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REVIEW 3 major objections 4 minor 297 references

A gauge-invariant 3D EFT formalism for the sphaleron rate replaces the heuristic baryon-preservation criterion vc/Tc > 1 with a computable condition on x = λ₃/g₃², and constrains real-triplet extensions of the Standard Model.

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2026-07-31 23:14 UTC pith:XGBD3C3T

load-bearing objection A serious thesis with a genuinely new 3D-EFT sphaleron-rate formalism and a plausible replacement for v/T, but the central first-order approximation is a fitted constant that needs independent confirmation before the x-criterion is trusted. the 3 major comments →

arxiv 2607.24026 v1 pith:XGBD3C3T submitted 2026-07-27 hep-ph hep-th

Electroweak Baryogenesis: Advances in Sphaleron Rate Calculations and Implications of Thermal Phase Transitions

Yanda Wu This is my paper
classification hep-ph hep-th
keywords electroweak baryogenesissphaleron ratethree-dimensional effective field theorydimensional reductionbaryon-number violationfirst-order phase transitionprimordial black holestopological field configurations
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The thesis targets the washout problem in electroweak baryogenesis: after a first-order phase transition, sphaleron processes inside the broken-phase bubbles can erase the just-generated baryon asymmetry, and the usual practical criterion for stopping that — vc/Tc ≳ 1 — is both approximate and gauge-sensitive. The thesis develops a gauge-invariant formalism for the sphaleron rate in SU(2)+Higgs theory using three-dimensional thermal effective field theory, valid for first-order transitions and accurate to O(g⁴) in power counting, and shows that the baryon-preservation condition becomes a statement about the dimensionless ratio x = λ₃/g₃². A calibrated numerical approximation — the rescaled sphaleron action is essentially 29·v₃(x,y), with v₃ the gauge-invariant scalar minimum — turns the washout exponent into a computable quantity. Applying this to the real-triplet extension of the Standard Model yields large washout over much of the parameter space, strongly constraining the model; supplementary results extend sphaleron/monopole constructions to arbitrary SU(2) multiplets and link delayed phase transitions to primordial black hole production.

Core claim

The central claim is that the sphaleron rate during a first-order electroweak phase transition can be computed gauge-invariantly to O(g⁴) in the 3D EFT of SU(2)+Higgs theory, even though the barrier that sustains the transition comes from integrating out the spatial gauge fields. After rescaling by the gauge-invariant scalar minimum v₃(x,y), the sphaleron action is kinetic-dominated and well approximated by S₃D ≈ 29·v₃(x,y), with x = λ₃/g₃² and y = μ₃²/g₃⁴; the washout exponent is then integrated accurately, replacing vc/Tc ≳ 1 with a gauge-invariant condition on x. In the real-triplet extension, the resulting washout is large across most of the parameter space, strongly constraining such mo

What carries the argument

The load-bearing object is the rescaled 3D sphaleron action S₃D = v₃(x,y)·C_sph(x,y), where v₃(x,y) is the minimizer of the leading-order scalar potential in the 3D EFT and x = λ₃/g₃², y = μ₃²/g₃⁴ are the two dimensionless parameters. Three ingredients carry the argument: soft-scale power counting (k ~ gT), which puts the sphaleron at a scale decoupled from bubble nucleation; kinetic dominance, the numerical result that C_sph(x,y) ≈ 29 is nearly constant across the (x,y) plane, so all parameter dependence enters through v₃(x,y); and normalization by g₃², which is positive-definite and gauge invariant, replacing the gauge-dependent v-normalization of earlier work. Gauge invariance to O(g⁴) re

Load-bearing premise

The computation treats the scalar potential as only a boundary-condition sector: the sphaleron action is approximated by its kinetic part, S₃D ≈ 29·v₃(x,y), on the numerical observation that 'the dominant contribution to the sphaleron action comes from the kinetic terms,' and the rate's dynamical prefactor is assumed (A_dyn ~ T) rather than computed — if kinetic dominance fails near the critical line or once U(1) and A₀ modes are added, the fitted constant and the x-criterion

What would settle it

A 3D lattice computation of the Chern–Simons diffusion rate in the first-order region (x ≲ 0.1, y > 0) would settle the claim: if the measured exponential suppression disagrees with exp(−29·v₃(x,y)) beyond the claimed O(g⁴) and prefactor uncertainties, kinetic dominance fails. A cheaper check is fully numerical — solve the full sphaleron equations with U(1) and A₀ modes included and verify that S₃D/v₃(x,y) stays within a few percent of 29 across the (x,y) region of interest; the thesis's own residual map shows where deviations already reach order one.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Baryon preservation after a first-order transition is decided by a gauge-invariant condition on x = λ₃/g₃², not by vc/Tc ≳ 1; washout exponents can be computed, so a model can overproduce the asymmetry via CP violation and then wash it down to the observed value.
  • The real-triplet extension of the Standard Model is strongly constrained: much of its parameter space exhibits large baryon washout, and consistent results require two-loop thermal matching of the 3D EFT parameters.
  • For a general SU(2) multiplet, nonzero hypercharge yields a sphaleron while zero hypercharge yields a monopole; the sphaleron one-form is representation-independent, and monopole masses can significantly exceed the SM sphaleron energy in two-step transitions, changing when baryon number can be violated.
  • A delayed first-order phase transition can produce primordial black holes with a relic abundance that is super-exponentially sensitive to phase-transition parameters (Eq. 7.22).
  • Collider searches for exotic Higgs decays at future lepton colliders can indirectly probe the strong first-order phase transition required by electroweak baryogenesis, covering a large portion of the relevant parameter space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If kinetic dominance survives the inclusion of U(1) and A₀ modes, the S₃D ≈ 29·v₃(x,y) approximation is portable: any BSM model that maps onto the same SU(2)+Higgs 3D EFT gets a nearly parameter-free washout computation, reducing electroweak baryogenesis to a scan over x and the CP source.
  • A sharp, testable extension would be a 3D lattice measurement of the Chern-Simons diffusion rate in the first-order region (x ≲ 0.1, y > 0): agreement with exp(−29·v₃(x,y)) would substantiate the whole procedure, while the fit residuals the thesis itself reports near the critical line mark where the approximation is most exposed.
  • The monopole results suggest a concrete two-step chronology: an asymmetry generated during an intermediate monopole phase could be preserved by a heavy monopole mass, only to face full sphaleron washout in the second, Higgs-breaking step — a sequence the thesis's washout formalism makes computable for specific models.
  • Taking the thesis at its word, the baryon asymmetry becomes a computable function of 3D EFT parameters rather than a heuristic threshold; this implies EWBG model building can be inverted — fixing x and the CP-violating couplings from the observed asymmetry — a shift that would sharpen collider targets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This is a PhD-thesis-style manuscript addressing electroweak baryogenesis, focused on two claims: (i) a gauge-invariant 3D EFT formalism for the sphaleron rate in SU(2)+Higgs theory, claimed to be gauge invariant under power counting up to O(g^4); and (ii) a replacement of the traditional baryon-preservation criterion vc/Tc ≳ 1 by a new gauge-invariant criterion on x = λ3/g3^2 (Secs. 1.1, 6.1–6.2, 6.5.2). The central quantitative step is the calibrated fit S3D ≈ 29 v3(x,y) (Eq. 6.27), where v3 is the minimum of the effective cubic potential (6.12)/(6.15); the kinetic-dominance argument justifies using the potential only for boundary conditions, and the fit is validated numerically over the (x,y)-plane in Fig. 16. The dynamical prefactor Adyn is assumed to be ~T on dimensional grounds (Sec. 6.1). The formalism is benchmarked against the SM crossover lattice rate (Fig. 17) and applied to the real-triplet extension, yielding strong washout constraints (Sec. 6.6.2). Additional contributions include the general-SU(2)-multiplet sphaleron/monopole classification and construction (Sec. 3.4), monopole-catalyzed BNV (Sec. 4.3), PBH production from delayed transitions (Sec. 7.1), and CEPC collider probes (Sec. 7.2). A large fraction of the manuscript is pedagogical review.

Significance. If the central claims hold, the manuscript supplies a formal, gauge-invariant replacement for the heuristic vc/Tc criterion and a concrete phenomenological output — the real-triplet exclusion from washout. The checkable strengths are real: the general-multiplet construction is verified for J ∈ {1,3/2,2,5/2,3}; the SM crossover benchmark agrees with lattice (Fig. 17); the calibrated fit (6.27) is validated over a stated parameter region; and the main approximations (Adyn, U(1)/A0 omission, the sign correction in footnote 37) are disclosed in the text rather than hidden. The contribution is conditional: the fitted-action residual (±6–7% if the Fig. 16 colorbar is in units of Csph ≈ 29) and the assumed prefactor both enter the washout exponent, and the manuscript does not yet translate them into an uncertainty on the x-criterion. Because the judgment 'this model is excluded by washout' depends on that translation, the quantitative significance is not yet fully established.

major comments (3)
  1. [Sec. 6.2.2 (Eq. 6.27; Fig. 16)] The x-criterion of Sec. 6.5.2 and the real-triplet constraints of Sec. 6.6.2 inherit the calibrated approximation S3D ≈ 29 v3(x,y). The bottom row of Fig. 16 reports 'fit − exact' residuals spanning about −1.78 to 2.07 over the (x,y) plane, but the caption does not state the plotted units or the location of the critical line y_c(x). If these are units of Csph (≈29), the residual is ~±6–7% in the washout exponent; since the rate is exponential in S3D, this shifts the decoupling temperature and the derived x-criterion by an amount comparable to the criterion's discriminating power. Please state the units, report the residual in physical units of ΔS3D/T over the washout-relevant region (including near y_c(x), where the cubic term matters most), and propagate it into the baryon-preservation boundary (Fig. 18) and the triplet exclusion — or moderate the precision claim. The disclosed U(1)Y an
  2. [Sec. 6.1 (Eq. 6.1)] The decomposition Γsph = Adyn × Astatic with Adyn ~ T assumed on dimensional grounds is disclosed explicitly, and I credit the disclosure. Nevertheless, the abstract and Sec. 1.1 present the washout computation as quantitative ('the washout can be computed precisely'). The prefactor enters the washout condition Γsph ≈ H only logarithmically (ln(Adyn/H) = S3D), so an O(1) coefficient error in Adyn shifts the required action by O(1) — smaller than, but comparable to, the Fig. 16 residual effect. Please quote the resulting uncertainty in the decoupling temperature and in the x-criterion, or state explicitly that the criterion controls only the exponential part of the rate.
  3. [Sec. 1.1; Sec. 6.2.2] The Introduction's second bullet can be read as attributing O(g4) accuracy to the full first-order-transition sphaleron rate. The gauge-invariance-under-power-counting property is established for the 3D EFT matching (Sec. 5.3), not for the calibrated fit (6.27) or the assumed prefactor in Sec. 6.1. Please add a sentence distinguishing the power-counting property of the EFT from the numerical accuracy of the FOPT action fit, so that the 'new gauge-invariant criterion' is presented with an explicit accuracy statement rather than an implicit O(g4) one.
minor comments (4)
  1. [Fig. 16] Label the colorbar units of the residual panel, and overlay the critical line y_c(x) so the washout-relevant region is identifiable.
  2. [Sec. 5.3.1, footnote 37] The sign correction relative to Ref. [263] is welcome, but please show the corrected derivation or state explicitly that the matching result for λ3 in Eq. (5.34) is unchanged by the correction.
  3. [Sec. 6.1 (Eq. 6.7)] Notation: the parameter x = λ3/g3^2 is also used for a spatial coordinate in Chs. 3–4, and y (mass parameter) clashes with hypercharge Y in Sec. 3.4. Please use a distinct font or symbol in the published version.
  4. [Sec. 4.2.1 (Eq. 4.38)] The heuristic Γsph ~ T^4 exp(−Esph/T) is a pedagogical scaling estimate; please state explicitly that it is superseded by the 3D EFT computation of Sec. 6 and does not determine the prefactor.

Circularity Check

0 steps flagged

No circular reduction found; the sphaleron-rate derivation is internally computed and benchmarked against lattice results.

full rationale

The central chain is: 3D EFT action (6.2), dimensionless rescaling, numerical solution of the sphaleron equations (6.21)-(6.22), the numerically observed near-constancy of C_sph leading to the fit S_3D ≈ 29 v3(x,y) (6.27), and validation against the full numerical action (Fig. 16) and the lattice SM sphaleron rate (Fig. 17). The parameter x = λ3/g3^2 is an EFT input, not defined in terms of the washout result; the washout condition is obtained by exponentiating the computed action. Citations to Refs. [94] and [230] are disclosed reproductions of the author's own prior work, but the thesis reproduces the scaling derivation, equations of motion, numerical fits, and benchmarks rather than relying on an unverified self-citation as the sole support. The paper itself flags genuine limitations: 'we assume Adyn ∼ T on dimensional grounds' (Sec. 6.1) and questions whether it is legitimate to integrate out the spatial gauge fields (Sec. 6.1). These are correctness/robustness concerns, not circular reductions: they do not make the predicted washout equal to the input by construction. No step was found in which a quantity defined in terms of the target result is later presented as a prediction of that target.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

Central claims rest on (i) numerical fits carrying the washout predictions (Eqs. 6.25-6.27); (ii) standard 3D-EFT matching assumptions, including gauge invariance to O(g4), integration of A0 as a heavy mode, and the assumed Adyn ~ T; (iii) an ansatz whose representation-independence is verified only up to J=3. No new particles, forces, or dimensions are introduced; the monopole/sphaleron configurations are standard topological objects extended to general SU(2) representations. The thesis's benchmark numbers (Bsph ~ 1.9, Bmon = 5.001145, Figs. 17-20) are reproduced from the author's own prior papers, so their independence from this manuscript is limited.

free parameters (4)
  • Csph fit constants (A, B, C, D) = A = 26.12, B = -2.145, C = 0.4237, D = 0.00717
    Eqs. (6.25)-(6.26): four constants fitted to numerically computed sphaleron actions for the second-order (crossover) case; this fit supplies the action used in the SM-crossover benchmark and in washout estimates.
  • FOPT action proportionality constant (the '29' in S_3D ~ 29*v3) = 29
    Eq. (6.27): the first-order-transition sphaleron action is approximated as 29*v3(x,y), calibrated to the numerical action over a scan in (x,y). This is load-bearing for all FOPT washout rates and the x-criterion.
  • Monopole BNV cross-section constant c = unspecified
    Eq. (4.55): sigma_{DeltaB!=0} ~ v^{-1} c / E_f^2 inherits an undetermined constant from Ref. [434]; quoted, not fitted here, but it normalizes the monopole-catalyzed washout channel.
  • Dynamical rate prefactor Adyn = ~ T (assumed)
    Sec. 6.1, Eq. (6.1): 'we assume Adyn ~ T on dimensional grounds, and focus on the static part Astatic.' This assumption sets the absolute normalization of every sphaleron rate in the framework.
axioms (5)
  • standard math Standard homotopy results: pi_n(S^n) = Z, pi_n(S^m) = 0 for n < m, pi_2(G/H) = pi_1(H), pi_n(Maps_0(S^q -> S^m)) = pi_{n+q}(S^m)
    Sec. 3.1 uses these to classify strings, monopoles, sphalerons, instantons; proofs are deferred ('The proof ... lies beyond the scope of this thesis', Refs. [385, 388]).
  • domain assumption Sphaleron dynamics is captured by the zero-Matsubara 3D EFT with O(g4)-matched couplings, and the temporal gauge field A0 is parametrically heavier than the sphaleron scale so it can be integrated out
    Sec. 5.2 Table 2 and footnote 38: the whole rate computation lives in the 3D EFT; if the scale separation fails near the transition, the dimensional reduction and the computed rates are invalid.
  • domain assumption Static/dynamic factorization of the sphaleron rate with Adyn ~ T
    Sec. 6.1, Eq. (6.1): the dynamical part is assumed on dimensional grounds; only the static part is computed.
  • domain assumption Kinetic terms dominate the sphaleron action, so the detailed scalar potential only sets boundary conditions through v3; this justifies integrating out spatial gauge fields in the first-order case
    Sec. 6.1 (after Eq. 6.9) and Sec. 6.2.2: the author explicitly raises 'whether it is legitimate to integrate out the spatial gauge fields' and justifies it numerically; this is load-bearing for Eq. (6.27).
  • domain assumption The sphaleron ansatz with radial profiles (f, f3, f0, h) and representation-independent one-forms Fa
    Eqs. (3.74)-(3.77), (3.92)-(3.95): standard Manton-Klinkhamer ansatz assumed; representation independence of Fa is explicitly checked only for J in {1, 3/2, 2, 5/2, 3}, not proven for all J.

pith-pipeline@v1.3.0-alltime-deepseek · 58534 in / 20760 out tokens · 173013 ms · 2026-07-31T23:14:22.695428+00:00 · methodology

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read the original abstract

This thesis reviews recent advances in calculating the sphaleron rate, with particular emphasis on electroweak baryogenesis. It also provides pedagogical introductions to sphaleron- and instanton-induced baryon-number violation, the vacuum structure of non-Abelian gauge theories, and other topological field configurations, and suggests "an zi" as a possible Chinese term for "sphaleron" (see p. 3). Broader implications of first-order phase transitions for collider searches and primordial black hole formation are also discussed.

Figures

Figures reproduced from arXiv: 2607.24026 by Yanda Wu.

Figure 1
Figure 1. Figure 1: Visualization of the homotopy group π1(S 1 ). The black circle represents the target space S 1 , and the pink curve represents a map from the domain S 1 to the target S 1 . The black dot at the bottom of the circle denotes the base point. Panel (a) is homotopic to the trivial map and therefore corresponds to the identity element 0 in π1(S 1 ). Panels (b) and (c) are homotopic to each other and correspond t… view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of the homotopy group π1(S 2 ). The blue sphere represents the target space S 2 , and the pink curve represents a map from S 1 to S 2 . The black dot on the sphere denotes the base point. All panels (a), (b), (c), and (d) are homotopic to each other, since each pink curve can be continuously deformed to the trivial map. They therefore all correspond to the same element 0 in the group π1(S 2 )… view at source ↗
Figure 3
Figure 3. Figure 3: Visualization of the homotopy group π1(Maps0 (S 1 → S 2 )). Here the target space is the space of maps from S 1 to S 2 , represented by the colored loops on the blue sphere. Although each individual loop is contractible to the base point, all loops together need not be simultaneously contractible. This configuration corresponds to the element 1 in the homotopy group π1(Maps0 (S 1 → S 2 )). provided that G … view at source ↗
Figure 4
Figure 4. Figure 4: Sphaleron radial-profile solutions for the pure Standard Model. The horizontal axis denotes the dimensionless radial coordinate ξ ≡ gΩr, with Ω = 246 GeV and g the SU(2) gauge coupling. This figure is reproduced from Ref. [94]. multiplet with the largest dimension satisfying both the unitarity constraint and viable dark matter criterion is the complex septuplet with J = 3 [326]. This multiplet naturally em… view at source ↗
Figure 5
Figure 5. Figure 5: Schematic illustration of the three scenarios for the electroweak phase transition. In each panel, the horizontal axis represents the Higgs field, while the vertical axis represents the septuplet field. Panel (a) represents the one-step phase transition from the symmetric phase O to the electroweak minimum X; (b) represents a one-step phase transition to the mixed minimum Z, where both fields acquire nonze… view at source ↗
Figure 6
Figure 6. Figure 6: Radial profile functions when only the septuplet obtains a vev during a two-step EWPT, for v = 0 GeV, vϕ = 500 GeV, λ13 = 0.05, and λs = 0.005. At this stage, the Higgs profile does not appear because its vev vanishes, and the monopole-mass value is Bmon = 5.001145. This figure is reproduced from Ref. [94]. obtains a vev. For this intermediate phase, the relevant homotopy group is the monopole one, as show… view at source ↗
Figure 7
Figure 7. Figure 7: Monopole mass Bmon when only the septuplet obtains a vev during a two-step EWPT. The horizontal axis is the septuplet vev vϕ in GeV, and the vertical axis is Bmon. The viable monopole-mass region is the part of the orange band to the right of the dashed vertical line at λ13. The olive-drab horizontal dashed line indicates the SM sphaleron benchmark Bsph ≈ 1.9. The orange band shows Bmon across scanned valu… view at source ↗
Figure 8
Figure 8. Figure 8: Illustration of the spacetime structure as a cylinder. Each horizontal slice represents a spatial slice at fixed time, while the vertical direction corresponds to imaginary time. The spatial boundary at infinity (|x| = R with R → ∞), together with the imaginary￾time direction, forms the surface of the cylinder, which has the topology of S 3 (1). At τ = −T and τ = +T, the fields are in vacuum configurations… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the instanton and sphaleron mechanisms for baryon-number vio￾lation. (a) Instanton mechanism: quantum tunneling from the vacuum with Chern-Simons number j = 1 to the vacuum with Chern-Simons number j = 2. (b) Sphaleron mechanism: thermal transition over the energy barrier from the vacuum with Chern-Simons number j = 1 to the vacuum with Chern-Simons number j = 2. The static sphaleron solution… view at source ↗
Figure 10
Figure 10. Figure 10: Finite-temperature barrier crossing in one-dimensional quantum mechanics with a double-well potential. According to the WKB approximation in quantum mechanics, the tunneling amplitude is dominated by the exponential of the Euclidean action. Therefore, the leading-order tunneling rate is [421] Γtunneling ∼ e −2S0 , (4.29) where S0 is the Euclidean action we will discuss below. Recalling that the particle h… view at source ↗
Figure 11
Figure 11. Figure 11: Schematic stages of electroweak baryogenesis: (i) a first-order phase transition proceeds through bubble nucleation from the false vacuum to the true vacuum; (ii) CP￾violating interactions with the expanding wall generate a nonzero left-handed fermion density; (iii) in the symmetric phase, rapid sphalerons convert the left-handed bias into a baryon excess that is swept into the bubbles; (iv) inside the bu… view at source ↗
Figure 12
Figure 12. Figure 12: One-loop diagrams for the two-point Green’s function of the scalar field that contribute to scalar wave-function renormalization. The dashed, wavy, and solid lines denote the scalar, gauge, and fermion fields, respectively. The first two diagrams arise from the non-zero Matsubara modes, while the last diagram is the counterterm contribution. details of these computations can be found in [260], and we quot… view at source ↗
Figure 13
Figure 13. Figure 13: Hierarchy of thermal scales together with the corresponding sphaleron and bubble￾nucleation processes. [264, 268], where x = λ3/g 2 3 . This ultrasoft scale is also the characteristic scale of a first￾order phase transition. Note that the static spatial gauge fields are integrated out at the bubble-nucleation scale, leaving only the scalar field. As mentioned in the previous section, integrating out the s… view at source ↗
Figure 14
Figure 14. Figure 14: Phase diagram in the x-y plane, with x and y defined in Eq. (6.7). The blue line denotes the critical-temperature line, which is very close to the nucleation-temperature line when the phase transition does not involve significant supercooling. As the temperature decreases, the system evolves from the upper part of the phase diagram to the lower part, as indicated by the red and purple arrows in cases (a) … view at source ↗
Figure 15
Figure 15. Figure 15: Rescaled sphaleron action Csph as a function of x. The black points show the exact numerical values of Csph, while the red curve shows the fit in Eq. (6.25). The agreement is so good that the two are nearly indistinguishable. The dashed curves indicate the sepa￾rate contributions to Csph: blue for the Yang–Mills term, purple for the covariant-derivative term, and orange for the scalar-potential term in Eq… view at source ↗
Figure 16
Figure 16. Figure 16: Top row: v3-rescaled sphaleron action for the cubic potential. Left: Csph as a function of x at fixed y = 0.008. The red curve shows the total action, the magenta curve shows the combined Yang–Mills and covariant-derivative contributions, and the orange curve shows the scalar-potential contribution. Right: Csph as a function of y at fixed x = 0.03, with the same color coding. Bottom row: Difference betwee… view at source ↗
Figure 17
Figure 17. Figure 17: Sphaleron rate for the Standard Model crossover. The blue curve shows the LO sphaleron rate in Eq. (6.52), while the red curve shows the NLO sphaleron rate in Eq. (6.53), including the zero-mode contribution. The black curve is the lattice result from Ref. [250], and the yellow band indicates its uncertainty. The purple dash-dotted curve shows the earlier perturbative result of Ref. [311]. This figure is … view at source ↗
Figure 18
Figure 18. Figure 18: Phase-transition diagram and sphaleron-decoupling line in the (x, y) plane. The solid blue and dashed red curves denote the critical- and nucleation-temperature lines, re￾spectively. The black dash-dotted curve marks sphaleron decoupling, defined by ΓB = H. As the Universe cools, y decreases while x remains approximately constant, so the system moves downward in the diagram. Crossing the nucleation line t… view at source ↗
Figure 19
Figure 19. Figure 19: Phase structure and baryon-number preservation condition in the (a2, MΣ) plane from the one-loop result for λ3 in the ΣSM. The solid black line denotes µ 2 Σ = 0. Below this line, µ 2 Σ > 0, where our EFT is valid; the white region corresponds to µ 2 Σ < 0, where our EFT is not valid. The pink region denotes a two-step phase transition, which is not the focus of this work. The grey region indicates that t… view at source ↗
Figure 20
Figure 20. Figure 20: Phase structure and baryon-number preservation condition in the (a2, MΣ) plane from the two-loop result for λ3 in the ΣSM. The meaning of the different color regions is the same as in the one-loop result shown in [PITH_FULL_IMAGE:figures/full_fig_p122_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Illustration of PBH production from a delayed first-order phase transition. The blue (green) background represents the false (true) vacuum, as shown in the left panel of the effective potential. The true-vacuum regions arise from bubble nucleation, as indicated by the green circles. In the right panels, (a) and (b), for illustration, the Universe is divided into nine patches, where the middle patch is the… view at source ↗
Figure 22
Figure 22. Figure 22: fpbh as a function of µ3 for the benchmark points listed in [PITH_FULL_IMAGE:figures/full_fig_p131_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Feynman diagram for e +e − → Zh2, followed by Z → ℓ +ℓ − (ℓ = e, µ), h2 → h1h1, and h1 → b ¯b, giving the final state ℓ +ℓ −b ¯bb¯b . and SM-Higgs-like mass eigenstates, respectively. This implies that we take m2 ≈ 125 GeV. We also take cos θ = 0.01 as our benchmark choice [85]. Since our goal is to investigate collider searches for the light scalar singlet, we mainly focus on the sensitivity of future le… view at source ↗
Figure 24
Figure 24. Figure 24: The vertical axis shows the scaled branching ratio Br(h2 → h1h1), and the horizontal axis shows the mass of the singlet-like scalar eigenstate h1. The black dotted curve gives the expected BDT sensitivity, with the 1σ and 2σ uncertainty bands shown in yellow and green, respectively. The blue shaded region indicates parameter points that yield a strong first-order electroweak phase transition with successf… view at source ↗
Figure 25
Figure 25. Figure 25: Triangle diagrams for π 0 → γγ. The dashed, solid, and wavy lines denote the pion, proton, and photon fields, respectively. Panels (a1) and (a2) contribute to T µν in Eq. (A.6), while panels (b1)–(b4) contribute to T ρµν in Eq. (A.9). The crossed vertex denotes the insertion of the axial current: (b1) and (b2) correspond to the first term of Eq. (A.3), while (b3) and (b4) correspond to the third term of E… view at source ↗
Figure 26
Figure 26. Figure 26: Triangle diagram (b1) of [PITH_FULL_IMAGE:figures/full_fig_p151_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Three example potentials, with the minima normalized to V = 0. (a) A quadratic potential centered at x = 0. (b) A double-well potential with minima at x = ±a. (c) A periodic potential with multiple minima at x = na for integer n. 155 [PITH_FULL_IMAGE:figures/full_fig_p155_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Instanton-like trajectory interpolating from x = −a to x = a in imaginary time (blue solid curve). The red dashed curve shows the anti-instanton trajectory interpolating from x = a to x = −a. (ii) For the double-well potential in [PITH_FULL_IMAGE:figures/full_fig_p157_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Two example instanton/anti-instanton configurations contributing to the ampli￾tude ⟨jf = 2|e −Hτ |ji = 0⟩ in the periodic potential, where the initial minimum is at ji = 0 (green point) and the final minimum is at jf = 2 (orange point). (a) Two instantons, 2n. (b) Three instantons and one anti-instanton, 3n + 1n¯. We now evaluate the amplitude ⟨jf |e −Hτ |ji⟩. We must sum over all possible instanton 46For… view at source ↗
Figure 30
Figure 30. Figure 30: Instanton-induced baryon number violation in the Standard Model. The JB and JL denote the baryon and lepton sources, respectively. For each quark and lepton, the upper index r denotes the QCD color, and the lower index L and R denote the chirality. Each baryonic source is a color singlet and can be written as JB(xi) ∝ ϵrstQr B (xi)Qs B (xi)Qt B (xi), where Qr B (xi) denotes a source carrying color index r… view at source ↗

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