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Unravelling the Scalar Sector of Grand Unification: Phenomenology & Implications

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Pith's one-line read Scalar fields in SO(10) GUTs give proton decay a kaon-rich signature, with p→K+ν dominant and p→e+π0 suppressed, opposite to gauge-boson mediation.

desk verdict A thorough SO(10)/SU(5) scalar-sector thesis with real synthesis value, but the abstract overstates the proton-decay mode and the mass-splitting assumption carries more weight than the framing admits. read the letter →

arxiv 2507.16605 v1 pith:UBALSRAX submitted 2025-07-22 hep-ph hep-th

Saurabh K. Shukla This is my paper
classification hep-phhep-th PACS 12.10.Dm13.30.-a
keywords grandunifiedtheoriesSO(10)protondecayscalarleptoquarksneutron-antineutronoscillationbaryogenesishierarchyproblemYukawacouplings
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This thesis works out what the scalar fields necessarily present in SO(10) and SU(5) grand unified theories do to observable processes, and uses the resulting bounds to locate the scalars in mass. The central claim is that, among the sixty B−L-charged scalars in the renormalisable and non-renormalisable SO(10) Yukawa sectors, only three colour-triplet types (plus conjugates) mediate leading dimension-six proton decay, while three more mediate dimension-seven modes. In a realistic SO(10) model built from 10H and 126H, scalar-mediated proton decay is predicted to be dominated by p→K+ν and p→µ+K0, with p→e+π0 strongly suppressed—opposite to the pattern from gauge-boson exchange. The same scalars are then subjected to proton-decay, n-nbar oscillation, flavour-violation, and baryogenesis constraints, yielding mass bounds, some below the GUT scale and some near the electroweak scale. The thesis also shows that one-loop corrections from heavy scalars can lift the bad SU(5) relation Yd=YeT, reducing the need for extra Higgs representations.

What carries the argument

The workhorse is the oscillator-expansion decomposition of SO(10) Yukawa invariants: every coupling $16\,16\,\phi$ is expanded in an SU(5) basis, canonically normalised, and matched onto SM-invariant four-fermion operators $O_1$–$O_4$ and their dimension-seven analogues. The selection rule is that a scalar induces tree-level proton decay only if it has both diquark and leptoquark couplings; the surviving candidates are integrated out to produce Wilson coefficients, which are converted into partial widths with lattice hadronic matrix elements. Flavour enters through hierarchical Yukawa matrices of the form $H\sim\lambda^4\,\mathrm{diag}(\lambda^7,\lambda^3,1)$ and $F$ with powers of the Cabibbo angle $\lambda=0.23$, which make second-generation couplings dominate and produce the $K^+\nu$/$K^0\mu^+$ pattern.

What would settle it

If a next-generation water-Cherenkov or liquid-scintillator detector records proton decay with p→e+π0 dominating and no significant p→K+ν, the scalar-mediation claim with hierarchical Yukawa couplings is ruled out; observing p→K+ν with suppressed p→e+π0 would confirm it. The sextet mass window is independently testable by projected neutron-antineutron oscillation searches.

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Extended reading notes

Core claim

The central discovery is a complete inventory of which scalars in the SO(10) Yukawa sector can actually destabilise the proton. Of the sixty B−L-charged scalars in $10_H$, $120_H$, $\overline{126}_H$, and $16_H$, only the colour triplets $T(3,1,1/3)$, $T(3,1,4/3)$, and $T(3,3,1/3)$ (with conjugates) have both the diquark and leptoquark vertices needed for tree-level dimension-six decay, and only $\Theta(3,1,2/3)$, $\Delta(3,2,1/6)$, and $\Omega(3,2,7/6)$ mediate the dimension-seven $\Delta(B-L)=2$ modes. In a realistic minimal SO(10) model built from $10_H$ and $\overline{126}_H$, the proton then decays preferentially into second-generation mesons: $p\to K^+\nu$ dominates when the antitriplet is lighter, $p\to\mu^+K^0$ when the triplet is lighter, while $p\to e^+\pi^0$ is suppressed by the hierarchical Yukawa texture. This is the opposite of gauge-boson-mediated decay, which favours $p\to e^+\pi^0$ and $p\to\nu\pi^+$. The same machinery constrains the coloured sextets via meson-antimeson oscillation, neutron-antineutron oscillation, and baryogenesis, and shows that one-loop scalar corrections can repair the minimal SU(5) relation $Y_d=Y_e^T$.

Load-bearing premise

The load-bearing premise is that scalars inside a single GUT irrep can be fine-tuned to distinct masses far below the GUT scale, beyond the Extended Survival Hypothesis, together with the hierarchical Yukawa texture inherited from fermion-mass fits; if either gives way, the quoted mass bounds and the kaon-rich decay pattern lose force.

Editorial extensions

If this is right

  • If scalar mediation dominates, future proton decay experiments should observe mostly p→K+ν and p→µ+K0, with p→e+π0 rare; the channel p→νK+ is precisely the mode the most sensitive upcoming detectors target.
  • The colour triplets T and T can lie as low as 10^10–10^11 GeV and still satisfy proton-decay limits, so scalar mediators need not sit at the 10^16 GeV gauge-unification scale.
  • In non-renormalisable 16H-based models, some scalar pairs escape proton-decay bounds down to near the electroweak scale, implying possible low-energy signals.
  • The dimension-seven B−L-violating mode constrains Δ to roughly 10^6–10^7 GeV for B−L breaking near 10^11 GeV, complementing bounds from neutron-antineutron oscillation.
  • The coloured sextet scalars in 120H and 126H can generate meson-antimeson and n-nbar oscillations and baryogenesis in a common mass window, linking flavour, baryon number, and cosmology.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the kaon-rich pattern is driven by second-generation Yukawa couplings rather than by group theory, the same classification would produce third-generation or first-generation signatures under different flavour textures; this is an extension the paper does not itself explore.
  • If proton decay is observed in both scalar-like and gauge-like channels, ratios such as BR(p→K+ν)/BR(p→e+π0) would measure the flavour hierarchy at the GUT scale and test the fitted texture independently of mass fits.
  • The fine-tuning premise implies that a fully natural GUT scalar spectrum would push most of these scalars to the GUT scale, making the predicted signals unobservable; a null result at next-generation detectors would favour such natural spectra over the split-spectrum scenario.
  • The diquark/leptoquark classification could be applied to other unification groups, such as E6 or trinification, to identify which scalars mediate baryon-number violation before any model-specific calculation is done.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. This manuscript is a PhD thesis presenting a systematic analysis of the scalar sectors of SO(10) and SU(5) grand unified theories. It decomposes the SO(10)-invariant Yukawa couplings of the 10H, 120H, 126H (renormalisable) and 16H (non-renormalisable) scalar irreps into SM-invariant vertices, classifies the B−L-charged scalars by their diquark and leptoquark couplings, and identifies the colour triplets T(3,1,−1/3), T(3,1,−4/3), T(3,3,−1/3) and their conjugates as the tree-level D=6 proton-decay mediators, with Θ, ∆, and Ω mediating D=7 modes. Using a realistic 10H+126H SO(10) model with Yukawa textures fit to fermion masses and mixing (from the fits of [109]), the thesis computes proton branching fractions: scalar mediation favours second-generation final states (p→µ+K0 or p→νK+, depending on the T/T mass hierarchy), with BR(p→µ+π0) ≫ BR(p→e+π0), in contrast to gauge-boson mediation (p→e+π0, p→νπ+). It derives scalar mass bounds (MT at 10^10–10^11 GeV; M∆ ≳ 7×10^6 GeV for a benchmark D=7 operator) and extends the analysis to effective dimension-five couplings of 16H pairs in non-renormalisable models. Separate chapters treat colour-sextet scalars (meson-antimeson and neutron-antineutron oscillations, baryogenesis) and one-loop quantum corrections to the minimal SU(5) Yukawa relation Yd = Ye^T.

Significance. The central systematic result—the classification of which of the sixty B−L-charged scalars in the leading SO(10) Yukawa irreps can mediate proton decay at D=6 or D=7—appears complete and internally consistent, with the operator bases and Wilson coefficients given in closed form (Eqs. 3.29–3.39, 3.44–3.49) in terms of Yukawa matrices, mixing angles, and scalar masses. The prediction that scalar-mediated proton decay is kaon-rich and muon-rich relative to gauge-mediated decay is a falsifiable distinction for Hyper-Kamiokande and JUNO, and the branching pattern is shown to be robust across acceptable χ² solutions in the 10H+126H model class (Figs. 3.2–3.4). The calculation is genuinely predictive rather than circular: Yukawa couplings are fit to fermion masses and mixing, and the B- and L-violating rates are then computed and compared with experimental limits, with no proton-decay data used to set constants. The main caveats—the hierarchical texture of Eq. (3.60) and the non-ESH fine-tuned scalar spectrum assumed in Sec. 1.3.3—are declared in the body, although the abstract does not carry the corresponding qualifications.

major comments (2)
  1. [Abstract; Tables 3.2, 3.3; Sec. 3.9] The abstract (both the arXiv abstract and the thesis abstract on p. viii) states that 'in both renormalisable and non-renormalisable scenarios, the proton favours decay into second-generation mesons accompanied by charged or neutral leptons, and the dominant mode is p → K+ ν.' This is not supported by the branching tables in Chapter 3 without qualification. For 10H mediation (Table 3.2), the hierarchy M_T ≪ M_Tbar gives BR(p→µ+K0)=93% and BR(p→νK+)=0%, while dominance of p→νK+ (84%) holds only for M_T ≫ M_Tbar or M_T = M_Tbar; for 126H mediation (Table 3.3), the same pattern appears, with 88% for µ+K0 and 0% for νK+ when M_T1,2 ≪ M_Tbar, versus 78% for νK+ otherwise. Section 3.9 states the correct conditional result: 'proton decay primarily results in ν K+ or µ+ K0 if T or T is lighter, respectively.' The abstract should be revised to present the branching pattern as hierarchy-dependent (νK+ versus µ+K0), since the unqualified 'dominant mode is p→K+ν' claim is false over a substantial allowed portion of the scalar mass parameter space.
  2. [Sec. 1.3.3; Eqs. (3.66)–(3.70); Tables 7.1, 7.2] The premise stated in Sec. 1.3.3—that scalar particles within the same irrep can acquire distinct masses and remain significantly below the GUT scale, even if this requires fine-tuning beyond the ESH framework—is load-bearing for every derived mass bound (Eqs. 3.66–3.70, and the summary tables in Chapter 7) and for the abstract's claim that scalars can lie below the GUT scale while adhering to proton-decay constraints. The text declares this assumption honestly, which is to its credit, but the abstract and concluding tables present the bounds without it. I recommend that the abstract and the summary tables explicitly condition the quoted bounds and the low-scale accessibility of the scalar mediators on this assumption, or alternatively point to an explicit scalar-potential construction that realises the split spectrum.
minor comments (6)
  1. [Abstract; Tables 2.2, 3.1; Sec. 3.9] The hypercharge signs in the abstract (T with Y=1/3, 4/3, 1/3) are inconsistent with the body, which consistently uses T(3,1,−1/3), T(3,1,−4/3), T(3,3,−1/3) for the proton-decay mediators and their conjugates; please unify the notation.
  2. [Sec. 1.3.3; Sec. 3.3] 'Sphalleron' should be 'sphaleron' in both occurrences.
  3. [Sec. 1.1] 'Cabibo-Kobayashi-Masakawa' should be 'Cabibbo-Kobayashi-Maskawa'.
  4. [Sec. 3.7.3, Eq. (3.70)] The quoted lower bound M∆ ≳ 7×10^6 GeV is a benchmark estimate for λ=1, α2=0.1, vD=174 GeV, MT=1.1×10^11 GeV, and vσ=10^11 GeV; a one-line statement of the parametric scaling (M∆ ∝ (λ vσ)^(−1/2) with these choices) would make the bound usable for other benchmark points.
  5. [Fig. 5.5 caption] The caption states 'ϵmax = 10−7 is represented by a lighter gray shade, and ϵmax = 10−2 by a darker gray'; please clarify that ε_max is a scanned parameter spanning this range and that the two shades are the endpoints.
  6. [Chs. 3–6; Publications (p. xi)] Each chapter would benefit from a short preface stating which sections reproduce the author's published papers (JHEP 08 (2022) 042, PRD 107 (2023) 055008, PRD 109 (2024) 015007, Nucl. Phys. B 1018 (2025) 117034) and which parts are new synthesis, since the current text requires the reader to infer the attribution from the publication list.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: scalar-mediated proton-decay patterns are computed from Yukawa couplings fitted to fermion masses, with no proton-decay data used as input; the abstract-vs-Table 3.2 mismatch and the explicit fine-tuning assumption in Sec. 1.3.3 are correctness/robustness caveats, not circularity.

full rationale

The central chain — classifying B-L-charged scalars, computing their SM-invariant vertices, integrating them out to obtain D=6/D=7 operators, and then estimating branching ratios and mass bounds — is self-contained at each step. The branching fractions in Tables 3.2 and 3.3 are computed from the best-fit Yukawa matrices H and F obtained in [109] from fits to fermion masses and mixing; no proton-decay observable is used to set these couplings, so the kaon-rich/muon-rich pattern is a genuine prediction of the fitted flavor structure rather than a fit to itself. The mass bounds in Sec. 3.7.2 are rearrangements of experimental lifetime limits with model couplings, which is constraint-setting, not circularity. Self-citations are present — the thesis reproduces the author's published papers, and [109] supplies the fermion-mass fit — but the cited fit is an external benchmark whose target is the fermion spectrum, not the proton-decay branching pattern, so the self-citation is not load-bearing in a circular sense. Two non-circular caveats should be weighed. First, Sec. 1.3.3 explicitly assumes that scalars within one irrep can sit far below the GUT scale with fine-tuning, an admitted premise on which the mass bounds depend; this is a robustness limitation, not a circular step. Second, the abstract's unqualified statement that 'the dominant mode is p -> K+ nu' is contradicted by the paper's own Table 3.2, where M_T much less than M_Tbar gives BR(p -> mu+ K0) = 93% and BR(p -> nu K+) = 0%, while only the M_T greater than or similar to M_Tbar hierarchies give 77-84% for nu K+; Section 3.9 states the correct hierarchy-dependent conclusion. This is an internal-consistency or correctness issue, not an equivalence of outputs to inputs. No uniqueness theorem is invoked, and no prediction reduces by definition to a fitted parameter. Verdict: no significant circularity.

Assumptions & free parameters 7 free parameters · 8 assumptions · 0 invented entities

The central claims rest on standard GUT machinery (SO(10)/SU(5) group theory, oscillator expansion, effective operator analysis) plus several domain assumptions the thesis states: the specific scalar irreps 10H, 120H, 126H, and 16H; B-L selection rules for operator dimensions; hadronic matrix elements from lattice and chiral perturbation theory; and perturbativity of couplings in the fits. The main ad hoc element is the admitted fine-tuned mass splitting within irreps (Section 1.3.3). Free parameters consist of fitted Yukawa normalizations, chi-squared-fit couplings in the SU(5) extension, swept scalar mass hierarchies, and sweeping parameters such as epsilon_max, lambda, and v_sigma. The thesis introduces no new particles: every field is a standard GUT representation, and the heavier states gain falsifiability through the derived proton decay and oscillation bounds.

free parameters (7)
  • alpha1, alpha2: hierarchical Yukawa texture normalizations (H, F matrices) = benchmarked at 0.1
    Set the scale of the hierarchical textures of Eq. (3.60) in the 10H+126H model; mass bounds scale as (alpha/0.1)^4 while branching fractions are claimed to be independent of them.
  • Scalar mass hierarchy choices (MT vs MT, MT1,2 vs MT, MX vs MY) = swept as much less, much greater, or equal
    Branching tables 3.2 to 3.4 are computed at three extreme mass hierarchies; this is an input assumption, not a prediction.
  • Y1, Y2, Y3 (SU(5) singlet-fermion Yukawa couplings) = chi-squared minimized with |Y| below sqrt(4 pi)
    Chapter 6 one-loop matching: these new couplings are fit to fermion mass and mixing data, and the mass-relation improvement is then demonstrated on the fit.
  • MT and MN (triplet and singlet masses in the SU(5) extension) = chi-squared minimized, scanned in Fig. 6.9
    The claim that one-loop corrections lift Yd = Ye^T requires MT and MN to differ by at least two orders of magnitude (Section 6.6).
  • epsilon_max (maximal CP asymmetry parameter in baryogenesis) = swept between 10^-7 and 10^-2
    Section 5.6: the excluded-region plots for YB above 6 times 10^-10 are drawn for two arbitrary epsilon_max values.
  • lambda and vev benchmarks for D=7 operators = lambda = 1, vD = 174 GeV, v_sigma = 10^14 GeV
    Section 3.7.3: the Delta mass bound (about 7 times 10^6 GeV) is computed at these fixed inputs and scales directly with them.
  • Hadronic matrix element parameters (alpha, beta, D-tilde, F-tilde) = taken from lattice literature [172]
    Inputs to the partial-width formulas (Eqs. 3.53 and 3.57); their uncertainties are not propagated into the quoted branching fractions.
assumptions (8)
  • domain assumption SO(10) and SU(5) gauge unification with the given scalar irrep content (10H, 120H, 126H, 16H)
    Sections 2.2 and 2.3: the entire analysis assumes these GUT groups, the stated breaking chain, and the stated SM decompositions of the irreps.
  • standard math Oscillator expansion technique correctly decomposes SO(10) invariants into SM invariants
    Section 2.4.1: the method is imported from references [125, 129]; every effective coupling in the thesis rests on it.
  • domain assumption B-L selection rules: even-dimension operators conserve B-L, odd-dimension operators violate it by two units
    Section 3.3: the classification of D=6 versus D=7 proton decay operators relies on the known selection rules from references [155, 156].
  • ad hoc to paper Scalar masses within a single GUT irrep can be split arbitrarily with fine-tuning
    Section 1.3.3: explicitly stated as going beyond the Extended Survival Hypothesis; load-bearing for every light-scalar bound in Chapters 3 to 5.
  • domain assumption The hierarchical Yukawa texture of Eq. (3.60) is representative of the 10H+126H model class
    Section 3.7.1: the K+nu-dominant branching pattern follows from this texture, which is asserted from fits in [109] rather than derived from first principles.
  • domain assumption Hadronic matrix elements from chiral perturbation theory and lattice QCD are reliable for decay widths
    Section 3.6.1: decay widths use alpha, beta, D-tilde, F-tilde, and A from [172]; non-perturbative uncertainties are not quantified.
  • domain assumption Quartic couplings sigma D T Delta and similar terms exist with O(1) coefficients
    Section 3.5: the D=7 B-L violating operators assume these SO(10)-invariant quartic terms; the bound in Section 3.7.3 sets lambda = 1.
  • domain assumption Perturbativity |Y| below sqrt(4 pi) in the one-loop fits
    Section 6.7: chi-squared minimization is constrained by perturbativity; the viability conclusion of the SU(5) mechanism is conditional on it.

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Pith. "Pith review of Unravelling the Scalar Sector of Grand Unification: Phenomenology & Implications." pith.science (2026). https://pith.science/paper/UBALSRAX

@misc{pith2026250716605,
  author       = {Pith},
  title        = {Pith review of: Unravelling the Scalar Sector of Grand Unification: Phenomenology & Implications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UBALSRAX}},
  note         = {Machine review of arXiv:2507.16605}
}
abstract

Grand Unified Theories (GUTs) based on groups like $SO(10)$ and $SU(5)$ unify Standard Model (SM) fermions into irreducible representations (irreps), and predict additional scalar fields beyond the SM Higgs. In $SO(10)$ GUTs, the scalar fields can arise from irreps contributing to the Yukawa sector at the renormalisable level, such as $10_{\mathrm{H}}$, $120_{\mathrm{H}}$, and $\overline{126}_{\mathrm{H}}$, or from $16_{\mathrm{H}}$ in non-renormalisable interactions. The direct implications of these scalars include the violation of baryon and lepton number, enabling processes such as nucleon decays, neutron-antineutron oscillation, and potentially accounting for the observed baryon asymmetry of the universe. We systematically analyse their couplings to SM fermions, identifying diquark and leptoquark interactions vertices involving all scalars residing in $10_{\mathrm{H}}$, $120_{\mathrm{H}}$, $\overline{126}_{\mathrm{H}}$, and $16_{\mathrm{H}}$ and comprehensively assess their contributions to nucleon decay, neutron-antineutron oscillation, quark flavour violation, and baryogenesis. Constraints on the masses of these scalars, derived from experimental bounds on the aforementioned processes, are also estimated. Conventional GUTs rely on multiple scalar irreps to avoid unrealistic fermion mass relations; for example, minimal $SU(5)$ with $5_{\mathrm{H}}$ predicts degenerate down-quark and charged-lepton masses. We demonstrate that quantum corrections from heavy scalars in a minimally extended $SU(5)$ model can lift this degeneracy, thereby reducing the arbitrariness in the scalar sector, which has been called as indirect impact. This thesis provides a comprehensive examination of the scalar sector's role in GUTs, establishing connections between UV-complete models and observable phenomena.

Figures

Figures reproduced from arXiv: 2507.16605 by the authors.

Figure 1.1
Figure 1.1. Running of SM gauge couplings at one loop. [PITH_FULL_IMAGE:figures/full_fig_p035_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Scalar induced correction to Yukawa coupling. [PITH_FULL_IMAGE:figures/full_fig_p048_1_2.png] view at source ↗
Figure 2.1
Figure 2.1. Breaking pattern of SO(10) to SU(3) × SU(2) × U(1) via SU(5). The SM charges of scalars acquiring vev can be inferred from Tabs. (2.1 and 2.2) rank-five group, various breaking patterns can yield the SM gauge symmetry, highlighting the complexity of the Higgs sector of SO(10). Different break￾ing chains can influence the renormalisation group (RG) running of various observables, including the gauge couplings [105, 1… view at source ↗
Figures from the paper (23 more)
Figure 3.1
Figure 3.1. Figure 3.1: Topologies of B − L conserving (left Feynman graph) and B − L violating (right Feynman graph) proton decay, drawn using [157]. B − L violating diagram is drawn in the unbroken Electroweak phase. D = 6, 8, 10, ... [155, 156]. In B − L conserving modes, the proton deca…
Figure 3.2
Figure 3.2. Figure 3.2: The scalar-mediated proton decay spectrum within the framework of the [PITH_FULL_IMAGE:figures/full_fig_p112_3_2.png]
Figure 3.3
Figure 3.3. Figure 3.3: Similar to Fig. (3.2), but with the assumption that the lightest triplets predominantly originate from 126H. where Uν is matrix having all the elements of O(1). Uu,d,e could taken as CKM-type, leading to small quark mixing while the large mixing in the PMNS matrix ar…
Figure 3.4
Figure 3.4. Figure 3.4: The spectrum of proton decay induced by gauge bosons for several solu [PITH_FULL_IMAGE:figures/full_fig_p119_3_4.png]
Figure 4.1
Figure 4.1. Figure 4.1: Tree and loop level proton decay topologies generated by the pair [PITH_FULL_IMAGE:figures/full_fig_p132_4_1.png]
Figure 5.1
Figure 5.1. Figure 5.1: Feynman graph of flavour violation process induced by [PITH_FULL_IMAGE:figures/full_fig_p152_5_1.png]
Figure 5.2
Figure 5.2. Figure 5.2: Feynman graph depicting n-n oscillation, drawn using [157] spectively. Further, we assume that in Eqs. (5.16-5.19), all sextet fields are expressed in the mass basis. Eq. (5.16) identifies three distinct operators cor￾responding to i, j = 1, 2 with η12 = η21. Given t…
Figure 5.3
Figure 5.3. Figure 5.3: Diagram showing quartic coupling of Σ and a correction due to trilinear coupling among three colour sextet scalars. The vertex marked by a bullet point represents a trilinear coupling generated by the VEV of the B − L charged scalar σ. shown in Eqs. (5.16-5.19). We f…
Figure 5.4
Figure 5.4. Figure 5.4: The decays of the colour sextet scalar ( [PITH_FULL_IMAGE:figures/full_fig_p163_5_4.png]
Figure 5.5
Figure 5.5. Figure 5.5: Values of MΣ and |η|vσ that are ruled out by YB > 6.0 × 10−10 are indicated for two scenarios: ϵmax = 10−7 is represented by a lighter gray shade, and ϵmax = 10−2 by a darker gray. Further, for MSi ≪ MΣ, one finds Γ[Σ → S ∗ i S ∗ j ] ≃ |ηij| 2v 2 σ 16πMΣ . Substituti…
Figure 5.6
Figure 5.6. Figure 5.6: The figure illustrates constraints on the masses of [PITH_FULL_IMAGE:figures/full_fig_p170_5_6.png]
Figure 5.7
Figure 5.7. Figure 5.7: Constraints on the masses of Σ and S for the high (left panel) and interme￾diate (right panel) B − L breaking scale. The additional details are the same as those in the caption of Fig. (5.6). 5.7.2 Light Σ, S Assuming |η2| = 1 in Eq. (5.17), we evaluate constraints o…
Figure 5.8
Figure 5.8. Figure 5.8: Constraints on the masses of Σ and S for the high (left panel) and interme￾diate (right panel) B − L breaking scale. The additional details are the same as those in the caption of Fig. (5.6). couplings are involved. The constraints arising from n-n¯ oscillations and …
Figure 5.9
Figure 5.9. Figure 5.9: Depiction of constraints on the masses of [PITH_FULL_IMAGE:figures/full_fig_p174_5_9.png]
Figure 6.1
Figure 6.1. Figure 6.1: Running of the ratios of different generations of Yukawas of down-quarks [PITH_FULL_IMAGE:figures/full_fig_p180_6_1.png]
Figure 6.2
Figure 6.2. Figure 6.2: Feynman graphs of the various vertices appearing in Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p183_6_2.png]
Figure 6.3
Figure 6.3. Figure 6.3: One-loop correction diagrams for Yd , with similar diagrams applicable to Ye , Yu, and Yν C . However, ν C contributes exclusively to the corrections in Yd . where f = u, d,e, ν. Together with, Y 0 u = Y1 , Y 0 d = Y2 , Y 0 e = Y T 2 , Y 0 ν = Y3, (6.20) at µ = MGUT.…
Figure 6.4
Figure 6.4. Figure 6.4: Feynman diagrams depicting vertex Corrections to [PITH_FULL_IMAGE:figures/full_fig_p187_6_4.png]
Figure 6.5
Figure 6.5. Figure 6.5: Feynman diagrams depicting vertex corrections to [PITH_FULL_IMAGE:figures/full_fig_p188_6_5.png]
Figure 6.6
Figure 6.6. Figure 6.6: Wave function renormalisation and its connection to self-energy correction [PITH_FULL_IMAGE:figures/full_fig_p189_6_6.png]
Figure 6.7
Figure 6.7. Figure 6.7: Feynman diagrams depicting correction to the external leg [PITH_FULL_IMAGE:figures/full_fig_p190_6_7.png]
Figure 6.8
Figure 6.8. Figure 6.8: Contours of yb/yτ = 3/2 (red), yb/yτ = 1 (orange), and yb/yτ = 2/3 (green) are drawn using Eq. (6.32). These contours are based on parameters yt = 0.427, g = 0.53, and µ = MX = 1016 GeV, and for yν = √ 4π (solid lines) and yν = 2.7 (dashed lines). Extrapolating the G…
Figure 6.9
Figure 6.9. Figure 6.9: The distribution of minimized χ 2 values shown for variousMT and MN. The regions are color-coded as follows: green for χ 2 min ≤ 3, yellow for 3 < χ 2 min ≤ 9, and red for χ 2 min > 9. We have set µ = MX = 1016 GeV and g = 0.53, with the constraint that |(Y1,2,3)ij| …

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