Recoverable Identifier

arXiv:2605.15270 · detector doi_compliance · incontrovertible · 2026-05-19 15:41:11.678665+00:00

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The∆(3n 2)and∆(6n 2), withn >3, limit The potential corresponding to invariance under ∆(3n 2), ∆(6n2), withn >3, is simply obtained settingλ 4 →0 in Eq. (18). This change propagates straightforwardly to the charged scalar mass spectrum in Eq. (32). Concerning the neutral sector, the mass spectrum is again beyond analytic reach; the intermediate regime leading to Eqs. (43)-(45) in theA 4 scenario is much less interesting forλ 4 →0: one can notice that Eqs. (43) and (44) only give one non-vanishing eigenvalue (rather than all three), and the eigenvalues in Eq. (45) become degenerate. B. One∆(27)triplet with CP invariance The ∆(27) invariant quartic potential for an irreducible triplet is V4|∆(27)(Φ1,Φ 2,Φ 3) = λ1 2 3X a=1 Φ† aΦa 2 +λ 2 3X a,b=1 (Φ† aΦa)(Φ† bΦb) +λ 3 3X a,b=1 (Φ† aΦb)(Φ† bΦa) + λ∆ 2 h (Φ† 1Φ2)(Φ† 1Φ3) + (Φ† 2Φ3)(Φ† 2Φ1) + (Φ† 3Φ1)(Φ† 3Φ2) i + λ∗ ∆ 2 h (Φ† 2Φ1)(Φ† 3Φ1) + (Φ† 3Φ2)(Φ† 1Φ2) + (Φ† 1Φ3)(Φ† 2Φ3) i ,(46) and imposing CP invariance one hasλ ∗ ∆ =λ ∆. In the ∆(27) case, concerning analytic results, we exclusively focus on the charged sector since this scenario is not as transparent as theA 4 case in Sec. III A, but still of interest. The characteristic polynomial ofM 2 ± reads P±(x) = det(M 2 ± −x1) =−x(x 2 +c 2x+c 1).(47) It is straightforward to obtain the coefficientc 2 as −c2 = Tr M 2 ± =µ 2 + 1 2(λ1 + 6λ2 + 2λ3)v2 ,(48) while the expression forc 1 is quite cumbersome. The point is that it has the following form c1 =µ 2 λ3v2 +λ ∆O(v2) +O(λ 2v4),(49) w

Evidence payload

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  "printed_excerpt": "The\u2206(3n 2)and\u2206(6n 2), withn >3, limit The potential corresponding to invariance under \u2206(3n 2), \u2206(6n2), withn >3, is simply obtained setting\u03bb 4 \u21920 in Eq. (18). This change propagates straightforwardly to the charged scalar mass spectrum in E",
  "reconstructed_doi": "10.13039/501100011033/through",
  "ref_index": 9,
  "resolved_title": null,
  "verdict_class": "incontrovertible"
}