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REVIEW 4 major objections 5 minor 136 references

Resonant TeV-scale electromagnetic leptogenesis can reproduce the observed baryon asymmetry of the Universe, within the electroweak crossover window, using only gauge-invariant neutrino-dipole operators and coherent heavy-neutrino evolution

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2026-08-03 02:35 UTC pith:MGSY5DPA

load-bearing objection The formal machinery is real and the author is honest about approximations, but the abstract's headline numbers don't match the body's equations, and the central quantitative claim is unverified. the 4 major comments →

arxiv 2603.01652 v3 pith:MGSY5DPA submitted 2026-03-02 hep-ph

Resonant electromagnetic leptogenesis with coherent heavy-neutrino evolution

Rin Takada This is my paper
classification hep-ph
keywords electromagnetic leptogenesisneutrino dipole operatorsresonant leptogenesisdensity-matrix quantum kinetic equationsSchwinger–Keldysh formalismelectroweak crossoverheavy-neutrino coherenceνSMEFT
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.

This paper tries to establish that electromagnetic leptogenesis — generating the cosmic baryon asymmetry through CP-violating decays of heavy right-handed neutrinos via neutrino-dipole operators — can work at the TeV scale, despite a strong parametric suppression of the non-resonant source, if the heavy neutrinos are nearly degenerate so that the self-energy CP asymmetry is resonantly enhanced. It provides a closed EFT pipeline from a UV-complete model to the frozen-out baryon yield, with density-matrix quantum kinetic equations derived in the Schwinger–Keldysh formalism that treat decays, inverse decays, and ΔL=0 scatterings from a single collision term without double counting. For a benchmark dipole strength, the signed baryon yield crosses the observed value at two specific mass splittings (ΔM around 1–2×10⁻⁵ GeV), while in the oscillation-motivated region the undiluted yield exceeds the observed asymmetry by several orders of magnitude, requiring late-time entropy dilution.

Core claim

The central claim is that, in the quasi-degenerate limit, the dipole-induced self-energy CP asymmetry regulated by Dyson-resummed widths becomes resonantly enhanced, and the resulting CP-odd source combined with coherent flavor evolution among the heavy neutrinos produces a large lepton asymmetry in the electroweak window T≈130–160 GeV. Solving the density-matrix QKEs with three resolved flavor charges and a finite-rate sphaleron equation, the paper obtains for the benchmark effective electromagnetic neutrino mass m~_1^EM = 3.97×10⁻² eV a positive near-resonant maximum Y_B^FO = +5.50×10⁻⁵ at ΔM = 1.29×10⁻¹¹ GeV with the decay-only pole prescription, and +8.44×10⁻⁵ at ΔM = 1.83×10⁻¹¹ GeV with

What carries the argument

The central objects are the gauge-invariant dimension-six neutrino-dipole operators O_NB and O_NW in the νSMEFT, generated in a Z2-symmetric UV completion with soft breaking, matched at 10 TeV and RG-evolved to 150 GeV. After electroweak symmetry breaking these induce effective dipole couplings μ^V_{αi} to γ, Z, and W, which mediate two-body decays, inverse decays, and ΔL=0 scatterings. The argument is carried by two mechanisms: (i) the Dyson-resummed self-energy CP asymmetry, regulated by the total width in a Breit–Wigner fashion, which becomes resonantly enhanced for quasi-degenerate heavy neutrinos; and (ii) density-matrix quantum kinetic equations derived from the Schwinger–Keldysh/Kadan

Load-bearing premise

The numerical result relies on starting the evolution at T_EW = 160 GeV with imposed initial conditions (thermal or zero heavy-neutrino abundance, zero lepton charges) and treating the Higgs VEV as constant at 246 GeV, so any pre-existing asymmetry or heavy-neutrino abundance generated in the symmetric phase above 160 GeV is neglected.

What would settle it

Extend the density-matrix QKEs to T > 160 GeV with a temperature-dependent Higgs VEV and include the symmetric-phase 1↔3 and crossed 2↔2 reactions; if the resulting Y_B^FO at sphaleron freeze-out no longer crosses the observed Y_B^obs at any ΔM near 1e-5 GeV, the central claim that the electroweak-window contribution alone can reproduce the baryon asymmetry would be falsified.

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

If this is right

  • If the central claim is correct, TeV-scale dipole-dominated leptogenesis can reproduce the observed baryon-to-entropy ratio within the Standard-Model electroweak crossover, without relying on renormalizable neutrino Yukawa couplings.
  • In the oscillation-motivated region (m~_1^EM in the 0.01–0.05 eV range), the undiluted yield exceeds the observed asymmetry, so a late-time entropy dilution factor is a natural ingredient to reconcile these parameter points.
  • The ΔL=0 scatterings are numerically subdominant (∼10⁻⁵ relative to decays/inverse decays) in the benchmark, so they mainly rescale the efficiency; the dominant washout is from inverse decays.
  • The closed UV-to-observable pipeline means that the baryon asymmetry is a prediction of the EFT inputs, enabling direct correlation with low-energy dipole observables such as μ→eγ, the electron EDM, and (g−2)_μ.
  • The resonant regime requires the mass splitting to be comparable to the width that regulates the self-energy pole, which selects a narrow band of heavy-neutrino spectra for successful leptogenesis.

Where Pith is reading between the lines

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

  • A direct extension of the paper would be to compute the baryon asymmetry with a fully temperature-dependent Higgs VEV and matched symmetric-phase processes above T=160 GeV; such a calculation would test whether the electroweak-window-only treatment is self-consistent.
  • The same dipole operators that generate the baryon asymmetry also induce charged-lepton flavor violation and electric dipole moments at low energies, so the scenario can be probed or excluded by MEG II, JILA, and (g−2)_μ measurements—an implication the paper notes but does not fully quantify.
  • The result suggests that resonant leptogenesis can operate in parameter space where the seesaw contribution to light-neutrino masses is negligible, decoupling the baryogenesis scale from the neutrino-mass scale and opening broader model-building possibilities.
  • A scan over both the mass splitting ΔM and the effective dipole strength m~_1^EM, rather than fixing ΔM, could reveal regions where the observed asymmetry is reproduced without extra dilution; the paper's fixed-ΔM scan already indicates such regions exist.

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

4 major / 5 minor

Summary. The paper develops a UV-complete EFT description of TeV-scale electromagnetic leptogenesis. A softly Z2-broken model generates the νSMEFT dipole operators O_NB and O_NW; one-loop matching and RG running produce broken-phase dipole couplings to γ, Z, and W. The authors derive density-matrix QKEs from CTP/Kadanoff–Baym equations, with decays, inverse decays, and ΔL=0 scatterings included from the same self-energy structures, and solve them in the electroweak crossover window T≃130–160 GeV. The central claim is that, for quasi-degenerate heavy neutrinos, the resonantly enhanced CP source can produce a frozen-out baryon asymmetry of order or above the observed value, with the headline numerical results stated in the abstract: Y_B^FO=+5.50×10^-5 at ΔM=1.29×10^-11 GeV, crossings at 1.68×10^-5 and 2.34×10^-5 GeV, and a scattering-induced shift to +8.44×10^-5 at ΔM=1.83×10^-11 GeV.

Significance. If the numerical results are correct, the paper would demonstrate a complete and explicit EFT pipeline from a renormalizable UV model to a resonant leptogenesis mechanism, a substantial step beyond treatments that treat dipole couplings as free parameters. The formal derivation is extensive: one-loop matching, RG anomalous dimensions, Dyson-resummed self-energy CP asymmetries, and the reduction from Kadanoff–Baym to density-matrix QKEs are all presented in detail, and the paper explicitly includes the ΔL=0 scattering channels and checks their smallness. These are genuine strengths. However, the central quantitative output is not reproducible from the body as written, because the abstract reports a ΔM scan that the numerical section does not describe, and no implementation is provided. The paper also honestly acknowledges that only the electroweak-window contribution is computed, with imposed initial conditions and v(T)=246 GeV.

major comments (4)
  1. [Abstract vs §IX] The abstract's headline numbers cannot be obtained from the body. §IX A–B fix ΔM=5.0×10^-12 GeV (Eq. 654) and state that the scan is over ~m^EM_1 at fixed ΔM; Fig. 12 plots |Y_B^FO| versus ~m^EM_1, not a signed ΔM scan. The abstract quotes Y_B^FO=+5.50×10^-5 at ΔM=1.29×10^-11 GeV and crossings at 1.68×10^-5 and 2.34×10^-5 GeV. Unless a separate ΔM scan is documented, these numbers are not derivable from the presented equations and inputs. This is the core quantitative claim of the paper.
  2. [§IX F vs abstract] The abstract states that changing from decay-only to decay-plus-scattering shifts the maximum from 5.50×10^-5 to 8.44×10^-5 (~50%) and moves the resonant ΔM from 1.29×10^-11 to 1.83×10^-11 GeV (~40%). But §IX F reports S1/D1≃1.5×10^-5 and concludes that scatterings only rescale the efficiency at the 10^-5 level. A relative scattering rate of 10^-5 cannot produce a 50% shift in Y_B^FO or a 40% shift in the resonant ΔM. Either the abstract's numbers come from a different, undocumented set of inputs, or the body's own estimate is inconsistent with the abstract.
  3. [§IX C, §X C] The calculation covers only T∈[130,160] GeV, starts from Y_N(z_EW)=Y_N^eq or 0 and Y_Δα(z_EW)=0, and approximates v(T) by 246 GeV throughout. §X C explicitly defers temperature-dependent Higgs background and symmetric-phase reactions to future work. Therefore Y_B^FO is the electroweak-window contribution of one particular history, not a full prediction. If the symmetric phase generates a pre-existing asymmetry or a heavy-neutrino abundance, or if the evolving VEV changes decay kinematics, the final yield changes. The qualitative conclusion that 'EMLG can reproduce the observed BAU' should be stated with this caveat as a central limitation.
  4. [§IX C/Fig. 12, Eq. (665)] The conclusion (IV) states that Y_B^FO can reproduce Y_B^obs over a broad range of ~m^EM_1, but Fig. 12 and the text show that in the oscillation-motivated region the undiluted yield exceeds Y_B^obs by orders of magnitude. Matching to Y_B^obs then requires a late entropy dilution factor D in Eq. (665), which is an additional free parameter not predicted by the model. The paper therefore demonstrates that the mechanism can overproduce the BAU for some parameters, not that it predicts the observed value without tuning the scan inputs.
minor comments (5)
  1. [§IX B vs abstract benchmark] The abstract's benchmark ~m^EM_1=3.97×10^-2 eV is not used in the numerical section; §IX B uses ~m^EM_1=10^-3 eV for the evolution plots and notes that this point gives negative Y_B^FO. The benchmark underlying the abstract's numbers should be identified and its evolution, if different, shown.
  2. [§IX E text] The sentence 'the corresponding curves therefore exhibit an approximately linear [cf. Sec. IX E]' is self-contradictory: the subsection then derives a quadratic ~(mtilde)^2 scaling. Please correct.
  3. [Table IV and §VIII J] The susceptibility matrix and sphaleron factor are given at selected temperatures, but the numerical implementation uses v(T)=v(0) for all T in the window. This is stated, but a sentence indicating the expected size of the resulting systematic error (e.g., from v(160 GeV) vs v(246 GeV)) would be useful.
  4. [Eq. (643)] Y_N^eq is written with g⋆s in the denominator; earlier in §VIII H the entropy density s is used. Please define g⋆s consistently and state its value used in the numerics (the text uses g⋆=106.75 in Eq. 649 without specifying whether this is the relativistic degrees of freedom for entropy).
  5. [Data availability] The paper states 'Data availability' but does not provide code or the numerical data points of Fig. 12. Given that the abstract reports specific maxima and crossings, a public implementation or at least a table of the signed ΔM-scan results would greatly improve verifiability.

Circularity Check

0 steps flagged

No significant circularity: the EFT pipeline is self-contained; abstract/body numerical inconsistencies are a verification issue, not a circular reduction.

full rationale

The core derivation is not circular. The paper constructs a UV completion, matches it to the νSMEFT dipole operators, runs the Wilson coefficients with one-loop RGEs, computes CP asymmetries from explicit loop diagrams with Dyson-resummed widths, derives density-matrix QKEs from Kadanoff-Baym equations, and solves them over the electroweak window. The observed BAU enters only as a comparison line in Fig. 12 and as a benchmark against which overproduction is noted; the body explicitly presents a scan over \tilde m^EM_1 and states that successful points require either a particular range of this input or late-time dilution. That is parameter-space exploration, not fitting a parameter to the target and then renaming it a prediction. The abstract's quantitative claims (ΔM = 1.29×10^-11 GeV, crossings at 1.68×10^-5 and 2.34×10^-5 GeV, a ~50% scattering shift) are not reproduced by the body's stated implementation, which fixes ΔM = 5×10^-12 GeV and finds scatterings at the 10^-5 level; this is an internal-consistency/reproducibility problem, not a circular dependency. The explicit limitations (evolution only over T ≃ 130-160 GeV, v(T) approximated by 246 GeV, unmodeled dilution, deferred symmetric-phase matching) are caveats about the validity domain, not self-referential inputs. Citations such as Ref. [22] for the Z2-symmetric UV completion are external model assumptions, not a load-bearing self-citation chain, and no uniqueness theorem or ansatz is smuggled in via self-citation. No exhibited reduction of an output to an input by construction is present, so the circularity score is 0.

Axiom & Free-Parameter Ledger

9 free parameters · 9 axioms · 4 invented entities

The final Y_B^FO rests on a long chain of formal approximations (KB→QKE reduction, quasiparticle ansatz, momentum averaging, equilibrium internal lines, vacuum-width Breit-Wigner regulator) and on modeling choices (imposed initial conditions at 160 GeV, v(T) ≈ v(0), neglected symmetric phase, negligible ΔL=2 washout). The UV inputs are hand-picked benchmarks with free CP phases, and the overall dipole scale is scanned over seven decades; the observed BAU is found by selection within that scan rather than predicted.

free parameters (9)
  • Effective electromagnetic neutrino mass \tilde m^EM_1 = Benchmark 3.97×10^-2 eV (abstract); scanned over 10^-6–1 eV (body)
    Sets the overall scale of the dipole couplings and hence the decay width, CP source, and washout; the observed BAU is obtained inside this scan (Fig. 12).
  • Heavy-neutrino mass splitting ΔM = 5×10^-12 GeV in the body's fixed scan; abstract 'crossing' values 1.68×10^-5 and 2.34×10^-5 GeV
    Controls the Breit-Wigner resonant enhancement; in the abstract, ΔM values where Y_B^FO(ΔM) = Y_B^obs are selected to match the observed asymmetry.
  • UV Yukawa couplings y_H = (1.0, 1.0 e^{-0.7i})^T × 10^-3
    Hand-chosen benchmark carrying a CP-violating phase; sets the one-loop dipole amplitude (Eq. 658).
  • UV Yukawa couplings y_Σ = (10.0, 8.0 e^{-0.3i}, 6.0 e^{-1.1i})^T × 10^-4
    Hand-chosen benchmark with CP phases; enters the matching coefficients C_NB, C_NW (Eqs. 88-90).
  • Dimensionful scalar coupling μ_s = 10.0 e^{-0.4i} GeV
    Appears in the one-loop matching; its phase contributes to the CP-violating source.
  • Soft-Z2-breaking scale μ_soft = 10.0 GeV
    Sets the y_eff suppression via μ_soft/m_D ≃ 10^-6; chosen to enforce dipole dominance and technical naturalness.
  • Late entropy dilution factor D = Unspecified, > 1
    Invoked to reconcile Y_B^FO exceeding Y_B^obs by orders of magnitude; not modeled (Sec. IX C).
  • Debye regulator coefficient κ_D = 1/3
    IR regularization for the photon-exchange scattering cross section; the paper states the true electroweak coefficient is O(1) and would not change conclusions.
  • Benchmark scales M_eq, M_1 = M_eq = 10 TeV, M_1 = 1 TeV
    Heavy mediator scale and heavy-neutrino mass chosen for the TeV-scale scenario; results depend on this choice.
axioms (9)
  • standard math Closed-time-path / Kadanoff-Baym formalism with leading-gradient Wigner expansion reduces to the stated density-matrix QKEs
    Invoked in Secs. VI-VIII; standard QFT framework, though the reduction to these specific QKEs is the paper's construction.
  • domain assumption Dyson resummation of the quasi-degenerate heavy-neutrino propagator with the vacuum width Γ_m as Breit-Wigner regulator
    Sec. IV I; the Pilaftsis-Underwood treatment is assumed; thermal widths are not used in the regulator.
  • domain assumption Quasiparticle + Kadanoff-Baym ansatz and kinetic-equilibrium momentum-averaging ansatz f_N - f_N^eq = f_N^eq (Y_N - Y_N^eq)/Y_N^eq
    Sec. VIII H; closes the QKE system; its validity in the weak-washout resonant regime is asserted, not demonstrated.
  • domain assumption Internal lepton and gauge-boson lines are in equilibrium with Maxwell-Boltzmann statistics
    Secs. VII-VIII; fixes the collision kernels and washes out chemical-potential feedback in the source term.
  • ad hoc to paper v(T) ≈ v(0) = 246 GeV throughout the electroweak window
    Table IV note; used for x(T) = v/T in c_sph and A_ℓ; the crossover VEV evolution is ignored.
  • ad hoc to paper Initial conditions at T_EW = 160 GeV: Y_N = Y_N^eq or 0, Y_Δα = 0
    Sec. IX A; the entire computed asymmetry is generated inside the window from these imposed states.
  • domain assumption Pole vs Landau-damping spectral split of the gauge propagator removes double counting, with no RIS subtraction needed
    Secs. VII-VIII; central structural claim, argued from disjoint kinematic support rather than demonstrated on a cross-check.
  • ad hoc to paper The symmetric phase (T > 160 GeV) contributes negligibly to the final asymmetry
    Sec. X C; symmetric-phase three-body CP asymmetries are argued to be two-loop suppressed, but the crossover matching is explicitly left to future work.
  • domain assumption |ΔL|=2 washout from the Weinberg operator is negligible at T ≃ 130-160 GeV
    Sec. II A; valid for oscillation-scale m_ν and absent semileptonic ΔL=2 operators from the UV completion.
invented entities (4)
  • Right-handed Majorana neutrinos N_1, N_2 at M ≈ 1 TeV no independent evidence
    purpose: The decaying and oscillating species that generate the lepton asymmetry
    Standard leptogenesis ingredient; direct collider production is suppressed by small dipole couplings and no sharp signal prediction is made.
  • Vector-like charged lepton E at 10 TeV no independent evidence
    purpose: Loop mediator generating the dipole operators O_NB and O_NW
    No production signature or mass prediction tied to data is analyzed in this paper.
  • Extra Higgs doublets Σ and D at 10 TeV no independent evidence
    purpose: Complete the one-loop matching and enable the soft-Z2 suppression of the neutrino Yukawa
    No direct observable proposed; the μ_soft/m_D ratio is tuned to ~10^-6.
  • Charged scalar Φ+ at 10 TeV no independent evidence
    purpose: Completes the N-E-L loop in the UV matching
    No distinct collider or precision observable is analyzed.

pith-pipeline@v1.3.0-alltime-deepseek · 75519 in / 22840 out tokens · 222350 ms · 2026-08-03T02:35:32.834751+00:00 · methodology

0 comments
read the original abstract

We study TeV-scale electromagnetic leptogenesis generated by the gauge-invariant neutrino-dipole operators $O_{NB}$ and $O_{NW}$. The benchmark Wilson coefficients are specified at the renormalization scale $\kappa_N=M_1=1\,\mathrm{TeV}$ and related to their values at $\kappa_{\rm match}=3\,\mathrm{TeV}$ through the coupled one-loop $\nu$SMEFT evolution. The collision network contains the symmetric-phase $1\leftrightarrow3$ and crossed $2\leftrightarrow2$ processes generated by the linear field-strength vertices and the VEV-induced two-body $\gamma$, $Z$, and $W$ channels through the electroweak crossover. We use a factorized leading-moment, momentum-averaged Markovian treatment for two $2\times2$ heavy-flavor density matrices, one for each helicity. Pole-resummed two-state effective couplings enter the lepton and antilepton collision tensors, while the off-diagonal density matrices describe coherent evolution and collision damping. Three resolved flavor charges are coupled to a finite-rate sphaleron equation, yielding the frozen baryon yield $Y_B^{\rm FO}$. For the benchmark $\tilde m^{\rm EM}_1=3.97\times10^{-2}\,\mathrm{eV}$, the decay-only pole prescription, in which only decay contributions enter the absorptive pole matrix, gives the locally optimized positive near-resonant maximum $Y_B^{\rm FO}=+5.50\times10^{-5}$ at $\Delta M=1.29\times10^{-11}\,\mathrm{GeV}$. Adding the thermally averaged crossed-scattering moment to the pole matrix defines the decay-plus-scattering prescription and shifts the maximum to $Y_B^{\rm FO}=+8.44\times10^{-5}$ at $\Delta M=1.83\times10^{-11}\,\mathrm{GeV}$. The corresponding signed mass-splitting curves cross the observed value $Y_B^{\rm obs}$ at $\Delta M=1.68\times10^{-5}\,\mathrm{GeV}$ and $2.34\times10^{-5}\,\mathrm{GeV}$.

Figures

Figures reproduced from arXiv: 2603.01652 by Rin Takada.

Figure 1
Figure 1. Figure 1: FIG. 1. Representative one-loop matching diagrams generat [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Feynman diagrams for the two-body decays of [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Two vertex-type one-loop diagrams contributing to [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Cutkosky cut for the vertex contribution. The ver [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Cutkosky cut for the self-energy contributions. The [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Self-energy-type one-loop diagrams contributing to [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Loop functions [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Closed-time-path (CTP) contour [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Representative [PITH_FULL_IMAGE:figures/full_fig_p037_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Time evolution of the QKE solution for ˜m [PITH_FULL_IMAGE:figures/full_fig_p052_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: shows the time evolution obtained with the zero initial abundance in Eq. (652), plotted in the same format as [PITH_FULL_IMAGE:figures/full_fig_p053_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Seagull contribution induced by the quartic [PITH_FULL_IMAGE:figures/full_fig_p065_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. One-loop [PITH_FULL_IMAGE:figures/full_fig_p070_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. One-loop self-energy diagram for a Dirac fermion. [PITH_FULL_IMAGE:figures/full_fig_p072_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. One-loop diagrams contributing to [PITH_FULL_IMAGE:figures/full_fig_p073_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19 [PITH_FULL_IMAGE:figures/full_fig_p074_19.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21 [PITH_FULL_IMAGE:figures/full_fig_p075_21.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. One-loop [PITH_FULL_IMAGE:figures/full_fig_p076_23.png] view at source ↗

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Reference graph

Works this paper leans on

136 extracted references · 22 linked inside Pith

  1. [1]

    Cutkosky rules Collecting the couplings into Eq

    V ertex contribution (a) a. Cutkosky rules Collecting the couplings into Eq. (276), we write I (a) vert(V;U) =A V U µ,αimJ (a) vert(V;U),(D1) whereA V U µ,αim is a complex constant independent of kine- matics, andJ (a) vert(V;U) is given by J (a) vert(V;U) := i Z d4q1 (2π)4 N (a) vert(q1;k, q) (q2 3 −M 2m + iϵ)(q2 1 + iϵ)(q2 2 −m 2 U + iϵ). (D2) With this...

  2. [2]

    mass term

    V ertex contribution (b) The discontinuityJ (b) vert is given by puttingq 1 andq 2 on shell and carrying out the phase-space integral: DiscJ (b) vert(V;U) = Disc Z d4q1 (2π)4 T(b)|cut (q2 3 −M 2m + iϵ)(q2 1 + iϵ)(q2 2 −m 2 U + iϵ) = 4iM 8 i rV rU (1−r V )(1−r U ) × Z d4q1 (2π)4 (1−t 1)−2(r V +r U )(1 +t 1) − M 2 i 2 (D0 −D 1t1) ×(−2πi) 2δ(q 2 1)Θ(E1)δ(q 2...

  3. [3]

    (F14) Similarly, theZexchange amplitude is MZ = 2eµZ αi sinθ W cosθ W 1 t−m 2 Z × ¯uα(p′)σµνqνPRui(p) × ¯uf (k′)γµ gf LPL +g f RPR uf (k) .(F15)

    T ree-level amplitudes Thet-channel photon exchange amplitude is Mγ = 2eQf µγ αi t ¯uα(p′)σµνqνPRui(p) ¯uf (k′)γµuf (k) . (F14) Similarly, theZexchange amplitude is MZ = 2eµZ αi sinθ W cosθ W 1 t−m 2 Z × ¯uα(p′)σµνqνPRui(p) × ¯uf (k′)γµ gf LPL +g f RPR uf (k) .(F15)

  4. [4]

    The Dirac-fermion self-energy is obtained by noting that the fermion–gauge-boson vertex is igγ µT a (a= 1,2,3)

    Two-point function of a Dirac fermion Next, in order to determine the fermion field renormal- ization constantZ 2 for a Dirac fermionf, we compute the two-point function. The Dirac-fermion self-energy is obtained by noting that the fermion–gauge-boson vertex is igγ µT a (a= 1,2,3). We find iΣf = X a Z dnk (2π)n ×(igγ µT aPL) i(/k+m f ) k2 −m 2 f (igγ νT a...

  5. [5]

    The gauge-boson propagator is−iδ abgµν/k2, the FP- ghost propagator is iδ ab/k2, and the gauge–ghost–anti- ghost three-point vertex factor is−gf abcqµ

    Two-point function of an FP ghost Next, in order to determine the ghost-field renormal- ization constant ˜Z3, we compute the two-point function of the Faddeev–Popov ghost. The gauge-boson propagator is−iδ abgµν/k2, the FP- ghost propagator is iδ ab/k2, and the gauge–ghost–anti- ghost three-point vertex factor is−gf abcqµ. Therefore, the ghost two-point fu...

  6. [6]

    Three-point function of an FP ghost In order to determine the renormalization constant ˜Z1 for the gauge-ghost vertex, we compute the gauge–ghost– anti-ghost three-point functionΓ ¯ccW. As can be seen from the counterterm structure (˜Z1 −1) ×g∂ µ¯caf abcW b µcc, the divergent part of this vertex func- tion must be proportional to ¯ε−1 f abcqµ, whereq µ is...

  7. [7]

    NC differential cross sections Using the results above yields dσγ dt = e2Q2 f |µγ αi|2 4π∆2 ˜K(s, t) −t ,(F74) dσZ dt = e2 8π∆2 gf L 2 + gf R 2 sin2 θW cos2 θW |µZ αi|2 −t ˜K(s, t) (t−m 2 Z)2 (F75) dσγZ dt =− e2Qf gf L +g f R 4π∆2 Re(µγ αiµZ∗ αi ) sinθ W cosθ W ˜K(s, t) t−m 2 Z (F76) 88

  8. [8]

    ¯LβLα self-energy from aB+Lloop TheB-L-Lvertex factor is ig ′γµYL

    ¯LαLα self-energy a. ¯LβLα self-energy from aB+Lloop TheB-L-Lvertex factor is ig ′γµYL. From the fourth row of Table III, the hypercharge of the left-handed lep- ton doublet (ν, e)L isY L =−1/2. Unlike the Dirac- fermion loop contribution in Table II, which is summed over all fermion species, the present two-point function with external legs ¯Lβ andL α ha...

  9. [9]

    (C112), the result is ΣLL,W (q) = g2 16π2 ¯ε−1 3 4 /qPL .(C147) The SU(2) L counterterm Lagrangian is given in Eq

    ¯LβLα self-energy from aW I +Lloop Since the vertex factor is igγ µT aPL, we obtain iΣLL,W (q) = 3 2 g2 Z dnk (2π)n /kPL (k2 −m 2 ℓ )(k−q) 2 .(C146) Since this is identical to Eq. (C112), the result is ΣLL,W (q) = g2 16π2 ¯ε−1 3 4 /qPL .(C147) The SU(2) L counterterm Lagrangian is given in Eq. (C102). Thus the UV divergence ofΣ LL,W (q) must be canceled b...

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    ˜H † ˜Hself-energy from aB+ ˜Hloop TheB- ˜H- ˜Hvertex factor is ig ′(kµ +q µ)Y ˜H

    ˜H † ˜Hself-energy a. ˜H † ˜Hself-energy from aB+ ˜Hloop TheB- ˜H- ˜Hvertex factor is ig ′(kµ +q µ)Y ˜H . From the last row of Table III, the hypercharge of the Higgs doubletHisY H = +1/2. Therefore, for ˜H= iτ 2H ∗ we haveY ˜H =−1/2. Here iτ 2 is an SU(2) L matrix and hence commutes with U(1) Y . The one-loop integral is then iΣHH,B (q) =− 1 4 g′2 Z dnk ...

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    (C153) Applying the same Feynman-parameter formula as in Sec

    ˜H † ˜Hself-energy from aW I + ˜Hloop Since the scalar vertex factor is ig(k µ +q µ)T a (noP L appears for a scalar), we obtain iΣHH,W (q) =− 3 4 g2 Z dnk (2π)n k2 + 2k·q+q 2 (k−q) 2(k2 −m 2 H ) . (C153) Applying the same Feynman-parameter formula as in Sec. C 8 a, we obtain ΣHH,W (q) =− g2 16π2 ¯ε−1 3 2 q2 + 3 4 m2 H .(C154) The counterterm contribution ...

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    (C158) Herey t is the SM top Yukawa coupling

    ˜H † ˜Hself-energy from at R +Q 3 loop In the electroweak-symmetric phase, the Yukawa in- teraction of the right-handed top quarkt R, the third- generation quark doubletQ 3 = (tL bL)⊺, and the Higgs doubletH= (ϕ + ϕ0)⊺ is −Lt =y t ¯Q3 ˜HP Rt+y t¯tPL ˜H †Q3, ˜H= iσ 2H ∗, (C157) i.e., −Lt =y t ¯tPRϕ0∗ −¯bPRϕ− PRt+y t¯tPL ϕ0PLt−ϕ +PLb . (C158) Herey t is the...

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    Consequently, the RG evolu- tion overµ∈[µ ref , µmatch] closes within the couplings of this EFT

    Couplings to be evolved together withC N Band CN W In this work, we integrate out the heavy new parti- clesE,Σ, D,Φ + at the matching scaleµ match and work in an EFT in which the low-energy degrees of freedom are those ofνSMEFT supplemented by the dimension-six operatorsO N BandO N W. Consequently, the RG evolu- tion overµ∈[µ ref , µmatch] closes within t...

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    =δ (q0 1)2 −E 2 1 = 1 2E1 δ(q 0 1 −E 1) +δ(q 0 1 +E 1) . (D12) Therefore, δ(q 2 1)Θ(E1) = 1 2E1 δ(q 0 1 −E 1).(D13) Using this to integrate overq 0 1, we obtain Icut := Z d4q1 (2π)4 (2π)2δ(q 2 1)δ(q 2 2 −m 2 U )Θ(E1)Θ(E2)F(t 1) = Z d3q1 (2π)2 1 2E1 δ(q 2 2 −m 2 U )Θ(E2)F(t 1),(D14) wheret 1 := cosθ 1 := ˆq· ˆq1, andF(t 1) denotes the remain- ing cut integ...

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    The main difference is already visible in the one-loop part (corresponding to Γ 2 = 0)

    2PI effective action for fermions The construction of the 2PI effective action for fermionic fields proceeds along very similar lines as for bosons, with the crucial difference that one has to ac- count for the anti-commuting (Grassmann) nature of fermion fields. The main difference is already visible in the one-loop part (corresponding to Γ 2 = 0). For v...

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    F eynman rules and kinematics For an outgoing gauge-boson momentumq, the dipole vertex and its conjugate vertex forV=γ, Zare (V V )ρ αi(q) = 2µV αiσρλqλPR, ( ˜V V )ρ αi(q) = 2µV∗ αi σρλqλPL, (V=γ, Z).(F2) The gauge interactions offare −L ⊃eQf Aµ ¯f γµf + e sinθ W cosθ W Zµ ¯f γµ gf LPL +g f RPR f,(F3) withQ f the electric charge in proton units, and gf L ...

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    Mandelstam variables and inner products Define q :=k ′ −k=p−p ′, t :=q 2 <0, s := (p+k) 2.(F5) Using the massless approximationk 2 =k ′2 =p ′2 = 0 andp 2 =M 2 i , s= (p+k) 2 =p 2 + 2p·k+k 2 =M 2 i + 2p·k, t= (p−p ′)2 =p 2 −2p·p ′ +p ′2 =M 2 i −2p·p ′, u= (p−k ′)2 =p 2 −2p·k ′ +k ′2 =M 2 i −2p·k ′. (F6) Hence, p·k= s−M 2 i 2 .(F7) Squaring momentum conserv...

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    Then qµJ (γ) µ = ¯uf (k′) q·(k ′ +k) 2mf + i qµσµνqν 2mf uf (k).(F17) The first term vanishes becauseq·(k ′ +k) =k ′2 −k 2 = m2 f −m 2 f = 0

    Simplifyingσ µν qν using current conservation For a Dirac fermion of massm f , the Gordon identity gives ¯uf (k′)γµuf (k) = ¯uf (k′) (k′ +k) µ 2mf + iσµνqν 2mf uf (k) =: J (γ) µ , (F16) withq=k ′ −k. Then qµJ (γ) µ = ¯uf (k′) q·(k ′ +k) 2mf + i qµσµνqν 2mf uf (k).(F17) The first term vanishes becauseq·(k ′ +k) =k ′2 −k 2 = m2 f −m 2 f = 0. The second term...

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    Spin-averaged squared amplitude:γ,Z, andγZ We define the kinematic polynomial K(s, t):= 4 (p·k)(p ′ ·k ′) + (p·k ′)(p′ ·k) = 2s2 + 2st+t 2 −2M 2 i s−M 2 i t,(F26) and ˜K(s, t):=K(s, t) +M 4 i −t 2 = 2s2 + 2st−2M 2 i s−M 2 i t+M 4 i .(F27) a. Photon exchange We start from the identity (F20) and recall that the photon current of the light fermionf, J (γ) µ ...

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    (F65) 87 Usingp 1 ·p 2 = (s−m 2 1 −m 2 2)/2, we find F= 1 2 p λ(s, m2 1, m2 2),(F66) whereλis K¨ all´ en function λ(s, m2 1, m2

    Differential cross sectiondσ/dt For 2→2 scattering 1(p1) + 2(p2)→3(p 3) + 4(p4),(F62) the Lorentz-invariant differential cross section is dσ= 1 4F |M|2 dΦ2,(F63) with Møller invariant flux [84] F := p (p1 ·p 2)2 −m 2 1m2 2,(F64) and two-body phase space dΦ2 = d3p3 (2π)32E3 d3p4 (2π)32E4 (2π)4δ4(p1 +p 2 −p 3 −p 4). (F65) 87 Usingp 1 ·p 2 = (s−m 2 1 −m 2 2)...

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    (F67) In the center-of-mass frame, dΦ2 = ∥pf ∥ 8π√s dΩ,(F68) with ∥pi∥= p λ(s, m2 1, m2 2) 2√s ,∥p f ∥= p λ(s, m2 3, m2 4) 2√s

    =s 2 +m 4 1 +m 4 2 −2m 2 1s−2m 2 2s−2m 2 1m2 2. (F67) In the center-of-mass frame, dΦ2 = ∥pf ∥ 8π√s dΩ,(F68) with ∥pi∥= p λ(s, m2 1, m2 2) 2√s ,∥p f ∥= p λ(s, m2 3, m2 4) 2√s . (F69) Hence, dσ dΩ = |M|2 64π2s ∥pf ∥ ∥pi∥ .(F70) Using dΩ= 2πd cosθ, and t(cosθ) =m 2 1 +m 2 3 −2E 1E3 + 2∥p1∥∥p3∥cosθ ⇒ dt d cosθ = 2∥pi∥∥pf ∥, (F71) we obtain dσ dt = |M(s, t)|2...

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