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Proton Decay, Yukawa Couplings and Underlying Gauge Symmetry in String Theory

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Radu Tatar, Taizan Watari

High Energy Physics - Theory · Submitted 23 Feb 2006 · 81 references

Pith's verdict: A useful bottom-up classification of SU(5) string vacua whose E7/E8 minimality claim is probably right but rests on an unproved Lie-algebra exclusion.

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arXiv IDhep-th/0602238 (version 2)
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CitationRadu Tatar, Taizan Watari. "Proton Decay, Yukawa Couplings and Underlying Gauge Symmetry in String Theory." arXiv:hep-th/0602238 (2006). https://pith.science/paper/GIYTDATM
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The paper

The paper claims

Requiring all Yukawa couplings to come from one Yang-Mills interaction forces the underlying gauge symmetry of SU(5) string vacua to be E7 or E8, with E7 minimal.

The reading

Supersymmetric SU(5) grand unification has a tension: the same operator that gives down-quark and charged-lepton masses would, if the two SU(5) ¯5 multiplets had the same origin, also produce dimension-4 proton decay. The paper proposes that the matter multiplet ¯5=(D,L) and the Higgs multiplet ¯H(¯5) come from different irreducible components of the coset g/su(5) of an underlying gauge group G, and that all Yukawa couplings of quarks and leptons be generated by the super Yang-Mills interaction of G. With H=SU(5) fixed, the paper argues purely from Lie-algebra decompositions that the smallest possible G is E7, and that E8 is the only alternative. This is then implemented explicitly in heterotic, M-theory and F-theory compactifications, where the dimension-4 proton decay operator is absent automatically. If correct, this gives a sharp structural constraint on the string-theory landscape of SU(5) vacua.

What carries the argument

The load-bearing object is the coset space g/su(5): the adjoint representation of the underlying gauge group G decomposed under the unbroken SU(5)GUT. The argument works by irreducible decomposition; for example, E6 supplies the up-type Yukawa coupling through (∧25,2)·(∧25,2)·(∧45,∧22), but offers only one ¯5, forcing an enlargement to E7, whose coset contains (5,2)(1,-4), (5,1)(0,6) and (∧45,∧22)(2,4), giving two distinguishable ¯5 multiplets and generating all Yukawa couplings while leaving the dimension-4 proton decay operator a non-singlet under the preserved symmetries. The same coset analysis is implemented as vector-bundle splitting in heterotic theory, as 2-cycles with E7 or E8 intersection form in M-theory, and as matter curves on a discriminant locus in F-theory.

Extended reading notes

Core claim

The paper claims that in Georgi-Glashow SU(5) unification, requiring (i) the up-type, down-type and charged-lepton Yukawa couplings all to come from the super Yang-Mills interactions of an underlying gauge group G, and (ii) the matter multiplet ¯5=(D,L) and the Higgs multiplet ¯H(¯5) to be assigned to different irreducible components of g/su(5) so as to eliminate dimension-4 proton decay, forces G=E7 or E8. E7 is minimal: its coset e7/su(5) contains two distinct ¯5 components, while e6/su(5) contains only one. E8 is the only alternative, arising from embedding E7×SU(2)⊂E8, and is the only setting in which H(5) and ¯H(¯5) can become a hermitian-conjugate pair, as needed for a Wilson-line solution to doublet-triplet splitting. The argument is presented first as a Lie-algebraic theorem independent of string construction, then translated into heterotic vector-bundle language, M-theory on G2-holonomy manifolds, and F-theory on elliptic Calabi-Yau fourfolds.

Load-bearing premise

The argument's load-bearing premise is that only an Er-type (r≥6) singularity, meaning a group at least as large as E6, can generate the up-type quark Yukawa coupling 10·10·H(5) from the coset of the underlying group, so smaller subgroups like SU(N) or SO(N) cannot do the job.

Editorial extensions

If this is right

  • Every supersymmetric SU(5) string vacuum whose Yukawa couplings all come from Yang-Mills interactions and whose ¯5 and ¯H(¯5) have distinct origins must have underlying symmetry E7 or E8.
  • The dimension-4 proton decay operator ¯5·10·¯5 is automatically absent in the seven identified symmetry-breaking patterns, because it transforms as a doublet under an SU(2)2 symmetry or violates the U(1) charges of the underlying group.
  • Perturbative Type IIA and Type IIB string vacua with intersecting D-branes cannot generate the up-type quark Yukawa coupling of SU(5) GUT, since that coupling requires an Er-type singularity; one must use M-theory or F-theory.
  • In the minimal E7 model both dimension-4 and dimension-5 proton decay operators are forbidden, while in the E8 identification A only a U(1) anomaly can generate dimension-5 operators, giving natural suppression.
  • The constraint is independent of whether the vacuum is realized in heterotic, M-theory, or F-theory, so it applies across all three string-theoretic constructions.

Reading between the lines

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

  • Editorial inference: the same coset-decomposition logic would constrain other unification groups, such as Pati-Salam or flipped SU(5): any H must admit at least two distinct H-representations playing the matter and Higgs roles, which will typically force a minimal G above H.
  • Editorial inference: if future proton-decay searches tighten bounds further, the set of string vacua consistent with both Yukawa generation and proton stability shrinks to these seven symmetry-breaking islands, making the landscape constraint sharper.
  • Editorial inference: the M-theory statement that diagonal entries of the up-type quark Yukawa matrix tend to be exponentially suppressed could be tested statistically by scanning known G2-holonomy or F-theory geometries with SU(5) enhancement.
Original abstract

In string theory, massless particles often originate from a symmetry breaking of a large gauge symmetry G to its subgroup H. The absence of dimension-4 proton decay in supersymmetric theories suggests that (\bar{D},L) are different from \bar{H}(\bar{\bf 5}) in their origins. In this article, we consider a possibility that they come from different irreducible components in $\mathfrak{g}/\mathfrak{h}$. Requiring that all the Yukawa coupling constants of quarks and leptons be generated from the super Yang--Mills interactions of G, we found in the context of Georgi--Glashow H=SU(5) unification that the minimal choice of G is E_7 and E_8 is the only alternative. This idea is systematically implemented in Heterotic String, M theory and F theory, confirming the absence of dimension 4 proton decay operators. Not only H=SU(5) but also G constrain operators of effective field theories, providing non-trivial information.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,049 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.16 · compares with prior results 0.22 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

In string theory, massless particles often originate from a symmetry breaking of a large gauge symmetry G to its subgroup H. The absence of dimension-4 proton decay in supersymmetric theories suggests that (\bar{D},L) are different from \bar{H}(\bar{\bf 5}) in their origins. In this article, we consider a possibility that they come from different irreducible components in $\mathfrak{g}/\mathfrak{h}$. Requiring that all the Yukawa coupling constants of quarks and leptons be generated from the super Yang--Mills interactions of G, we found in the context of Georgi--Glashow H=SU(5) unification that the minimal choice of G is E_7 and E_8 is the only alternative. This idea is systematically implemented in Heterotic String, M theory and F theory, confirming the absence of dimension 4 proton decay operators. Not only H=SU(5) but also G constrain operators of effective field theories, providing non-trivial information.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In string theory, massless particles often originate from a symmetry breaking of a large gauge symmetry G to its subgroup H. background 1.00 0.04 0.03 asserted 0.68
s1 The absence of dimension-4 proton decay in supersymmetric theories suggests that (\bar{D},L) are different from \bar{H}(\bar{\bf 5}) in their origins. background 0.90 0.07 0.07 hedged possibility 0.98
s2 In this article, we consider a possibility that they come from different irreducible components in $\mathfrak{g}/\mathfrak{h}$. contribution 0.91 0.28 0.07 hedged possibility 0.94
s3 Requiring that all the Yukawa coupling constants of quarks and leptons be generated from the super Yang--Mills interactions of G, we found in the context of Georgi--Glashow H=SU(5) unification that the minimal choice of G is E_7 and E_8 is the only alternative. result 0.96 0.93 0.73 asserted 1.00
s4 This idea is systematically implemented in Heterotic String, M theory and F theory, confirming the absence of dimension 4 proton decay operators. result 0.70 0.65 0.25 asserted 1.00
s5 Not only H=SU(5) but also G constrain operators of effective field theories, providing non-trivial information. interpretation 0.91 0.55 0.30 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.88.

What was examined, and what was not

sources read
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The (extended) Dynkin diagram of E6, which not only describes the Lie algebra of E6 symmetry but also the intersection form of E6-type singularity. Each node is assigned a simple root αi and a 2-cycle Ci . The su(2)2 and su(5)GUT subalgebra of su(2)2 + u(1) + su(5)GUT ⊂ e6 in section 2 are generated by α1 and α3,4,5,6, respectively. The highest root θ is not linearly independent from other six simple roots: θ = α1 + 2α2 + 3α3 + 2α4 + α5 + 2α6. ~ζ 3 = ~ζ 4 = ~ζ 5 = ~ζ 6 = ~0, and the 2-cycles C3, C4, C5 and C6 are of zero size. Let ~y = (y1, y2, y3) be a set of local coordinates of
Figure 2. The (extended) Dynkin diagram of E7. The su(2)2 is generated by α1, and su(6)1 by α3,4,5,6,7 (su(5)GUT without α7). The highest root θ satisfies −θ + 2α1 + 3α2 + 4α3 + 3α4 + 2α5 + α6 + 2α7 = 0. 2-cycles −C2 −C2 + (C1 + 2C2) is not topologically trivial. Diagonal entries can be generated only by M2 branes that once sweep a broad region over the base manifold Q so that discrete holonomy between ~ζ 2 (~y) and ~ζ 1 (~y) + ~ζ 2 (~y) of the 2-fold cover can convert C2 to C1 + C2 or vice versa. Thus, when diagonal entries are generated, they are suppressed exponentially by the volume M2
Figure 3. The extended Dynkin diagram of E8 (I). The subalgebra su(2)2 + su(2)1 + su(6)1 is generated by the simple roots α1, −θ, and α3,4,5,6,7,8, respectively. α3 α4 α5 α6 α7 −θ α′ α1 α+
Figure 4. The extended Dynkin diagram of E8 (II), and its maximal subalgebra so(10)+su(4). α+ ≡ L0 − (L3 + L4 + L5). structure group of the vector bundle is generated by the simple roots −θ, α ′ and α1, where a root α ′ is defined in the appendix A. As the two rank-2 vector bundles 21 and 22 are extended to a rank-4 bundle U4 in Heterotic description, the two 2-fold cover data, or two doublets of 2-cycles (C7, C7 + C−θ) and (C2, C1 + C2) should be paired up into a quartet in M-theory description; indeed, the quartet should be (C7, C7 +C−θ, C7 +C−θ +Cα′, C7 +C−θ +Cα′ +C1) and since α7 + (−θ) +…
Figure 5. su(3)2 is generated by the roots α ′′ and −θ, and su(2)2 by α1 (see the appendix A). The replacing the bundle 1 ⊕ 21 by U3 [(22) by (30)] in the Heterotic side corresponds to mixing up the 2-cycle −(2C8 + 2C1 + 4C2) ≡ C7 − Cα ′′ with the S (1) 2 doublet (C7, C7 + C−θ) globally on Q to be a triplet of the Weyl group S3 generated by α ′′ and −θ. The data (−(2~ζ 8+2~ζ 1+4~ζ 2 ), ~ζ 7 , ~ζ 7+~ζ −θ ) gives a 3-fold cover of Q. The S (2) 2 -doublet (−(C1+C2), −C2) remains intact in the process of lifting the “2+2+1 bundles” to “3+2 bundles.” The transition function in (105)

Figures reproduced from arXiv:hep-th/0602238 by the authors.

References
How the literature uses it

25 arXiv papers cite this in 107 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 80 sentences without a confident label

Builds on it 2 papers
Uses its method or tool 1 paper
Uses its data or values 1 paper
Compares with it 1 paper
Cites it as support 5 papers
  • Without a fine-tuning of expectation values of gauge-field moduli, we can obtain $N^c ≠0$ while $N=0$; $N^c ≠0$ is crucial for neutrino masses as we have seen above, and $N=0$ is crucial for the absence of trilinear R-parity violation [ref] .

    A Theoretical Framework for R-parity Violation · 4+1 Model and 3+2 Model · 0.96

  • Either $N^c$ or $N$ develop non-vanishing expectation values to balance the non-vanishing Fayet–Iliopoulos parameter $ξ∝ω∧F$, and then the counter part acquire Majorana masses [ref] .

    Right-handed Neutrinos in F-theory Compactifications · sub-bundle with vanishing first Chern class in the fiber · 0.96

  • Note that Wilson lines can be a solution to the doublet–triplet splitting problem only when a pair of Higgs doublets $H_u$ and $H_d$ is completely vector like under the underlying gauge symmetry such as $E_8$ (and in fact, $E_8$ is the only candidate of the underlying gauge symmetry if we assume $(5)_GUT$ unification and the vector-like nature of $H_u$ and $H_d$; see [ref] ).

    GUT Relations from String Theory Compactifications · Introduction · 0.97

Cites it as background 10 papers

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