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arxiv: 2605.20794 · v1 · pith:SYSYY722new · submitted 2026-05-20 · ✦ hep-ph

The Alternative Left-Right Scenario: Unitarity, Vacuum Stability and RG Evolution

Hrishikesh Deka , Avnish , Sumit K. Garg , Poulose Poulose This is my paper

Pith reviewed 2026-05-21 04:38 UTC · model grok-4.3

classification ✦ hep-ph
keywords Alternative Left-Right Modelscalar sectorunitarity constraintsvacuum stabilityrenormalization group evolutionHiggs massesbeyond Standard Model
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The pith

Requiring consistency of the Alternative Left-Right Model up to 10^16 GeV imposes upper bounds on its new scalar masses.

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

The paper derives tree-level perturbative unitarity conditions on the quartic couplings of the Alternative Left-Right Model and combines them with bounded-from-below requirements and positive mass-squared eigenvalues. It then evolves the parameters via one-loop renormalization group equations from the electroweak scale to a high cutoff while preserving unitarity, vacuum stability, and perturbativity. This procedure yields concrete upper limits on the masses of the additional charged and neutral scalars that tighten as the cutoff rises. A sympathetic reader cares because the resulting bounds convert an open parameter space into a predictive target for collider searches of the extended Higgs sector.

Core claim

When the Alternative Left-Right Model parameters are evolved from the electroweak scale to 10^16 GeV under the joint requirements of unitarity, vacuum stability, and perturbativity, the allowed quartic couplings shrink sufficiently that, for a right-handed breaking scale v_R of 10 TeV, the physical scalars satisfy m_{H_1^±} ≲ 6.5 TeV, m_{H_2^±} ≲ 1.5 TeV, and m_{H_1^0} ≃ m_{A_1} ≲ 1.3 TeV, with all bounds scaling linearly with v_R.

What carries the argument

One-loop renormalization group equations for the quartic scalar couplings, subject to the full set of tree-level unitarity and bounded-from-below constraints.

If this is right

  • The allowed ranges for the quartic couplings become narrower at higher cutoff scales.
  • All physical scalar masses acquire upper limits that grow proportionally with the right-handed vacuum expectation value.
  • The model supplies definite mass targets for experimental searches of the extended Higgs sector.
  • The simultaneous imposition of unitarity and stability conditions at every scale along the flow produces tighter restrictions than either condition alone.

Where Pith is reading between the lines

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

  • The same running procedure could be applied to left-right models with different gauge embeddings to obtain comparable mass caps.
  • Lowering the cutoff scale to a few TeV would tighten the mass bounds further, potentially bringing some scalars into the direct reach of the LHC.
  • Non-observation of the predicted scalars up to the quoted limits would strengthen the case for the model's consistency assumptions.
  • The bounds could inform priority ordering in dedicated searches for charged versus neutral scalars.

Load-bearing premise

No additional new physics appears between the electroweak scale and the chosen high-energy cutoff, so that the one-loop running alone governs the evolution.

What would settle it

Observation at a collider of any scalar whose mass exceeds the derived upper limit for a measured value of v_R would violate the requirement that the model remain unitary and stable up to 10^16 GeV.

Figures

Figures reproduced from arXiv: 2605.20794 by Avnish, Hrishikesh Deka, Poulose Poulose, Sumit K. Garg.

Figure 1
Figure 1. Figure 1: Dependence of the charged Higgs mass mH ± 2 on (α12 − α13). (Left): The impact of µ3 for fixed tan β = 10 and vR = 10 TeV; (Right): The sensitivity of tan β for fixed µ3 = −1000 GeV and vR = 10 TeV. The negative values of (α12 − α13) are disallowed for the potential to be bounded from below. 3.2 Perturbative unitarity constraints At tree level, another theoretical constraint on the ALRM parameter space ema… view at source ↗
Figure 2
Figure 2. Figure 2: Correlation plots showing the allowed region by the perturbative unitarity (blue) and the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: Dependence of gBL and gR on the ratio parameter rg = gR(vR) gL(vR) , shown at vR = 10 TeV (solid lines) and at the GUT scale set as 1016 GeV (dashed lines). Right: gR and gBL at vR. In both panels, the shaded regions are excluded by the perturbative limit (gi ≤ √ 4π). as functions of rg, evaluated at vR (solid lines) and at the GUT scale (dashed lines). The contrasting behaviour of the two couplings … view at source ↗
Figure 4
Figure 4. Figure 4: The perturbativity bounds on the masses of the heavy gauge bosons [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: RGE running of the scalar quartic couplings from the input scale [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The maximum allowed initial values of the scalar quartic couplings [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Left: Mass spectrum of mH ± 2 (red), mH0 1 (blue), and mH ± 1 (black) as a function of µ3 for vR = 10 TeV, obtained from the full numerical scan with all theoretical constraints imposed up to 1016 GeV. Right: Maximum allowed scalar masses as a function of vR, derived by requiring theoretical consistency up to 1016 GeV. theoretically consistent up to 1016 GeV, it restricts |µ3| to approximately 1.3 TeV for … view at source ↗
read the original abstract

We study the theoretical constraints on the scalar sector of the Alternative Left-Right Model (ALRM), an $E_6$-motivated extension of the Standard Model based on the gauge group $\mathrm{SU}(3)_c \otimes \mathrm{SU}(2)_L \otimes \mathrm{SU}(2)_{R'} \otimes \mathrm{U}(1)_{B-L}$, supplemented by a global $\mathrm{U}(1)_S$ symmetry. We derive the complete set of tree-level perturbative unitarity constraints on the model, resulting in 14 independent conditions on the quartic scalar couplings. When combined with the boundedness-from-below conditions and the requirement of positive-definite scalar mass-squared eigenvalues, these constraints are found to be complementary, with their simultaneous imposition yielding significantly more stringent restrictions on the parameter space than either set alone. We then perform a one loop renormalization group analysis, evolving the model parameters from the electroweak scale up to a high energy cut-off scale, and requiring that the vacuum stability, the unitarity, and the perturbativity conditions are preserved throughout. The renormalisation group evolution is found to restrict the allowed parameter space considerably beyond the tree-level bounds, with the constraints on the quartic couplings becoming more stringent as the cut off scale is raised. Consequently, the physical scalar masses in the model acquire upper bounds. For the right-hand symmetry breaking scale, $v_R = 10$ TeV and requiring theoretical consistency up to $10^{16}$ GeV, we obtain $m_{H_1^\pm} \lesssim 6.5$ TeV, $m_{H_2^\pm} \lesssim 1.5$ TeV, and $m_{H_1^0} \simeq m_{A_1} \lesssim 1.3$ TeV, with all bounds scaling with $v_R$. These findings offer a predictive and falsifiable framework for searches of the extended Higgs sector of the ALRM at the current and future collider experiments.

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

1 major / 2 minor

Summary. The paper derives 14 independent tree-level perturbative unitarity conditions on the quartic scalar couplings of the Alternative Left-Right Model (ALRM) with gauge group SU(3)_c ⊗ SU(2)_L ⊗ SU(2)_{R'} ⊗ U(1)_{B-L} and global U(1)_S. These are combined with boundedness-from-below conditions and positive-definite scalar mass-squared eigenvalues to constrain the parameter space. A one-loop RG evolution is performed from the electroweak scale to high cutoffs (up to 10^16 GeV), requiring preservation of unitarity, vacuum stability, and perturbativity; this yields upper bounds on scalar masses that scale with the right-handed breaking scale v_R, e.g., for v_R = 10 TeV the bounds m_{H_1^±} ≲ 6.5 TeV, m_{H_2^±} ≲ 1.5 TeV, and m_{H_1^0} ≃ m_{A_1} ≲ 1.3 TeV.

Significance. If the RG implementation is accurate, the work supplies falsifiable upper limits on the ALRM scalar masses that complement collider searches and strengthen the model's predictivity. The explicit count of 14 unitarity conditions and their complementarity with BFB conditions is a clear technical contribution using standard QFT methods. The one-loop running and assumption of no intervening new physics up to the cutoff are standard but limit the robustness; the bounds' linear scaling with v_R is a useful feature for varying the breaking scale.

major comments (1)
  1. [RG Evolution] RG Evolution section: the one-loop beta functions of the unbroken ALRM appear to be used for the entire running from the electroweak scale to 10^16 GeV without a threshold-matching step or change in effective theory at v_R = 10 TeV. Because SU(2)_{R'} breaks at this scale, the heavy right-handed gauge bosons, scalars, and fermions decouple, altering the beta-function coefficients for the quartics below v_R. This directly affects the low-scale values of the couplings that set the quoted mass bounds (e.g., m_{H_1^±} ~ sqrt(λ) v_R), so the reported limits may shift once proper matching is included.
minor comments (2)
  1. [Abstract] The abstract states that 14 conditions are derived but does not preview their explicit forms or which quartics they constrain; a short enumeration or reference to the relevant equations would improve accessibility.
  2. [Introduction] Notation for the physical scalars (H_1^±, H_2^±, H_1^0, A_1) is introduced without an early summary table relating them to the underlying doublet and singlet fields; adding such a table would aid readers unfamiliar with the ALRM scalar content.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comment on the renormalization group analysis. We address the point in detail below and will incorporate appropriate revisions to improve the robustness of the results.

read point-by-point responses

  1. Referee: RG Evolution section: the one-loop beta functions of the unbroken ALRM appear to be used for the entire running from the electroweak scale to 10^16 GeV without a threshold-matching step or change in effective theory at v_R = 10 TeV. Because SU(2)_{R'} breaks at this scale, the heavy right-handed gauge bosons, scalars, and fermions decouple, altering the beta-function coefficients for the quartics below v_R. This directly affects the low-scale values of the couplings that set the quoted mass bounds (e.g., m_{H_1^±} ~ sqrt(λ) v_R), so the reported limits may shift once proper matching is included.

    Authors: We agree that a fully consistent treatment requires threshold matching at the SU(2)_{R'} breaking scale v_R. In the present analysis we evolved the quartic couplings using the one-loop beta functions of the complete ALRM from the electroweak scale to the cutoff, which is a standard approximation when the intermediate scale is not too close to the cutoff. This choice keeps the effective theory unchanged throughout the running and yields conservative upper bounds on the scalar masses. Nevertheless, the referee is correct that below v_R the heavy states decouple and the beta-function coefficients for the light quartics change; proper matching would therefore modify the allowed low-scale values of the couplings that enter the mass expressions. We will revise the manuscript to (i) explicitly state this approximation, (ii) discuss its expected quantitative impact for v_R = 10 TeV, and (iii) outline how a two-stage running with threshold corrections at v_R can be implemented in a follow-up study. These clarifications will be added in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard QFT constraints to the ALRM Lagrangian

full rationale

The paper first derives 14 independent tree-level unitarity conditions directly from the scalar potential quartics, combines them with bounded-from-below and positive-mass-squared requirements, then evolves the couplings via one-loop beta functions from the electroweak scale to a high cutoff while demanding the same conditions remain satisfied. All steps follow from the model Lagrangian and general renormalization-group equations without any reduction of the final mass bounds to fitted parameters, self-definitions, or load-bearing self-citations. The quoted upper limits on m_{H_1^±}, m_{H_2^±} and m_{H_1^0} are therefore independent outputs of the imposed theoretical consistency requirements rather than tautological restatements of inputs.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 1 invented entities

The central claim rests on standard quantum-field-theory assumptions plus the specific scalar content and potential of the ALRM. The free parameters are the initial quartic couplings at the electroweak scale together with the chosen values of v_R and the cutoff scale. The additional scalar fields are introduced by the model definition.

free parameters (3)
  • quartic scalar couplings
    Initial values at the electroweak scale are free parameters that are constrained by unitarity, stability, and RG evolution but not fixed to specific numbers.
  • v_R
    The right-handed symmetry breaking scale is chosen as 10 TeV to obtain the quoted numerical bounds.
  • cutoff scale
    The high-energy cutoff is set to 10^16 GeV to derive the mass upper limits.
axioms (3)
  • domain assumption Tree-level perturbative unitarity holds for 2-to-2 scalar scattering
    Invoked to derive the 14 independent conditions on the quartic couplings.
  • domain assumption One-loop renormalization group equations are sufficient for evolution to high scales
    Used to track the running of couplings from the electroweak scale to the cutoff.
  • standard math The scalar potential must be bounded from below and yield positive mass-squared eigenvalues
    Standard requirement for a stable vacuum, combined with unitarity conditions.
invented entities (1)
  • Additional scalar fields of the ALRM no independent evidence
    purpose: Extend the Higgs sector to break the enlarged gauge symmetry
    Introduced by the model definition; no independent experimental evidence is provided.

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