Pith. Sign in

REVIEW

A Model of Muon Anomalies

PaperPDF from arXiv

data files

ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Admir Greljo, Peter Stangl, Anders Eller Thomsen

High Energy Physics - Phenomenology · Submitted 25 Mar 2021 · 148 references

Pith's verdict: A well-built synthesis of gauged B−3L_mu with two muoquarks; the proton-decay argument is the real original contribution, and the 'evades all other constraints' line leans on a flavor assumption the paper discloses but does not demonstrate.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 31, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

  1. T0 AI review An AI read the paper and answered a fixed list of questions.
  2. T1 Audit Automatic checks that the references and files the paper relies on exist and match.
  3. T2 Compute The paper's code or calculations re-run with the same results.
  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 12 minor · Ask about this paper · The compiled record · Read the full review →
Challenge this review Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID2103.13991 (version 2)
Pith IDDFTM4S5S, Pith's permanent record of this paper and its review
CitationAdmir Greljo, Anders Eller Thomsen, Peter Stangl. "A Model of Muon Anomalies." arXiv:2103.13991 (2021). https://pith.science/paper/DFTM4S5S
Cite this reviewBibTeX for Pith's review of this paper
Claim paper

Why Pith says this

Highlight any text to comment on it.

Referee report

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

A single gauged muon-number symmetry plus two scalar leptoquarks can account for the muon magnetic moment, the $R_{K^{(*)}}$ ratios, and the $b to s mu^+ mu^-$ angular data at once, without proton decay or charged lepton flavor violation at

The reading

The paper asks whether three muon-flavored tensions — the anomalous magnetic moment of the muon, the $R_{K^{(*)}}$ ratios in rare $B$ decays, and the angular distributions of $b o s mu^+ mu^-$ — can come from one renormalizable extension of the Standard Model rather than three unrelated fixes. The proposed answer is to gauge $U(1)_{B-3L_mu}$, a symmetry that counts baryon number minus three times muon number, and to add two scalar leptoquarks that carry its charge and therefore couple only to second-generation leptons; the authors call such fields muoquarks. Because the gauge symmetry forbids the quark-quark-leptoquark couplings that normally let leptoquarks destroy the proton, and forbids the couplings that normally generate charged lepton flavor violation, the model keeps the Standard Model's accidental symmetries at the renormalizable level while still permitting lepton-flavor-universality violation. A global fit to low-energy data prefers couplings far from the Standard Model limit, with a $Delta chi^2 simeq 62$ improvement at 3 TeV leptoquark masses, and the authors argue the same parameter point survives renormalization-group running to the Planck scale.

What carries the argument

The load-bearing object is the muoquark: a scalar leptoquark whose $U(1)_{B-3L_mu}$ charge forbids any coupling to first- or third-generation leptons, so that its Yukawa couplings are muon-only by symmetry rather than by hand. That single assignment does double duty — it supplies the lepton-flavor-universality violation needed for $R_{K^{(*)}}$ and $(g-2)_mu$ while making charged lepton flavor violation a loop-suppressed accident, and it removes the $qqS$ and $qell S$ operators that would otherwise mediate proton decay at dimension four. The quark flavor structure of the same couplings is then assumed to follow the $U(2)^3$ minimal-flavor-violation spurion pattern of the quark sector, with left-handed couplings of order $O(V)oplus 1$ and right-handed ones of order $O(Delta^dagger V)oplus 1$, which is what suppresses $K_L o mu^+mu^-$, meson mixing, and the other complementary observables. The quantitative case is built by matching the $S_1+S_3$ Lagrangian onto SMEFT at one loop, feeding the Wilson coefficients through a chain of running and low-energy matching tools, and fitting to a large observable set; the ultraviolet case is built by running the full model with three-loop gauge and two-loop Yukawa and quartic $\beta$ functions.

Extended reading notes

Core claim

The central claim is that a renormalizable model — the Standard Model plus three right-handed neutrinos, a gauged $U(1)_{B-3L_mu}$ broken above the TeV scale by a singlet scalar $Phi$, and two scalar leptoquarks $S_1=(bar{mathbf 3},mathbf 1,1/3)$ and $S_3=(bar{mathbf 3},mathbf 3,1/3)$ carrying $+8/3$ under the new symmetry — simultaneously fits the muon $(g-2)$, $R_{K^{(*)}}$, and $b o s mu^+ mu^-$ anomalies while evading every other constraint the authors check. The division of labor is clean: tree-level $S_3$ exchange produces the semileptonic $b o s mu^+ mu^-$ Wilson coefficients that drive $R_{K^{(*)}}$, while a loop of $S_1$ with a top quark in the loop supplies the chirally enhanced contribution to $(g-2)_mu$. At $M_1=M_3=3$ TeV the best-fit point is $(eta_{3L3},,eta_{1L3}=eta_{1R3}) simeq (0.43,,0.12)$, and the fit is quoted as a $Delta chi^2 simeq 62$ improvement over the Standard Model. The authors further claim that proton decay is pushed to dimension six, that the neutrino sector works through a type-I seesaw with a two-zero-minor-like structure, and that all couplings can be run to the Planck scale without Landau poles or vacuum instability.

Load-bearing premise

The claim that the model evades all other constraints rests on assuming the muoquark Yukawa couplings inherit the $U(2)^3$ minimal-flavor-violation pattern of the quark sector — an assumption the authors call reasonable rather than derive — and on setting the right-handed coupling $tildeeta_{1R}$ to zero by hand.

Editorial extensions

If this is right

  • Direct leptoquark searches should find $S_3$ and $S_1$ within the reach of the high-luminosity LHC, since the naturalness argument from the Higgs mass and muon Yukawa corrections pushes both masses toward the TeV scale rather than far above it.
  • The model predicts no observable charged lepton flavor violation — $mu o e gamma$ and its relatives stay suppressed by the residual $U(1)_{L_mu}$ — so any future signal in those channels would count against this framework.
  • Proton decay remains absent up to dimension-six operators, so improved proton-lifetime limits test the construction only weakly, but they do discriminate it from leptoquark models built on other lepton-flavored gauge groups such as $U(1)_{L_mu-L_tau}$.
  • The same gauge symmetry can host a light vector boson explanation of $(g-2)_mu$ instead of $S_1$, giving a sharply predicted window near $m_X sim 20$ MeV and $g_X sim 5imes 10^{-4}$ that upcoming beam-dump and neutrino experiments can probe.
  • Adding a tau-philic $S_1$ extends the framework to the $b o c tau u$ charged-current anomaly without the tuned cancellation against $tau o mu gamma$ that earlier combined fits required.

Reading between the lines

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

  • The paper's real structural bet is that a gauge symmetry can substitute for the flavor puzzle: instead of explaining why leptoquark couplings happen to be muon-only and quark-hierarchical, it makes the first automatic and borrows the second from an assumed $U(2)^3$ structure — a template that could be reused for any future anomaly localized in one lepton flavor.
  • If the $U(2)$ assumption is the weak link, then the model's fate may be decided less by leptoquark searches than by precision measurements of $K_L o mu^+mu^-$ and $B_s$ mixing, which are exactly the observables that would grow if the quark couplings were generic; those measurements are the sharpest indirect probe of the flavor assumption rather than of the gauge symmetry.
  • Because the new symmetry is anomaly-free only with the specific charge assignment $B-3L_mu$, the construction suggests a small classification problem the authors only begin: which anomaly-free lepton-flavored $U(1)$ groups admit a stable proton together with a viable type-I seesaw, and $U(1)_{L_mu-L_tau}$ appears to fail that test in its minimal form.
  • The radiative-stability result cuts both ways: the same large leptoquark charges that make the running to the Planck scale interesting also make the quartic couplings hit Landau poles for gauge couplings above roughly $0.15$, so a future discovery of a relatively strongly coupled $X$ boson in this framework would be in tension with the ultraviolet completion the authors claim.
Original abstract

The Standard Model (SM) is augmented with a $\mathrm{U}(1)_{B-3L_\mu} $ gauge symmetry spontaneously broken above the TeV scale when an SM-singlet scalar condenses. Scalar leptoquarks $S_{1(3)} = (\overline{\mathbf{3}},\, \mathbf{1} (\mathbf{3}),\, ^1\!/_3)$ charged under $\mathrm{U}(1)_{B-3L_\mu} $ mediate the intriguing effects observed in muon $(g-2)$, $R_{K^{(*)}}$ and $b \to s \mu^+ \mu^-$, while generically evading all other phenomenological constraints. The fermionic sector is minimally extended with three right-handed neutrinos, and a successful type-I seesaw mechanism is realized. Charged lepton flavor violation is effectively suppressed, and proton decay - a common prediction of leptoquarks - is postponed to the dimension-6 effective Lagrangian. Unavoidable radiative corrections in the Higgs mass and muon Yukawa favor leptoquark masses interesting for collider searches. The parameters of the model are radiatively stable and can be evolved by the renormalization group to the Planck scale without inconsistencies. Alternative lepton-flavored gauge extensions of the SM, under which leptoquarks become muoquarks, are proposed for comparison.

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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,882 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.27 · human participants 0.02 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.50 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

The Standard Model (SM) is augmented with a $\mathrm{U}(1)_{B-3L_\mu} $ gauge symmetry spontaneously broken above the TeV scale when an SM-singlet scalar condenses. Scalar leptoquarks $S_{1(3)} = (\overline{\mathbf{3}},\, \mathbf{1} (\mathbf{3}),\, ^1\!/_3)$ charged under $\mathrm{U}(1)_{B-3L_\mu} $ mediate the intriguing effects observed in muon $(g-2)$, $R_{K^{(*)}}$ and $b \to s \mu^+ \mu^-$, while generically evading all other phenomenological constraints. The fermionic sector is minimally extended with three right-handed neutrinos, and a successful type-I seesaw mechanism is realized. Charged lepton flavor violation is effectively suppressed, and proton decay - a common prediction of leptoquarks - is postponed to the dimension-6 effective Lagrangian. Unavoidable radiative corrections in the Higgs mass and muon Yukawa favor leptoquark masses interesting for collider searches. The parameters of the model are radiatively stable and can be evolved by the renormalization group to the Planck scale without inconsistencies. Alternative lepton-flavored gauge extensions of the SM, under which leptoquarks become muoquarks, are proposed for comparison.

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 Standard Model (SM) is augmented with a $\mathrm{U}(1)_{B-3L_\mu} $ gauge symmetry spontaneously broken above the TeV scale when an SM-singlet scalar condenses. contribution 0.76 0.47 0.08 asserted 1.00
s1 Scalar leptoquarks $S_{1(3)} = (\overline{\mathbf{3}},\, \mathbf{1} (\mathbf{3}),\, ^1\!/_3)$ charged under $\mathrm{U}(1)_{B-3L_\mu} $ mediate the intriguing effects observed in muon $(g-2)$, $R_{K^{(*)}}$ and $b \to s \mu^+ \mu^-$, while generically evading all other phenomenological constraints. contribution 0.79 0.57 0.63 asserted 0.97
s2 The fermionic sector is minimally extended with three right-handed neutrinos, and a successful type-I seesaw mechanism is realized. contribution 0.70 0.66 0.06 asserted 1.00
s3 Charged lepton flavor violation is effectively suppressed, and proton decay - a common prediction of leptoquarks - is postponed to the dimension-6 effective Lagrangian. result 0.91 0.59 0.07 asserted 1.00
s4 Unavoidable radiative corrections in the Higgs mass and muon Yukawa favor leptoquark masses interesting for collider searches. result 0.56 0.64 0.07 asserted 0.98
s5 The parameters of the model are radiatively stable and can be evolved by the renormalization group to the Planck scale without inconsistencies. result 0.93 0.77 0.11 asserted 0.96
s6 Alternative lepton-flavored gauge extensions of the SM, under which leptoquarks become muoquarks, are proposed for comparison. contribution 0.78 0.65 0.04 asserted 0.88

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from argument 0.63.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
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. FIG. 1. The preferred muoquark Yukawa couplings from
Figure 2. FIG. 2. The RG flow of a selection of couplings from the
Figure 3. FIG. 3. Allowed parameter space for the light
Figure 4. FIG. 4. Preferred regions for the Yukawa couplings

Figures reproduced from arXiv:2103.13991 by the authors.

References
How the literature uses it

52 arXiv papers cite this in 86 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 whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

Papers citing it · 68 sentences without a confident label

Builds on it 1 paper
Cites it as support 3 papers
Cites it as background 12 papers

What it cites · 152 sentences without a confident label

Builds on 4 papers
Uses the method or tool of 12 papers
  • After we pass the SMEFT Wilson coefficients, which we compute from the parameters of our Lagrangian [ref] at the matching scale $μ_M$, to smelli, this tool automatically takes care of the renormalization group running down to the meson scale as well as the intermediate matching to the low-energy EFT [ref] thanks to the wilson [ref] package.

    Renormalization Group Evolution of the Standard Model Dimension Six Operators II: Yukawa Dependence · Muoquark solution of the muon anomalies · 0.97

  • After we pass the SMEFT Wilson coefficients, which we compute from the parameters of our Lagrangian [ref] at the matching scale $μ_M$, to smelli, this tool automatically takes care of the renormalization group running down to the meson scale as well as the intermediate matching to the low-energy EFT [ref] thanks to the wilson [ref] package.

    Renormalization Group Evolution of the Standard Model Dimension Six Operators I: Formalism and lambda Dependence · Muoquark solution of the muon anomalies · 0.97

  • After we pass the SMEFT Wilson coefficients, which we compute from the parameters of our Lagrangian [ref] at the matching scale $μ_M$, to smelli, this tool automatically takes care of the renormalization group running down to the meson scale as well as the intermediate matching to the low-energy EFT [ref] thanks to the wilson [ref] package.

    Low-energy effective field theory below the electroweak scale: matching at one loop · Muoquark solution of the muon anomalies · 0.97

Uses data or values from 6 papers
  • We update the measurements included in smelli and take into account the most recent results for $R_K$ [ref] and $(g-2)_μ$ [ref] as well as the current world average of BR$(B_s,d→μμ)$ from [ref] , which includes the most recent LHCb measurement [ref] .

    2103.11769 · Muoquark solution of the muon anomalies · 1.00

  • We update the measurements included in smelli and take into account the most recent results for $R_K$ [ref] and $(g-2)_μ$ [ref] as well as the current world average of BR$(B_s,d→μμ)$ from [ref] , which includes the most recent LHCb measurement [ref] .

    New Physics in Rare B Decays after Moriond 2021 · Muoquark solution of the muon anomalies · 1.00

  • We update the measurements included in smelli and take into account the most recent results for $R_K$ [ref] and $(g-2)_μ$ [ref] as well as the current world average of BR$(B_s,d→μμ)$ from [ref] , which includes the most recent LHCb measurement [ref] .

    2104.03281 · Muoquark solution of the muon anomalies · 1.00

Cites as support 1 paper
Cites as background 9 papers
  • The SM Yukawa interactions exhibit a good approximate flavor symmetry $(2)_q ×(2)_u ×(2)_d$ under which the first two generations transform as doublets, while the third generation is a singlet [ref] (see also [ref] ).

    General Minimal Flavor Violation · Muoquark solution of the muon anomalies · 0.83

  • The SM Yukawa interactions exhibit a good approximate flavor symmetry $(2)_q ×(2)_u ×(2)_d$ under which the first two generations transform as doublets, while the third generation is a singlet [ref] (see also [ref] ).

    U(2) and Minimal Flavour Violation in Supersymmetry · Muoquark solution of the muon anomalies · 0.83

  • To name a few examples of explicit models, Ref. [ref] is based on the gauge group whereas Ref. [ref] works with $(1)_B_3 - L_2$.

    $U(1)_{B_3-L_2}$ Explanation of the Neutral Current $B-$Anomalies · The scenarios for muon anomalies · 0.84

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.