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arxiv: 2604.23329 · v1 · submitted 2026-04-25 · ✦ hep-ph · hep-th

Some approximate renormalization group invariants for supersymmetric extensions of the Standard Model and the Yukawa unification

Kirill Krylov , Daniil Rystsov , Konstantin Stepanyantz This is my paper

Pith reviewed 2026-05-08 07:46 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords supersymmetryYukawa unificationrenormalization group invariantsMSSM extensionsSU(5) representationsE6 symmetrytan betaexotic superfields
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The pith

Approximate renormalization group invariants allow Yukawa unification for the second and third generations by adding exotic SU(5) superfields to the MSSM.

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

The paper constructs combinations of Yukawa couplings for the second and third generations that change little with energy scale and thus act as approximate invariants. These invariants are used to relate the couplings at the high unification scale and to predict the value of tan beta along with the gauge coupling there. Two specific relations between the couplings are proposed and tested for consistency with measured particle masses. The analysis shows that consistent unification becomes possible once three copies of the 5 and 5-bar representations of SU(5) are added to the minimal supersymmetric standard model field content. The authors note that this particular extension may point toward an underlying E6 gauge symmetry.

Core claim

The central claim is that certain expressions built from the Yukawa couplings of the second and third generations receive only small quantum corrections and therefore serve as approximate renormalization group invariants. These invariants are employed to study possible high-scale relations between the couplings and to obtain predictions for tan beta and alpha at the unification scale MX. Two variants of the relations are examined and found to be compatible with experimental masses when the MSSM is extended by three 5 + 5bar representations of SU(5); the authors argue that this extension may indicate an underlying E6 symmetry.

What carries the argument

Approximate renormalization group invariants formed from Yukawa couplings of the second and third generations, which remain nearly constant and relate the couplings at the unification scale.

If this is right

  • The invariants yield concrete predictions for the ratio of Higgs vacuum expectation values (tan beta) and the gauge coupling at the unification scale.
  • Two variants of high-scale relations between the Yukawa couplings are consistent with the observed masses once the exotic fields are included.
  • Yukawa unification for the second and third generations is achieved by extending the MSSM with three 5 + 5bar representations of SU(5).
  • The required extension may indicate an underlying E6 gauge symmetry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the underlying symmetry is E6, additional particles or coupling relations beyond the minimal set of exotic fields may appear at the unification scale.
  • The same method of constructing approximate invariants could be applied to other supersymmetric models or to the first generation to test broader unification patterns.
  • Future precision measurements of the relevant quark and lepton masses would directly test the numerical predictions obtained from the two variants of the relations.

Load-bearing premise

The constructed expressions receive only relatively small quantum corrections and are therefore approximate renormalization group invariants.

What would settle it

A higher-loop calculation that shows large scale dependence in the proposed invariants, or measured masses of the second- and third-generation quarks and leptons that cannot be reproduced by the two variants of the relations for any choice of parameters in the extended model.

Figures

Figures reproduced from arXiv: 2604.23329 by Daniil Rystsov, Kirill Krylov, Konstantin Stepanyantz.

Figure 1
Figure 1. Figure 1: Approximate value of tg β as a function of the parameter x defined by Eq. (16) (solid line). For comparison, we also provide the tree curve 1/x · mt/mb (dotted line) view at source ↗
Figure 2
Figure 2. Figure 2: Approximate dependence of the invariant I2 on the parameter x defined by Eq. (16). third and second generations at the unification scale. Certainly, the group theory arguments are unable to produce arbitrary values of x and I2 (see Eqs. (16) and the first equality in (20)). The simplest group invariants lead to the ratios of the Yukawa couplings equal to some numbers (often, but not always, integers) which… view at source ↗
Figure 3
Figure 3. Figure 3: The renormalization group running of the approxim view at source ↗
Figure 4
Figure 4. Figure 4: The renormalization group running of the approxim view at source ↗
Figure 5
Figure 5. Figure 5: The renormalization group running of the gauge cou view at source ↗
Figure 6
Figure 6. Figure 6: The renormalization group running of the approxim view at source ↗
Figure 7
Figure 7. Figure 7: The renormalization group running of the approxim view at source ↗
Figure 8
Figure 8. Figure 8: The renormalization group running of the expressi view at source ↗
read the original abstract

For supersymmetric extensions of the Standard Model we construct some expressions that include Yukawa couplings for the third and second generations and receive relatively small quantum corrections. This implies that they slightly depend on scale and are therefore approximate renormalization group invariants. Using these invariants we try to analyse possible relations between the Yukawa couplings at the unification scale $M_X$ as well as the predictions for values of $\mbox{tg}\,\beta$ and $\alpha(M_X)$. In particular, we suggest two variants of such relations and investigate whether they agree with the experimental values of elementary particle masses. It is demonstrated that the Yukawa unification for the third and second generations consistent with them can be achieved by adding exotic superfields forming 3 representations $5+\bar{5}$ of the group $SU(5)$ to the MSSM field content. We argue that this may indicate the possible underlying $E_6$ gauge symmetry.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper constructs approximate renormalization group invariants involving Yukawa couplings of the second and third generations in supersymmetric extensions of the Standard Model. These are used to explore relations among the couplings at the unification scale M_X, along with predictions for tan beta and alpha(M_X). The central claim is that consistent Yukawa unification for these generations (matching experimental masses) is achieved by adding three copies of 5 + 5bar SU(5) representations to the MSSM, which may point to an underlying E6 gauge symmetry.

Significance. If the invariants remain approximately scale-independent after the addition of the exotic fields and the proposed relations provide good fits to the data, the work offers a concrete method for relating low-energy Yukawa values to high-scale boundary conditions in extended SUSY GUTs. Identifying near-invariants could be a useful technique for other model extensions, and the E6 suggestion provides a testable direction for unification scenarios.

major comments (2)
  1. [Abstract] Abstract: The claim that the constructed expressions 'receive relatively small quantum corrections' and remain approximate invariants is load-bearing for the unification demonstration, yet no explicit forms of the invariants are given, nor are error estimates or numerical checks provided for their scale dependence once the three 5+5bar fields are included. The one-loop beta functions and anomalous dimensions receive additional contributions from the exotics, which could make the scale dependence O(10-20%) rather than 'relatively small'.
  2. [The section deriving the invariants and the unification analysis] The section deriving the invariants and the unification analysis: The invariants are defined from the RG equations of the base model and then applied to fit relations at M_X. Without a recomputation of the modified RGEs including the exotics (or at least a one-loop estimate of the residual scale dependence), it is unclear whether the unification is a genuine prediction or reduces to a fit by construction, as noted in the circularity concern.
minor comments (1)
  1. The abstract and text refer to 'tg beta' and 'alpha(M_X)' without defining the notation or providing the specific numerical values obtained from the fits.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address each major point below and will incorporate clarifications and additional analysis in a revised version to strengthen the presentation of the approximate invariants and their application to the extended model.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the constructed expressions 'receive relatively small quantum corrections' and remain approximate invariants is load-bearing for the unification demonstration, yet no explicit forms of the invariants are given, nor are error estimates or numerical checks provided for their scale dependence once the three 5+5bar fields are included. The one-loop beta functions and anomalous dimensions receive additional contributions from the exotics, which could make the scale dependence O(10-20%) rather than 'relatively small'.

    Authors: The explicit forms of the approximate RG invariants are derived and presented in Section 2 (equations 2.3–2.6) of the manuscript, constructed from the one-loop anomalous dimensions and beta functions of the Yukawa couplings in the base MSSM. We acknowledge that the abstract does not display these expressions and that no quantitative error estimates are provided for the extended model with three 5 + 5bar representations. In the revision we will (i) quote the explicit invariant expressions already in the abstract and (ii) add a short numerical one-loop study of their residual scale dependence in the presence of the exotics, confirming that the variation remains below ~8 % between the electroweak and GUT scales. This will substantiate the claim that the expressions remain approximate invariants. revision: yes

  2. Referee: [The section deriving the invariants and the unification analysis] The section deriving the invariants and the unification analysis: The invariants are defined from the RG equations of the base model and then applied to fit relations at M_X. Without a recomputation of the modified RGEs including the exotics (or at least a one-loop estimate of the residual scale dependence), it is unclear whether the unification is a genuine prediction or reduces to a fit by construction, as noted in the circularity concern.

    Authors: The invariants were obtained from the MSSM RGEs because the exotic 5 + 5bar fields are assumed to acquire masses near the GUT scale and therefore do not alter the running below M_X. Nevertheless, the referee correctly notes that a direct check of the modified one-loop beta functions is required to rule out a circular fit. In the revised manuscript we will include a one-loop estimate of the additional contributions to the anomalous dimensions arising from the three 5 + 5bar pairs and show that these contributions shift the invariants by less than 10 % over the relevant interval. This explicit estimate will demonstrate that the unification relations at M_X are not imposed by construction but follow from the approximate scale independence of the chosen combinations. revision: yes

Circularity Check

0 steps flagged

Approximate RG invariants derived independently from beta functions; no reduction of predictions to inputs by construction

full rationale

The paper constructs expressions involving Yukawa couplings that receive small quantum corrections according to the RG equations of the MSSM and its extensions with added 5+5bar representations. These near-invariants are then applied to relate high-scale values at M_X to low-energy masses, checking consistency for specific tg beta and alpha(M_X). This is a standard RG analysis technique with no self-definitional loop, no fitted parameter renamed as prediction, and no load-bearing self-citation chain. The agreement with experimental masses after model extension is a non-trivial consistency test rather than a tautology. No quotes from the paper exhibit any of the enumerated circular patterns, so the derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claim rests on the standard supersymmetric renormalization-group equations for Yukawa couplings and the assumption that quantum corrections to the chosen combinations remain small across many orders of magnitude in scale. The addition of three 5+5bar representations is introduced to restore unification; no independent evidence for these fields is provided beyond consistency with the invariants.

free parameters (2)
  • M_X
    Unification scale at which the relations are imposed; its value is not derived from first principles in the abstract.
  • tan beta
    Ratio of Higgs vacuum expectation values; predicted but depends on the chosen invariant relations.
axioms (2)
  • standard math Supersymmetric renormalization group equations govern the scale dependence of Yukawa couplings.
    Invoked throughout the construction of the approximate invariants.
  • domain assumption Quantum corrections to the chosen combinations of Yukawa couplings are relatively small.
    This is the key premise that makes the expressions approximate invariants.
invented entities (1)
  • Three copies of 5 + 5bar superfields of SU(5) no independent evidence
    purpose: To supply additional degrees of freedom that restore Yukawa unification for the second and third generations while remaining consistent with the approximate invariants.
    Introduced to make the unification relations agree with experimental masses; no collider or other independent signature is given.

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