Vector-like quarks with non-renormalizable interactions

Juan Carlos Criado; Manuel Perez-Victoria

REVIEW 1 major objections 6 minor 57 references

Vector-like quarks with non-renormalizable interactions

T0 review · 1 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Dropping the assumption of renormalizability turns some hypothetical heavy quarks into long-lived particles that escape the usual prompt-decay searches.

desk verdict Solid, systematic EFT paper that finds five vector-like quark multiplets whose leading couplings are dimension-5 and whose lifetimes can become long; the qualitative result is convincing, but the quantitative width table needs the missing formulas before publication. read the letter →

arxiv 1908.08964 v2 pith:HSXYL7N3 submitted 2019-08-23 hep-ph

Juan Carlos Criado , Manuel Perez-Victoria This is my paper · ORCID

classification hep-ph

keywords vector-likequarksnon-renormalizableinteractionsdimension-fiveoperatorseffectivefieldtheorylong-livedparticlesR-hadronsLHCphenomenologybranchingratios

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 paper asks what changes if vector-like quarks—heavy hypothetical quarks common in extensions of the Standard Model—are allowed to interact through non-renormalizable terms as well as the usual renormalizable ones. It argues that once dimension-five operators are included, five new multiplets of vector-like quarks, which the paper calls NRVLQ, can mix with and decay into Standard Model quarks even though they have no dimension-four Yukawa coupling. Because their leading interactions are suppressed by inverse powers of the cutoff scale Λ, their decay widths shrink as Λ grows; for Λ around or above 5 TeV the quarks hadronize before decaying, and for much larger Λ they become long-lived. The paper works out the resulting collider phenomenology: pair production remains, new single-production and decay channels appear, branching ratios no longer obey the standard sum rule, and long-lived signatures such as displaced vertices, anomalous ionization, and delayed time-of-flight replace the prompt decays that current searches assume.

What carries the argument

The key machinery is the dimension-five effective Lagrangian for a single extra quark multiplet, truncated at order $1/\Lambda$, with two classes of operators: Yukawa-type $\bar Q q \phi \phi$ and magnetic-type $\bar Q \sigma^{\mu\nu} q F_{\mu\nu}$, plus the quadratic $\bar Q Q \phi \phi$ and $\bar Q \sigma^{\mu\nu} Q F_{\mu\nu}$ terms. The classification of which multiplets can have linear couplings rests on the representation condition $T + Y + 1/3 \in \mathbb{Z}$ for colour triplets, proved in the paper for products of Standard Model fields; applying it at dimension five adds the five NRVLQ multiplets to the seven renormalizable ones. The power-counting mechanism that carries the long-lifetime claim is that every leading NRVLQ interaction carries at least one inverse power of $\Lambda$, so decay amplitudes and mixing angles receive a suppression of order $v^2/(\Lambda M)$ or $v/\Lambda$; the total width is a decreasing function of $\Lambda$, crossing the QCD scale and then the displaced-vertex and detector-stability scales as shown in Figure 7 and Table 6 of the paper.

What would settle it

Measure the decay length of pair-produced heavy quarks in the T4 or F5 models with M=2 TeV and 1/y≈5 TeV: the paper predicts widths at or below Λ_QCD≈0.2 GeV, so events should show hadronization, displaced vertices, or late ionizing tracks; observing prompt decays at these parameters would falsify the extrapolated widths. A direct computation of the full partial widths (two-body and three-body) for these multiplets at the Table 6 values of 1/y would also settle the crossing points.

Watch

Extended reading notes

Core claim

The central claim is that relaxing renormalizability while keeping gauge invariance and a cutoff Λ above the heavy-quark mass M enlarges the list of vector-like quark multiplets with linear couplings to Standard Model fields from seven at dimension four to twelve at dimension five. The five new multiplets—two triplets and three quadruplets, denoted T4, T5, F1, F5, F7—have no renormalizable linear interaction, so their leading couplings are dimension-five operators of the forms $\bar Q q \phi \phi$ and $\bar Q \sigma^{\mu\nu} q F_{\mu\nu}$. These operators generate mixing with the third generation suppressed by $y v^2/M$ (with $y$ a coupling of dimension inverse mass), so the new quarks evade precision constraints without tuning, while pair production via QCD remains unsuppressed. The decisive consequence is the cutoff dependence of the lifetime: for natural couplings and $\Lambda \gtrsim 5$ TeV the width falls below the QCD scale, so hadronization precedes decay, and for $\Lambda$ around $10^6$ TeV the resulting R-hadrons are long-lived within detector distances and would appear as tracks with anomalous ionization, long time of flight, or displaced vertices rather than as prompt decays. The paper also shows that non-renormalizable operators open new single-production channels and new decay modes, so the standard branching-ratio triangle is replaced by a tetrahedron, and provides the recasting formula $M_\Sigma = (M_1^{1/2} + f^{1/2}\log\Sigma)^2$ for LHC mass limits when the sum of branching ratios to $Hq$, $Zq$, $W^\pm q'$ is $\Sigma<1$.

Load-bearing premise

The long-lifetime claim rests on the assumed parametric size and scaling of the NRVLQ decay widths, whose sub-QCD values in Table 6 are obtained by extrapolation; if the true widths are larger (from neglected four-fermion operators, non-natural couplings, or higher-order corrections), the lifetimes shorten and the boundary where usual searches stop working moves to higher Λ or disappears.

Editorial extensions

If this is right

Pair production of NRVLQ through QCD remains the main production channel up to about 3.5 TeV, so LHC searches for pair-produced heavy quarks can probe them even when single production is suppressed.
For NRVLQ, the branching ratios to $Hq$, $Zq$ and $W^\pm q'$ no longer sum to one; the recasting formula $M_\Sigma = (M_1^{1/2}+f^{1/2}\log\Sigma)^2$ converts existing LHC mass limits into bounds for the enlarged parameter space.
For $\Lambda \gtrsim 10^6$ TeV, the hadrons containing NRVLQ are effectively stable in the detector, so searches for long-lived coloured particles—anomalous ionization, time-of-flight, displaced vertices—become the relevant probes, with an estimated lower mass bound near 1.5 TeV from reinterpreted LHC limits.
Non-renormalizable interactions also modify Higgs physics: the $\bar Q Q \phi \phi$ coupling $Y_1$ contributes to $gg\to H$ and $H\to gg$ at one loop with a coefficient unsuppressed by mixing, giving a bound $|{\rm Re}\,Y_1|/M \lesssim 1/[(2T+1)(1.25\ {\rm TeV})^2]$.
New decay channels such as $T\to b W^+ Z$, $T\to b H W^+$, $B\to t H W^-$, $B\to t Z W^-$, and $\gamma$/$g$ plus a third-generation quark can have branching ratios above 0.01 and alter the standard branching-ratio triangle.

Reading between the lines

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

Editorial extension: if the long-lifetime regime is realized, existing searches for long-lived supersymmetric particles can be reinterpreted to constrain vector-like quarks, but the distinctive final states (e.g., $Ht$, $Zt$, $W b$) require dedicated displaced-vertex analyses; the paper's estimate of a ~1.5 TeV lower bound is only a first approximation.
Editorial extension: the long-lifetime conclusion is sensitive to the assumed scaling of the width; dimension-six four-fermion operators of the form $qqqQ$, which the paper leaves for future work, could provide additional decay channels and shorten the lifetime, so the window $\Lambda \gtrsim 10^6$ TeV is robust only if those operators are suppressed.
Editorial extension: the approximate equality of $Hq$ and $Zq$ branching ratios, traced in the paper to an isospin condition that holds for all multiplets except $F_1$, could be used experimentally as a quantum-number diagnostic: measuring that ratio distinguishes the $F_1$ quadruplet from the other NRVLQ candidates.
Editorial extension: the NRVLQ framework naturally embeds in pseudo-Goldstone composite Higgs models with $\Lambda=f$; in low-$f$ versions the width is not suppressed enough for long lifetimes, so the long-lived window selects UV completions where the dimension-five operators are generated at a genuinely high scale, such as tree-level heavy-scalar exchange.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing to know: this is a solid EFT paper doing something genuinely new. It classifies all dimension-5 interactions of vector-like quarks, identifies five multiplets (T4, T5, F1, F5, F7) that only have linear couplings at that order, and shows their decay widths can be suppressed enough to make R-hadron, displaced-vertex, and long-lived signatures relevant. The reader's conditional verdict is right. The machinery is standard but applied carefully: mass matrices, SMEFT matching, branching-ratio triangles, and a genuinely useful formula to recast LHC pair-production limits when the branching ratios no longer sum to one. I checked that recast logic; the exponential approximation for the production cross section with f = 20.5 GeV is a reasonable interpolation, and the claimed 3% agreement over the relevant mass range is credible.

Credit where it is due: the classification itself, the inclusion of the magnetic operators, the treatment of the dimension-5 Yukawa operators, and the observation that the no-dimension-4 multiplets have widths set by y^2 v^2 M/(16 pi), so lifetimes grow roughly as 1/y^2. The qualitative long-lifetime conclusion follows from power counting and is robust to factors of a few in the width. The paper also states its limitations honestly about extrapolating below the QCD scale.

The soft spot is exactly where the reader put it. Figure 7 and Table 6 quote widths and 1/y thresholds, but the width formulas are not shown. The text says the sub-QCD values are obtained by extrapolation from larger couplings. I do not think the central claim collapses: a factor of a few in the width shifts the thresholds by a factor of 2-10, not orders of magnitude. But Table 6 is presented as quantitative, and right now it is not independently checkable. No code or data are shipped. This should be fixable by putting the total-width expressions in an appendix or providing a small calculation script. One minor slip: the conclusions say the five NRVLQs are "two triplets and four quadruplets"; the table shows three quadruplets. Cosmetic, but it will annoy a careful referee. The citation pattern is fine; the self-citations are to earlier dictionary/matching results, not inputs making this circular.

Bottom line: this is a BSM phenomenology paper for LHC exotica people and EFT practitioners. It deserves a serious referee. My recommendation: send it to peer review, and ask the authors to document the width calculation and correct the multiplet count before acceptance.

Referee Report

1 major / 6 minor

Summary. The paper develops a model-independent effective field theory for Standard Model extensions with vector-like quarks, truncated at canonical dimension 5. It classifies the multiplets that can have linear couplings to SM fields at that order, identifying seven 'renormalizable' vector-like quarks (RVLQ) and five 'non-renormalizable' vector-like quarks (NRVLQ). For each multiplet the authors present mass matrices, mixing angles, indirect constraints from electroweak and Higgs observables, LHC production cross sections, decay branching ratios, and a formula for recasting pair-production mass limits when additional decay channels are present. The central new claim is that NRVLQs, whose leading interactions are dimension-5, have widths suppressed by the cutoff scale and become long-lived for large Λ, so that the hadrons they form evade prompt-decay searches and give R-hadron-like signatures such as anomalous ionization, time-of-flight signals, and displaced vertices.

Significance. If the quantitative claims hold, this is a useful systematic addition to the vector-like quark literature. The paper provides a complete classification of multiplets and dimension-5 operators, explicit mass matrices, SMEFT-matching results, and a practical mass-limit recast formula in Eq. (49). The prediction that certain vector-like quarks can be long-lived for Λ around or above 10^6 TeV is falsifiable and differs qualitatively from the standard prompt-decay assumption. The qualitative lifetime suppression follows from power counting and is robust, and the paper is careful to distinguish renormalizable and non-renormalizable multiplets. The main weakness is that the quantitative thresholds in Table 6 and Figure 7, which are the numerical basis for the long-lifetime claim, are not backed by explicit width formulas or a documented extrapolation procedure.

major comments (1)

[Section 6, Figure 7, Table 6] The numerical backbone of the long-lifetime claim is not documented. Table 6 gives values of 1/y at which the total width equals Λ_QCD, 10^-12 GeV, and 10^-16 GeV (for example, 5.3 TeV for a T5 T, and 10^6-10^8 TeV for the displaced-vertex and detector-stable thresholds), and Figure 7 plots the total widths, but no partial- or total-width formulas are shown. The text says only that the sub-QCD values were 'obtained by extrapolation of the results calculated for larger couplings.' Since the abstract and conclusions present the long lifetime as the most dramatic effect, the authors should provide the explicit width formulas used, specify the extrapolation method, and quantify the sensitivity to neglected dimension-6 operators and to O(Λ_QCD/M) corrections. Without this, the numerical thresholds in Table 6 cannot be checked or reproduced.

minor comments (6)

[Section 4] The statement that NRVLQs contribute to the SMEFT at tree level only from dimension 8 is not correct as stated: connecting two dimension-5 Qbar-q-phi-phi type vertices with a heavy-quark propagator generates a dimension-7 operator of the form qbar-q-phi^4, not dimension 8. This does not change the qualitative conclusion that the indirect effects are small, but the sentence should be corrected or qualified.
[Table 1] The symbols in the last three columns of Table 1 are not legible in the typeset text and their meaning is not defined; please use explicit check and cross symbols or add a legend.
[Figure 7] The caption of Figure 7 says 'total decay width of T and B', but the six panels display X', X, T, B, Y, and Y'; the caption should be updated to describe all panels.
[Equations (36) and (49)] The typesetting of the recast formula obscures the exponents; it should read M_Sigma = (M_1^{1/2} + f^{1/2} log Sigma)^2.
[Section 6] Please state the proper decay length corresponding to the thresholds Lambda_disp and Lambda_longlived (for example, c tau values) so that the relation between Table 6 and the detector-stability discussion is transparent.
[Various] There are several typographical errors: 'loosing' near the end of Section 2, 'the later eventually becomes' in Section 5, 'proceses' in Section 7, 'statisﬁed' in Appendix C, and 'The always decay' in Section 6.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the long-lifetime prediction follows from the explicit dimension-5 Lagrangian and is not fitted to the data it explains.

full rationale

The paper's central claims are derived from an explicit effective Lagrangian (eqs. 3-16) truncated at dimension 5, with the NRVLQ multiplets defined by the absence of dimension-4 linear couplings. The long-lifetime prediction follows from the parametric suppression of dimension-5 couplings in decay widths, which are computed from this Lagrangian rather than fitted to lifetime data. The sub-QCD width values in Table 6 are explicitly flagged as an extrapolation of results calculated for larger couplings, and the figure and table are presented as outputs of the width calculation; the missing width formulas are a reproducibility and documentation concern, not a circular one. The recast mass-limit formula in Appendix B takes an external LHC bound M1 and a MadGraph-based cross-section fit (f = 20.5 GeV) as inputs, and no parameter of that formula is fitted to the predicted branching ratios or lifetimes. Self-citations (refs. [4], [10], [26]) provide prior independent results for the renormalizable vector-like quark framework, electroweak precision limits, and SMEFT matching; they are used as tools or cross-checks, not as the justification for the paper's novel conclusions. No equation in the paper is equivalent to its own input by construction, so the derivation is self-contained and not circular.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claims are conditional on the EFT assumptions listed above. The most load-bearing are naturalness of couplings, the dimension-5 truncation, and the width calculations behind the lifetime thresholds. No new particles or entities are invented beyond the existing vector-like quark framework.

free parameters (6)

lambda (dim-4 Yukawa coupling) = saturates electroweak precision bounds, not numerically quoted
Sets mixing angles and branching ratios for RVLQ; treated as a free input in Table 7 and the branching-ratio figures.
y (dim-5 Yukawa-type coupling) = (2 TeV)^-1 in most plots and tables; (4 TeV)^-1 in some production plots
Controls NRVLQ widths, lifetimes, and branching ratios; it is the key parameter behind the long-lived conclusions.
w (dim-5 magnetic dipole coupling) = (2 TeV)^-1 or (4 TeV)^-1 in examples; loop-suppressed values also discussed
Generates new single-production and decay channels; no independent measured value is available.
Y1, Y2 (dim-5 Q-Q-H-H couplings) = constrained by eq. (32), otherwise free
Shift heavy-quark masses and contribute to Higgs gluon fusion; free parameters of the effective theory.
f (cross-section exponential scale) = 20.5 GeV
Used in the mass-limit recast formula in Appendix B; fitted to MadGraph output over the mass range 0.8 to 1.4 TeV.
cutoff Lambda = at least 2 TeV in numerical examples; at least 10^6 TeV for the long-lived regime
Sets the suppression of all dimension-5 effects; chosen, not fitted, and central to the lifetime statements.

assumptions (6)

domain assumption The effective theory is valid below a cutoff Lambda larger than all masses, and SM gauge symmetry is linearly realized.
Invoked in Section 2 to justify the 1/Lambda expansion and the operator basis.
domain assumption The new quarks are vector-like color triplets with at least one linear gauge-invariant coupling to SM fields.
This restriction selects the finite list of multiplets in Table 1; stated in Section 2.
domain assumption Only couplings to the third SM family are kept; couplings to the first two families are set to zero.
Stated in Section 2 to reduce parameters and satisfy flavor limits; this choice affects single-production rates.
domain assumption The Lagrangian is truncated at dimension 5, and dimension-6 four-fermion operators are neglected.
Section 4 states the neglect of dimension-6 terms and assumes no cancellations with the tree-level matching; Section 7 mentions qqqQ operators as future work.
domain assumption Mass-matrix mixing is dominated by one off-diagonal element and CP phases are set to zero.
Used in Section 3 and Appendix C to derive simple mixing angles and the approximate equality of Hq and Zq branching ratios.
domain assumption Couplings are natural: dimensionless couplings are order one and dimensionful couplings are order 1/Lambda, with w loop suppressed in weakly coupled completions.
Underlies the central long-lifetime estimate; stated in Sections 1, 2, and 5.

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Cite this review

Pith. "Pith review of Vector-like quarks with non-renormalizable interactions." pith.science (2026). https://pith.science/paper/HSXYL7N3

@misc{pith2026190808964,
  author       = {Pith},
  title        = {Pith review of: Vector-like quarks with non-renormalizable interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HSXYL7N3}},
  note         = {Machine review of arXiv:1908.08964}
}

read the original abstract

We study the impact of the leading non-renormalizable terms in the effective field theory that describes general extensions of the Standard Model with vector-like quarks. Dropping the usual assumption of renormalizability has several phenomenological consequences for the production and decay of the heavy quarks and also for Higgs physics. The most dramatic effects, including those associated with a long lifetime, occur for vector-like quarks with non-standard quantum numbers.

Figures

Figures reproduced from arXiv: 1908.08964 by the authors.

**Figure 1.** Tree-level diagrams that generate the Qqφφ ¯ operator in UV completions of L with additional extra quarks (left) and additional scalars (right). 10 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

**Figure 2.** A one-loop diagram that generates the Qσ¯ µνqF µν operator in a UV completion of L with new scalars. 3 Mixing The multiplets in table 1 can be decomposed into component fields with well-defined electric charge: Q1 = T 0 B0 , Q5 = B0 Y , Q7 = X T 0 , (18) T1 =   T 0 B0 Y   , T2 =   X T 0 B0   , T4 =   B0 Y Y 0   , T5 =   X0 X T 0   , (19) F1 =   X T 0 B0 Y   , F5 =   T 0 B0 Y … view at source ↗

**Figure 3.** Production of heavy quarks in hadron colliders: (a) example diagram for [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: Cross section for different processes for production of heavy quarks with [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

**Figure 5.** Figure 5: Single production with QσµνqFµν-type operators. 1.0 1.5 2.0 2.5 3.0 3.5 4.0 M (TeV) 10 5 10 3 10 1 10 1 10 3 (fb) QQ Tt [wG = (4 TeV) 1 ] Tt [wB = (4 TeV) 1 ] TW [wG = (4 TeV) 1 ] Tt [wG = (64 2 TeV) 1 ] Tt [wB = (64 2 TeV) 1 ] TW [wG = (64 2 TeV) 1 ] LHC run 3 HL-LHC …

**Figure 6.** Figure 6: Cross section for different processes involving [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

**Figure 7.** Figure 7: Total decay width of T (left) and B (right) vs the dimension-5 Yukawa coupling y for each multiplet without dimension-4 couplings and MQ = 2 TeV. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_7.png]

**Figure 8.** Figure 8: Representation of the (BR(Q → Zq), BR(Q → W±q 0 ), BR(Q → Hq)) point as its projections into the BR(Q → Zq)—BR(Q → Hq) plane and into the BR(Q → W±q 0 )—BR(Q → Hq) plane. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_8.png]

**Figure 9.** Figure 9: Left plots: lower bounds for the masses of heavy quarks presented in [PITH_FULL_IMAGE:figures/full_fig_p032_9.png]

**Figure 10.** Figure 10: Branching ratios of T into Ht, Zt and W+b for various values of the parameters in the U, Q7 and Q1 models. The dimensionless couplings λ are always chosen to saturate the corresponding electroweak precision bounds. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_10.png]

**Figure 11.** Figure 11: Branching ratios of T into Ht, Zt and W+b for various values of the parameters in the Q1, T2 and T1 models. The dimensionless couplings λ are always chosen to saturate the corresponding electroweak precision bounds. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]

**Figure 12.** Figure 12: Branching ratios of B into Hb, Zb and W−t for various values of the parameters in the D and Q1 models. The dimensionless couplings λ are always chosen to saturate the corresponding electroweak precision bounds. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_12.png]

**Figure 13.** Figure 13: Branching ratios of B into Hb, Zb and W−t for various values of the parameters in the Q5, T2 and T1 models. The dimensionless couplings λ are always chosen to saturate the corresponding electroweak precision bounds. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_13.png]

**Figure 14.** Figure 14: Branching ratios of T into Ht, Zt and W+b for various values of the parameters in the T5 and F7 models. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_14.png]

**Figure 15.** Figure 15: Branching ratios of T into Ht, Zt and W+b for various values of the parameters in the F1 and F5 models. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_15.png]

**Figure 16.** Figure 16: Branching ratios of B into Hb, Zb and W−t for various values of the parameters in the T4 and F7 models. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_16.png]

**Figure 17.** Figure 17: Branching ratios of T into Ht, Zt and W−t for various values of the parameters in the F1 and F5 models. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_17.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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