the vacuum driving pressure and the hydrodynamic backreaction from temperature gradients across the wall. The identification follows from conservation of an energy function for the scalar field once the plasma temperature is treated as a function of the field","pith_inferences":["The pseudopotential degeneracy condition might be reinterpreted as a variational principle or an extremal principle for steady-state interfaces, which could lead to existence proofs for stationary solutions in more general hydrodynamic settings.","The method's accuracy depends on the smallness of the second-order gradient correction to the temperature; a direct calculation of that correction for models with strong mass variation could reveal how far the 0.5% accuracy extends across parameter space.","The paper notes the difficulty in probing stationary hybrid solutions due to numerical sensitivity near the sound-speed condition; a more robust root-finding approach for that constraint might uncover stationary hybrids in parameter regions not explored here.","If out-of-equilibrium corrections to the scalar equation can be expressed as field-dependent mass terms (as the paper suggests for certain condensate contributions), the pseudopotential approach could be extended to non-LTE regimes, giving a cheap estimate of how non-equilibrium physics shifts wall velocities."],"forward_implications":["Bubble velocities in local equilibrium can be computed by scanning the wall velocity and upstream temperature until the pseudopotential extrema become degenerate, a much faster procedure than solving the scalar field equation at every step.","The method works directly from the finite-temperature effective potential, removing the need for the bag equation of state or other simplified parametrisations of the plasma, and without imposing a tanh ansatz for the wall profile.","The computed net outward pressure as a function of wall velocity exhibits a peak near the sound speed for deflagrations; the slope of the pressure around the stationary solutions indicates that deflagrations are stable while detonations are unstable, in line with earlier simulation results.","In the benchmark model, predicted wall velocities agree with full equation-of-motion solutions to about 0.5%, and the temperature correction from field gradients stays below the per-mille level along the wall profile.","The method provides a practical tool for scanning particle-physics model parameters to predict gravitational-wave signals and baryogenesis efficiencies without expensive dynamical simulations."],"fun_headline_variants":["Bubble velocity from pseudopotential degeneracy","Degenerate pseudopotential sets bubble wall speed","Pseudopotential trick pins bubble terminal velocity","No EOM needed: pseudopotential gives bubble speed","Pseudopotential extrema dictate bubble wall motion"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The method assumes that the plasma temperature obtained from the hydrodynamic equations depends only on the scalar field and not on its spatial gradient; if gradient dependence is non-negligible, the pseudopotential energy is not conserved and the degenerate-extrema condition no longer identifies stationary walls.","fun_headline_variants_meta":{"raw":{"variants":["Bubble velocity from pseudopotential degeneracy","Degenerate pseudopotential sets bubble wall speed","Pseudopotential trick pins bubble terminal velocity","No EOM needed: pseudopotential gives bubble speed","Pseudopotential extrema dictate bubble wall motion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00023,"raw_usage":{"total_tokens":1318,"prompt_tokens":743,"completion_tokens":575,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":503}},"tokens_in":487,"tokens_out":575,"duration_ms":4751,"temperature":1.0,"reasoning_tokens":503,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T19:42:00.417601+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a local-equilibrium model with a first-order transition and strong coupling, the dimensionless quantity ΔT = (½ d²T/d(φ')²|φ'=0 φ'²)/T(φ,0) along the wall profile; if this quantity is not small (say, above a few percent), the terminal velocity from ΔV=0 should deviate from the full solution of the scalar equation of motion by more than 0.5%, contradicting the paper's central claim.","supporting_citations":[],"review_version":1}