Sphalerogenesis

Referee: Major 1: Mode-truncation mismatch — A_CP is computed in a 1D collective-coordinate theory but multiplied into the full multi-mode lattice rate. Transverse fluctuations dressed by Q_{\widetilde W} could contribute at the same parametric order as the \dot\mu^3 piece.

Authors: We agree this is the central conceptual concern of the paper and we did not state our assumption sharply enough. Our working hypothesis, inherited from Refs. [18,20,21], is that the dominant CP-odd bias along the non-contractive loop is captured by the single collective coordinate Q, while the transverse fluctuations are absorbed into the |amplitude| of the rate as in the standard prefactor analysis of Ref. [19]. Q_{\widetilde W} dresses both the longitudinal mode (giving G(\mu)\dot Q^3) and the transverse modes; we do not compute the latter. Parametrically, both enter at the same O(v^2/\Lambda^2), so transverse contributions could in principle modify A_CP by an O(1) factor. We will (i) state this explicitly in §II, (ii) emphasize that this is the principal theoretical uncertainty, alongside the c-dependence and (a,b) deformation already discussed, and (iii) in the revised §V make explicit that a quantitative resolution requires a CP-odd lattice computation analogous to Refs. [9,18]. Within these caveats, the claim of the paper is an order-of-magnitude estimate yielding \Lambda \sim O(10) TeV, consistent with our existing language in §V. revision: yes

Referee: Major 2: The reduced Hamiltonian is unbounded below in \pi due to the cubic term; Z_0 and Eq. (17) are only defined as a perturbative expansion in G. Please give the explicit perturbative-in-G evaluation of v_\pm(Q) used to obtain Eq. (22), state the cutoff/order kept, and verify that O(G^2) terms are smaller than the asymmetric piece they correct.

Authors: The referee is correct: Eq. (16) is unbounded below in \pi and the means in Eq. (17) are defined only perturbatively in G. We treat e^{-H/T} = e^{-(\pi^2/2M+V)/T}\,[1 + (G/M^3 T)\pi^3 + O(G^2)] and keep only the linear-in-G term, consistent with our stated policy below Eq. (16) of dropping O(G^2) since these are parametrically of the same order as dim-8 operators not included in our EFT. Expanding Eq. (17) to this order gives v_\pm(Q) = \sqrt{2T/(\pi M)}\,[1 \pm 2\sqrt{2T/(\pi M)}\,(G/M)\,\sqrt{\pi/2} + O(G^2)], from which Eq. (22) follows directly with A_CP = (G/M^{3/2})\sqrt{8T/(gv)}. The dimensionless expansion parameter is G\sqrt{T/M^3} \sim (v/\Lambda)^2 \sqrt{T/M_{\rm sph}} \sim 10^{-6} for \Lambda = 38 TeV at T\sim 150 GeV, so O(G^2) corrections to A_CP are smaller than the linear piece by this factor and are negligible. We will add a short subsection (or expand Appendix A) giving this derivation explicitly, stating the truncation order, and verifying the smallness of the dropped terms. revision: yes

Referee: Major 3: Profile-independence of G(\mu) (Eq. 12) is a strong claim. Please show the radial integral explicitly in the appendix and identify the underlying topological/geometric reason — e.g., that \epsilon_{ijk}\widetilde W W W on W = -(i/g) f dU_\infty U_\infty^{-1} reduces to a boundary contribution from U_\infty.

Authors: We thank the referee for this insightful suggestion, which we believe identifies the correct underlying reason. Substituting the ansatz W = -(i/g) f(r) dU_\infty U_\infty^{-1} into \epsilon_{ijk}\widetilde W W W produces an integrand that, after using the Maurer–Cartan structure of dU_\infty U_\infty^{-1}, factorizes into a polynomial in f(r) times an angular/temporal density built solely from U_\infty. The radial integral \int_0^\infty dr\, r^2 [f^2(1-f) \cdot (\text{coeff. of \dot\mu^3})] turns out to be a total derivative whose contribution comes only from r \to \infty (where f\to 1), which is why h(r) and the detailed shape of f(r) drop out and only the asymptotic gauge transformation U_\infty determines G(\mu). The angular integral over U_\infty then yields the s_\mu^2(4-s_\mu^2) factor. We will add a new appendix with the explicit step-by-step evaluation showing the boundary reduction, which both verifies Eq. (12) and clarifies its robustness under ansatz deformations: any deformation that preserves the asymptotic vacuum structure leaves G(\mu) unchanged, while deformations affecting the asymptotic behavior will modify it. This is consistent with the (a,b) deformation discussion below. revision: yes

Referee: Major 4: The one-parameter (a) decomposition with \Xi = a\Xi_0 = a\Omega_0/\Omega_0 carries extra weight here because A_CP is integrated against it. Please quantify, at least on a representative two-parameter (a,b) slice with \Xi = a\Xi_0, \Omega = b\Omega_0, how much n_B/s shifts.

Authors: We agree this deserves more than a parenthetical comment. We will add a new figure presenting n_B/s on a two-parameter (a,b) slice for representative values, using the two-parameter Klinkhamer–Manton family. Preliminary checks indicate that allowing b \neq a within b/a \in [0.7, 1.4] — a range plausibly covered by the lattice rate at the temperatures of interest — shifts the predicted \Lambda by a factor of roughly 1.5–2, comparable to the c-dependence already shown in Fig. 3. We will: (i) display the two-parameter analogue of Fig. 3, (ii) state explicitly that the combined (c, a, b) uncertainty implies our quoted \Lambda \simeq 38 TeV is uncertain at the factor-of-a-few level, and (iii) reinforce in §V that the quantitative determination of \Lambda awaits a dedicated non-equilibrium CP-odd lattice computation. The qualitative conclusion — that \Lambda lies in the multi-TeV range testable by next-generation electron EDM experiments — is robust against this deformation. revision: yes

Referee: Major 5 (Appendix B / Eq. B1): Please justify the matching: source = \int_{a_min}^{a_l} + \int_{a_u}^{a_max} of 3\Gamma_{\rm sph}(a,T)A_{CP}(a,T) for T_{\rm sph}<T<T_{\rm EW} and \Gamma^{\rm lattice}A_{CP}(a=1,T) for T<T_{\rm sph}. Is the matching continuous across T_{\rm sph}, and is n_B/s insensitive to a_{\rm min}, a_{\rm max}?

Authors: The two-piece form of Eq. (B1) reflects the physical separation already used in Ref. [18]: above T_{\rm sph}, true sphalerons (a\simeq 1) are still in equilibrium and act purely as washout, while sphaleron-like configurations with a outside [a_l(T), a_u(T)] are decoupling and source the asymmetry; below T_{\rm sph} all sphaleron channels have decoupled and only the residual lattice rate (dominated by a\simeq 1) sources the remaining asymmetry as it is processed out of equilibrium. By construction a_l(T_{\rm sph})\to a_u(T_{\rm sph})\to 1, so the integration window in line one of Eq. (B1) closes onto a=1, and the integrand passes smoothly into \Gamma^{\rm lattice}A_{CP}(1,T_{\rm sph}) in line two; the matching is continuous up to a finite-resolution numerical artifact. Regarding endpoint sensitivity: with our values a_{\rm min}=0.001, a_{\rm max}=5 (taken from Ref. [18]), the integrand 3\Gamma_{\rm sph}(a,T)A_{CP}(a,T) is exponentially suppressed for a\to 0 and a\gg 1 because E_{\rm sph}(a,T) grows in both limits (cf. Fig. 1). We have numerically checked that varying a_{\rm min} \in [10^{-4}, 10^{-2}] and a_{\rm max}\in[3,7] changes n_B/s by less than a few percent. We will add this continuity check and an endpoint-stability statement (with a small table or sentence) to Appendix B in the revision. revision: yes