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Realistic Yukawa Couplings through Instantons in Intersecting Brane Worlds

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Steven A. Abel, Mark D. Goodsell

High Energy Physics - Theory · Submitted 12 Dec 2006 · 50 references

Pith's verdict: A genuine new amplitude with a plausible but unproven rank-three claim; the technical appendices are the durable part.

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arXiv IDhep-th/0612110 (version 3)
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CitationMark D. Goodsell, Steven A. Abel. "Realistic Yukawa Couplings through Instantons in Intersecting Brane Worlds." arXiv:hep-th/0612110 (2006). https://pith.science/paper/DFGXBGPP
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The paper

The paper claims

The paper argues that a one-loop E2-instanton contribution can turn the rank-one Yukawa matrix of simple intersecting D6-brane models into a rank-three matrix, giving the lighter generations masses and mixing.

The reading

The simplest intersecting D6-brane models on toroidal orientifolds have a rank-one Yukawa matrix: only one generation gets a mass and there is no mixing. This paper asks whether a Euclidean D-brane instanton, an E2-brane wrapping a three-cycle, can repair that by contributing at one loop through an annulus diagram with three matter fields on the D6-brane boundary. The authors argue that the instanton correction is a sum of six factorized rank-one terms, so the full Yukawa matrix becomes rank three, and that the correction is exponentially suppressed by the instanton volume. If correct, the result would tie the observed mass hierarchy to the geometry of the extra dimensions; the paper does not yet explain the splitting between the first two generations.

What carries the argument

The calculation combines E2-instanton calculus with one-loop intersecting-brane amplitude technology. The annulus correlator contains two fermionic supersymmetry zero modes and three matter vertex operators, and three picture-changing operators must act on different tori to cancel the internal $H$-charge. The worldsheet instanton action and the displacement vectors enter through monodromies of $partial bar X$; each of the six cyclic assignments of picture-changing operators gives a separate factorized contribution of the form $A_i^a B_i^b$. The paper also supplies the Pfaffian, determinant, and partition-function factors needed for instanton calculus in this class of background.

Extended reading notes

Core claim

The central claim is that an E2 instanton which does not intersect any D6 brane can contribute to the Yukawa superpotential at one loop through an annulus whose inner boundary is the instanton and whose outer boundary carries three matter vertex operators. The resulting correction is written as $Y_{ab}=Y^0_{ab}+Y^{np}_{ab}$, with $Y^{np}_{ab}$ a sum of six factorized rank-one contributions, $Y^{np}_{ab} = sum_{i=1}^{6} A_i^a B_i^b$, so that the full matrix has rank three even when the tree-level $Y^0$ has rank one. The correction is weighted by $e^{-S^0_{E2}}$, where $S^0_{E2}$ is the instanton action, so it is naturally smaller than the perturbative coupling and can produce a hierarchy.

Load-bearing premise

The load-bearing premise is that the six terms in $Y^{np}_{ab} = sum_{i=1}^{6} A_i^a B_i^b$ are linearly independent rank-one matrices; the paper labels its master amplitude in Eq. (3.9) 'schematically written' and does not evaluate the moduli integrals that would establish this independence.

Editorial extensions

If this is right

  • The corrected Yukawa matrix has rank three, so the first and second generations acquire nonzero masses and the mixing matrix can be non-trivial.
  • The nonperturbative correction is exponentially smaller than the tree-level top Yukawa, giving a geometric reason for the top-to-light mass hierarchy.
  • The same instanton volume controls both the light-generation suppression and the Majorana neutrino masses, so the two hierarchies should be related.
  • Yukawa measurements would carry information about the sizes of the extra dimensions and the positions of branes.
  • Models must contain an E2 brane with no intersections with the D6 branes, $b_1(Pi_{E2})=0$, and orientifold invariance, which is a new model-building constraint.

Reading between the lines

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

  • The decisive check is whether the six permutation terms are linearly independent; if they are, the mechanism is a genuine rank-three deformation, and if not the rank-three conclusion has to be weakened.
  • Because the correction is exponential in a cycle volume, a measured pattern of light-generation masses could be read backwards as a constraint on the compactification geometry.
  • The same annulus topology could contribute to other three-field superpotential couplings, such as neutrino mass terms, whenever the E2 zero-mode and intersection conditions are satisfied.
  • A model-building search could ask whether the required non-intersecting E2 brane with $b_1(Pi_{E2})=0$ and orientifold invariance is common or rare among realistic toroidal orientifolds.
Original abstract

The Yukawa couplings of the simpler models of D-branes on toroidal orientifolds suffer from the so-called ``rank one'' problem -- there is only a single non-zero mass and no mixing. We consider the one-loop contribution of E2-instantons to Yukawa couplings on intersecting D6-branes, and show that they can solve the rank one problem. In addition they have the potential to provide a geometric explanation for the hierarchies observed in the Yukawa coupling. In order to do this we provide the necessary quantities for instanton calculus in this class of background.

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, 3,613 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.67 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

The Yukawa couplings of the simpler models of D-branes on toroidal orientifolds suffer from the so-called ``rank one'' problem -- there is only a single non-zero mass and no mixing. We consider the one-loop contribution of E2-instantons to Yukawa couplings on intersecting D6-branes, and show that they can solve the rank one problem. In addition they have the potential to provide a geometric explanation for the hierarchies observed in the Yukawa coupling. In order to do this we provide the necessary quantities for instanton calculus in this class of background.

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 The Yukawa couplings of the simpler models of D-branes on toroidal orientifolds suffer from the so-called ``rank one'' problem -- there is only a single non-zero mass and no mixing. background 0.96 0.06 0.03 asserted 1.00
s1 We consider the one-loop contribution of E2-instantons to Yukawa couplings on intersecting D6-branes, and show that they can solve the rank one problem. contribution 0.70 0.95 0.87 asserted 1.00
s2 In addition they have the potential to provide a geometric explanation for the hierarchies observed in the Yukawa coupling. interpretation 0.87 0.51 0.12 hedged possibility 1.00
s3 In order to do this we provide the necessary quantities for instanton calculus in this class of background. contribution 0.51 0.90 0.03 asserted 0.94

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

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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. Contributions to 3 point functions: tree level and one-loop instanton. terms are induced by E2 instantons which do not intersect the D6-branes. The tree and one-loop contributions to Yukawa couplings are shown in figure 1; the tree level diagram consists of the usual disc diagram with three vertices, the one loop diagram is an annulus with the inside boundary being an E2 brane and three vertices on the D6- brane boundary. By explicit computation of these one-loop instanton contributions we show that the corrected Yukawas are of the form Yab = Y 0 ab + Y np ab , where the nonpertur

Figures reproduced from arXiv:hep-th/0612110 by the authors.

References
How the literature uses it

49 arXiv papers cite this in 78 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 · 53 sentences without a confident label

Builds on it 2 papers
  • * We generalize the analysis of [ref] (see also [ref] ) to show that for any compactification of type II string theory on Calabi-Yau orientifolds preserving $=1$ supersymmetry, the partition functions $Z_A$ and $Z_M$ associated with D-instanton corrections are equal to the threshold correction to the gauge coupling when we replace the D-instanton by a space-filling D-brane.

    D-instanton Induced Superpotential · Introduction · 1.00

  • What we have found is a generalization to the quartic term of the relation between instantonic one-loop amplitudes and gauge threshold corrections to the quadratic action, originally found in [ref] and further elaborated in [ref] .

    Classical solutions for exotic instantons? · Background fields vs D-instantons · 0.86

Uses its method or tool 1 paper
  • Regularization with Wilson lines We now briefly describe the effect of introducing Wilson lines on the torus $𝒯_2^(3)$ which can act as IR regulators [ref] for the integral $I$ in Eq. ([ref]).

    Instantons in N=2 magnetized D-brane worlds · Calculation of the integral $I$ · 0.88

Compares with it 1 paper
Cites it as support 3 papers
  • The interpretation of the D(–1)'s as a classical configuration of action ([ref]) is further confirmed by the relation between instantonic one-loop amplitudes and gauge threshold corrections to the quadratic action, as originally found in [ref] .

    Classical solutions for exotic instantons? · The D7/D(–1) system in Type I$^'$ · 0.99

  • This correspondence was also explicitly checked for some cases in [ref] .

    D-instanton Induced Superpotential · Holomorphy of the superpotential · 0.99

  • (The upper index $^⋆$ in ([ref]) will be explained momentarily.) In [ref] it was shown that these vacuum diagrams are related to the gauge threshold corrections associated with the auxiliary D-brane $D_E$.

    D-brane Instantons in Type II String Theory · Superpotential calculus · 0.96

Cites it as background 17 papers
  • In recent years new ways to compute non-perturbative effects in string theory were developed, based on Euclidean $p$-branes ($Ep$-branes) wrapping various cycles of the internal manifold of string compactifications [ref] .

    String instantons, fluxes and moduli stabilization · Introduction and Conclusions · 0.98

  • In fact, during the last year there has been some progress towards a better understanding of non-perturbative effects in $N=1$ supersymmetric four-dimensional string compactifications on Calabi-Yau orientifolds ( [ref] and further developments in [ref] ; for closely related earlier work see [ref] ).

    Non-perturbative Yukawa Couplings from String Instantons · Introduction · 0.83

  • Various applications of these novel non-perturbative effects in different branches of the string landscape have appeared [ref] [For two reviews on the subject of novel stringy instanton effects see [ref] .

    Stringy Instantons and Yukawa Couplings in MSSM-like Orientifold Models · Introduction · 0.89

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