Generalised P and CP transformations in the 3-Higgs-doublet model

M. Maniatis; O. Nachtmann

REVIEW 1 major objections 4 minor 41 references

Generalised P and CP transformations in the 3-Higgs-doublet model

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that every generalized parity or CP transformation in the three-Higgs-doublet model that acts linearly on the bilinears and squares to the identity falls into exactly three equivalence classes: two parity classes and one…

desk verdict A clean, mostly hand-checkable classification of order-two generalized P and CP in the 3HDM, but the load-bearing proof of (5.6) rests on an unpublished computer check that a referee should ask to see. read the letter →

arxiv 1908.04303 v1 pith:UI75UIQZ submitted 2019-08-12 hep-ph

M. Maniatis , O. Nachtmann This is my paper · ORCID

classification hep-ph PACS 11.30.Er12.60.Fr

keywords three-Higgs-doubletmodelgeneralizedCPtransformationsparitybilinearformalismHiggspotentialsymmetriesn-Higgs-doubletSU(n)generatorsdiscrete

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

The paper asks which discrete space-time symmetries a model with three Higgs doublets can possess. Working with the eight real bilinears that encode the gauge-invariant products of the doublets, it classifies all generalized parity (P) and CP transformations that are linear, preserve the allowed space, and return to the identity when applied twice. The central result is that, up to Higgs-basis changes, there are exactly two classes of such parity transformations and exactly one class of CP transformations; in particular every generalized CP transformation is equivalent to the standard one. This matters because it settles, for the scalar sector of the 3HDM, which discrete symmetries a potential may be forced to obey, and it gives the explicit parameter conditions for each class. The paper further shows that in an $n$-doublet model the standard CP transformation on the bilinears is a diagonal matrix of $\pm 1$ entries, with minus signs exactly on the $n(n-1)/2$ antisymmetric generator directions.

What carries the argument

The central object is the eight-component vector $K$ of bilinears, obtained by decomposing the hermitian matrix $K=\phi\phi^\dagger$ (with $\phi$ the $3\times 2$ matrix of doublet fields) in a basis of SU(3) generators. A generalized P or CP transformation is a real $8\times 8$ matrix $C$ acting linearly on $K$, preserving the length of $K$ and the allowed orbit space, and squaring to the identity; the constraints force $C$ to be symmetric and orthogonal and to satisfy the SU(3) $d$-symbol condition $d_{a'b'c'}C_{a'a}C_{b'b}C_{c'c}=d_{abc}$. Flavour transformations act by $C\to R(U)C\,R(U)^T$ with $R(U)\in\mathrm{SO}(8)$, and the proof uses these to bring $C$ to diagonal form, reducing the classification to an eigenvalue problem plus the $d$-symbol condition, whose solutions are the eight matrices of table I.

What would settle it

A direct computer search over real symmetric $8\times 8$ matrices $C$ with $C^2=1$ satisfying $d_{a'b'c'}C_{a'a}C_{b'b}C_{c'c}=d_{abc}$ would settle the completeness claim: any solution whose eigenvalue pattern is not one of the eight rows of table I, or any explicit transformation satisfying (4.11)–(4.14) that mixes $K_0$ with the $K_a$, would refute the claimed exhaustive classification.

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Extended reading notes

Core claim

For the 3HDM with only Higgs and gauge fields, the complete set of generalized P and CP transformations satisfying the conditions (4.11)–(4.14) consists of the eight matrices (S1)–(S8). Under flavour transformations these form exactly two equivalence classes of parity transformations, represented by the identity and by $\mathrm{diag}(1,1,1,-1,-1,-1,-1,1)$, and one equivalence class of CP transformations, represented by $\mathrm{diag}(1,-1,1,1,-1,1,-1,1)$. A potential is invariant under such a transformation precisely when its bilinear parameters satisfy $\xi=C\xi$, $\eta=C\eta$, and $E=CE\,C^T$; the known criterion that a potential is CP-invariant exactly when a basis with all real parameters exists is recovered from these conditions. In the nHDM the standard CP transformation acts on the $n^2-1$ bilinears as the diagonal matrix (E26), whose $-1$ entries sit precisely on the $n(n-1)/2$ antisymmetric generalized SU($n$) generator directions; for $n=4$ this matrix has determinant $+1$, so the reflection picture familiar from the two- and three-doublet cases does not extend.

Load-bearing premise

The classification assumes the symmetry is a linear map on the eight Higgs-bilinear variables that keeps the overall scale direction fixed, preserves the allowed space, and returns to the identity when applied twice; any symmetry that mixes the scale direction in, or that needs four applications to close, would be missed.

Editorial extensions

If this is right

Any 3HDM potential that is invariant under some generalized CP transformation is, in a suitable basis, invariant under the standard CP transformation, with the parameter restrictions $\xi_2=\xi_5=\xi_7=0$, $\eta_2=\eta_5=\eta_7=0$, and the corresponding zero pattern in $E$.
There is a genuinely distinct generalized parity class: a potential invariant under it must satisfy $\xi_1=\xi_2=\xi_4=\xi_5=0$, $\eta_1=\eta_2=\eta_4=\eta_5=0$, and the matching $E$ conditions, in the basis of the representative (S8).
The standard parity transformation is represented by the identity matrix on the bilinears, so every 3HDM potential built from Higgs and gauge fields is automatically invariant under standard parity.
In the nHDM the standard CP transformation is always a diagonal $\pm 1$ matrix on the bilinears, with exactly $n(n-1)/2$ minus signs; consequently the determinant is $+1$ when $n(n-1)/2$ is even, so for $n=4$ and higher the standard CP action is no longer a reflection in bilinear space.

Reading between the lines

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

The same flavour-equivalence strategy could be applied to transformations requiring four applications to close, such as the order-four CP of ref. [32]; extending the equations to $C^4=1$ would likely produce additional inequivalent classes and new potential constraints.
The count of antisymmetric directions, $n(n-1)/2$, suggests that the geometric intuition “CP is a reflection” fails specifically when $n(n-1)/2$ is even; this may change how spontaneous CP violation is diagnosed in models with four or more doublets.
One direct test of the classification is numerical: randomly generate symmetric orthogonal $8\times 8$ matrices with $\pm 1$ eigenvalues and check the $d$-symbol condition; any solution outside table I would signal an error in the proof, while agreement would lend confidence to the completeness claim.

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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

Maniatis and Nachtmann finish the classification of order-two generalized P and CP transformations in the 3HDM, and the answer is clean: two parity classes, one CP class, with explicit bilinear matrices. The nHDM result—standard CP acts as diag(±1) with minus entries on the antisymmetric generalized Gell-Mann directions—is simple and correct, and the consistency check reproducing the known real-basis CP theorem is reassuring.

The bilinear method is standard and the derivation is mostly careful. Sections 5 and 6, plus Appendix C, reduce the problem to eight candidate matrices and then show the flavour equivalences. That part is hand-checkable and I think it holds up. The explicit equivalence transformations in Section 6 are a nice touch.

The soft spot is exactly what the stress-test flags. Equation (5.6), the claim that any admissible transformation preserves the d-tensor, is proven in Appendix A only by comparing polynomials for eight special cases "with the help of a computer program." No code, no output, no explicit polynomial identities, not even the case (a) displayed for (5.6) the way it is for (5.2). That step is load-bearing: without it, the reduction to the algebraic condition (6.17) fails and the classification has a hole. The authors do show the hand-checkable case (a) for (5.2), and the 3HDM-to-THDM embeddings are plausible, but a referee cannot verify (5.6) from the text alone. This is a genuine reproducibility gap, not a manufactured one.

A smaller point: the completeness is explicitly scoped to transformations that keep K0 fixed, act linearly on the bilinears, and square to the identity—stated in (4.11)-(4.14) and acknowledged at the end. That is not a hidden assumption, but the abstract's "complete answer" could over-promise to casual readers.

Who is this for? Multi-Higgs model builders, especially 3HDM people who need to know which generalized P or CP symmetries a potential can support. They will cite this paper. It is not a framework-changer, but it is a useful reference result with a clear method.

My recommendation: if this lands on your desk, send it to a referee with experience in bilinear methods and computational algebra. Ask the authors to make the computer code available or provide hand-checkable polynomial identities for (5.6), ideally both. If they can satisfy that, the classification is solid and publishable. If not, the result stays conditional.

Referee Report

1 major / 4 minor

Summary. The manuscript classifies, within the bilinear formalism, all generalized parity and generalized CP transformations of the 3HDM that are linear on the eight bilinears, leave K0 fixed, preserve the allowed bilinear domain, and square to the identity (conditions (4.11)-(4.14)). The authors derive that such transformations are described by symmetric orthogonal 8x8 matrices C satisfying the d-tensor preservation condition (5.6), reduce C to diagonal form using flavour transformations, obtain exactly eight solutions (S1)-(S8) listed in Table I, and show that these fall into two equivalence classes for generalized P (representatives S1 and S8) and one class for generalized CP (representative S3). They also derive the conditions for the potential to be invariant under these transformations (7.4)-(7.6), recover the known criterion that CP invariance is equivalent to the existence of a conventional basis with all parameters real, and generalize the standard CP transformation on bilinears to the nHDM with the explicit matrix (E26).

Significance. If correct, the classification is a useful and nontrivial result for multi-Higgs-doublet model building. It provides a complete set of candidate generalized discrete symmetries within the stated class, gives explicit invariance conditions for the potential, and settles the equivalence-class structure under flavour transformations. The derivation is algebraic and parameter-free, and it reproduces the known THDM/3HDM CP-real-basis theorem as a consistency check. The nHDM formula (E26) is an explicit, easily applicable result. The main caveat is the reproducibility of the computer-assisted proof of (5.6), which is load-bearing for the central classification. The scope restriction to order-two, K0-preserving, linear transformations is explicit in Section 4 and acknowledged in Section 8, so the completeness claim is well defined, but it should not be overinterpreted as covering order-four transformations such as those of ref. [32].

major comments (1)

[Appendix A, Eqs. (A21)-(A26); used in Section 6] The proof of (5.6) is load-bearing for the classification, since it leads to (6.2) and then to condition (6.17), which selects the eight solutions in Table I. However, the verification that D_abc = d_abc for the special cases (a)-(h) is stated to be done "with the help of a computer program", and neither the program, its output, nor the explicit polynomial identities are provided in the manuscript. The same issue affects the assertion (A27)-(A28) used in Section 7 to derive the invariance conditions (7.4). As a result, the completeness of the eight-solution list and the necessity part of the potential-invariance conditions cannot be independently checked by the reader. Please supply the program as supplementary material, display the coefficient identities case by case, or give a hand proof of (5.6).

minor comments (4)

[Section 4, text after Eq. (4.10)] The sentence referring to "appendix C" for the standard P and CP transformations in the nHDM is incorrect; the relevant material is presented in Appendix E.
[Appendix A, after Eq. (A12)] The text says "We consider then six more special cases (b) ... (e)", but only four cases, (b), (c), (d), and (e), are listed; please correct the count.
[Section 5.2, Eq. (5.22)] The notation c_8 c^(8) c^(8)T in Eq. (5.22) is confusing because the scalar eigenvalue c_8 and the eigenvector c^(8) are denoted too similarly; please use a distinct symbol for the eigenvalue.
[Appendix E, Eq. (E24)] The set-builder notation for I_n^as is hard to parse for small k, since for k=2 the displayed sequence "k^2-2k+2, k^2-2k+4, ..., k^2-2" degenerates to a single term; rewriting the set as {k^2+2j-1 | k=1,...,n-1, j=1,...,k} would be unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central classification is a self-contained algebraic derivation, with only a reproducibility caveat in Appendix A that is not a circularity.

full rationale

The paper's central claim—that the generalized P and CP transformations satisfying (4.11)–(4.14) are exactly the eight solutions (S1)–(S8), forming two parity and one CP equivalence class—is derived from explicit algebraic conditions rather than assumed. Equations (5.2) and (5.6) are proved in Appendix A by reducing to THDM special cases and comparing polynomials; the classification then solves (6.1) and (6.2) with the case distinctions of Appendix C carried out by hand. The equivalence classes are established through explicit flavour rotations R(U), e.g. (6.23), (6.25), (6.29). The reproduction of the known CP-real-basis theorem in Appendix D is derived from the derived condition (7.5), not imported as an input. The paper does rely on the authors' prior work for the bilinear formalism, orbit-space constraints, and SU(3) d-constants, but these are mathematical tools used inside the derivation, not the target result; no fitted parameter is renamed as a prediction and no self-citation is load-bearing for the classification. One genuine caveat appears in Appendix A: the proof of (5.6) states that the polynomial comparison for special cases (f)–(h) is done 'with the help of a computer program' without providing code or explicit identities, so that step is not independently reproducible from the paper alone. That is a verification gap, however, not circularity: the claimed identity D_abc = d_abc is not equivalent to the desired classification by construction, and the surrounding hand-checked analysis would still be meaningful if the computer output were supplied. Overall, the central derivation is self-contained against the stated assumptions, and the circularity burden is not met.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted; the paper is a pure algebraic classification. The central claim rests on the established bilinear formalism for NHDMs and on standard SU(3) structure constants, plus a computer-assisted polynomial identity check.

assumptions (5)

domain assumption The space of bilinears of the 3HDM is exactly described by (2.12), including the rank two condition det K = 0.
Taken from refs. [9,14]; used in section 2 and in deriving the constraints on C.
domain assumption Any hermitian 3x3 matrix K of rank at most two corresponds uniquely to a set of Higgs doublets up to gauge transformations.
Theorem from ref. [9], invoked in section 2 to justify the bilinear parametrization.
domain assumption The most general renormalisable gauge-invariant 3HDM potential has the form (2.13) with real parameters.
Standard result from ref. [14]; used throughout sections 2, 7 and appendix D.
standard math Polynomial identities on the bilinear cone (2.12) can be established by checking special THDM embeddings, as done in appendix A.
The special cases (a)-(e) and (f)-(h) reduce the 3HDM to THDM fields; the comparison of polynomials on a full-dimensional cone is a standard argument, though the explicit checks for the d-symbol are delegated to a computer program.
standard math The SU(3) d-symbol constants and the trace identities (2.8) are correct as tabulated in appendix A.
Standard group theory; used in all calculations of section 6 and appendices.

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Pith. "Pith review of Generalised P and CP transformations in the 3-Higgs-doublet model." pith.science (2026). https://pith.science/paper/UI75UIQZ

@misc{pith2026190804303,
  author       = {Pith},
  title        = {Pith review of: Generalised P and CP transformations in the 3-Higgs-doublet model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UI75UIQZ}},
  note         = {Machine review of arXiv:1908.04303}
}

abstract

We study generalised P and CP transformations in the three-Higgs-doublet model (3HDM) with Higgs and gauge fields only. We find that there are two equivalence classes, with respect to flavour transformations, of generalised P transformations and there is only one class of CP transformations. We discuss the conditions the potential has to satisfy in order to be invariant under these transformations. We apply the method of bilinears which we briefly review. We discuss the relation to the conventional basis, where the potential is written in terms of scalar products of the doublet fields. In particular we reproduce the known result that a potential is invariant under CP transformations if and only if there is a conventional basis where all parameters are real. Eventually we study standard P and CP transformations in the $n$-Higgs-doublet model (nHDM). We show that for the bilinears of the nHDM the standard CP transformation corresponds to a diagonal linear transformation with only $\pm 1$ as diagonal elements. We give this matrix explicitly for arbitrary $n$.

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Works this paper leans on

41 extracted references · 23 canonical work pages

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Exploring the Quark Flavour Puzzle within the 3 Higgs Model

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Generalised P and CP transformations in the 3-Higgs-doublet model

INTRODUCTION One motivation, decades ago, to study models with an extension of the number of Higgs-boson doublets was to investigate possible sources of CP violation. In [1] it was shown that in a model with more than one Higgs ﬁeld one can have spontaneous CP violation. In [2] not only the famous Cabibbo-Kobayashi-Maskawa (CKM) matrix was introduced, gov...

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BILINEARS IN THE 3HDM We will consider models with three Higgs-boson doublets which all carry the same hypercharge y = +1/2 and denote the complex doublet ﬁelds by ϕi(x) = ( ϕ+ i (x) ϕ0 i (x) ) , i = 1, 2, 3. (2.1) We shall consider Yang-Mills-Higgs Lagrangians of the form L(x) =LYM(x) + 3∑ i=1 (Dµϕi(x))† (Dµϕi(x))−V (ϕi), (2.2) whereLYM(x) is the standar...

work page
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CHANGE OF BASIS Let us now study an arbitrary unitary mixing of the Higgs-boson doublets of the form (see section 3 of [14]) ϕi(x)→Uijϕj(x), i,j ∈{ 1, 2, 3}, (3.1) with U = (Uij) a unitary 3× 3 matrix, U∈U(3). This change of basis corresponds to the following transformations of the 3× 2 matrix φ(x) and of the 3× 3 matrix K(x) deﬁned in (2.4) and (2.5), re...

work page
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(4.2) For the bilinears we ﬁnd from (4.1) Kα(x) Ps −→K′ α(x) =Kα(x′)

GENERALISED P AND CP TRANSFORMATIONS IN THE 3HDM The standard parity transformation, P s, reads ϕi(x) Ps −→ϕi(x′), Wi λ(x) Ps −→Wiλ(x′) Bλ(x) Ps −→Bλ(x′), (4.1) where i∈{ 1, 2, 3} and x = ( t x ) , x ′ = ( t −x ) . (4.2) For the bilinears we ﬁnd from (4.1) Kα(x) Ps −→K′ α(x) =Kα(x′). (4.3) The Lagrangian (2.2) is invariant under P s. Of course, once we in...

work page
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STANDARD FORMS OF GENERALISED P AND CP TRANSFORMATIONS The problem is now to ﬁnd standard forms for the matrices ˆC which satisfy our conditions (1)-(3) to which general matrices ˆC can be brought using only the ﬂavour transformations (4.17). 5.1. The equations for the matrix ˆC We start from (4.11), (4.12), and (4.13) which imply K′ a(x)K′ a(x) = ˆCabKb(...

work page
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The equations to be solved are the following: we have from (4.14) and (5.2) ˆC ˆC = ˆC T ˆC = 1 8

THE SOLUTIONS FOR ˆC In this section we give the solutions for the matrices ˆC (4.12). The equations to be solved are the following: we have from (4.14) and (5.2) ˆC ˆC = ˆC T ˆC = 1 8. (6.1) Using this we can write (5.6) in the form dabc ˆCbr ˆCcs = ˆCaa′da′rs. (6.2) With the help of the ﬂavour transformations, as explained in section 5, we can, without ...

work page
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Note that all parameters (2.14) of the potential, written in this form, must be real

INVARIANT POTENTIALS We consider now the potential of the 3HDM in the form (2.13) respectively (2.15). Note that all parameters (2.14) of the potential, written in this form, must be real. We consider now a generalised parity (P g) or CP transformation (CPg) satisfying the conditions (1), (2), (3) of section 4; see (4.11)-(4.14). The potential V is invari...

work page

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We have investigated generalised P and CP transformations in this model

CONCLUSIONS In this paper we have considered the three-Higgs-doublet model (3HDM) with Higgs and gauge ﬁelds only. We have investigated generalised P and CP transformations in this model. We have shown that there are two equivalence classes 13 a 1 1 1 2 2 2 3 3 3 3 3 4 5 6 7 8...
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2018 arXiv
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2016 arXiv

Pith tools

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

[1] [32]

Exploring the Quark Flavour Puzzle within the 3 Higgs Model

D. Emmanuel-Costa, J. I. Silva-Marcos and N. R. Agostinho, “Exploring the quark ﬂavor puzzle within the three-Higgs doublet model,” Phys. Rev. D 96, no. 7, 073006 (2017) [arXiv:1705.09743 [hep-ph]]

work page Pith review arXiv 2017

[2] [1]

Generalised P and CP transformations in the 3-Higgs-doublet model

INTRODUCTION One motivation, decades ago, to study models with an extension of the number of Higgs-boson doublets was to investigate possible sources of CP violation. In [1] it was shown that in a model with more than one Higgs ﬁeld one can have spontaneous CP violation. In [2] not only the famous Cabibbo-Kobayashi-Maskawa (CKM) matrix was introduced, gov...

work page Pith review arXiv 1908

[3] [2]

BILINEARS IN THE 3HDM We will consider models with three Higgs-boson doublets which all carry the same hypercharge y = +1/2 and denote the complex doublet ﬁelds by ϕi(x) = ( ϕ+ i (x) ϕ0 i (x) ) , i = 1, 2, 3. (2.1) We shall consider Yang-Mills-Higgs Lagrangians of the form L(x) =LYM(x) + 3∑ i=1 (Dµϕi(x))† (Dµϕi(x))−V (ϕi), (2.2) whereLYM(x) is the standar...

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[4] [3]

CHANGE OF BASIS Let us now study an arbitrary unitary mixing of the Higgs-boson doublets of the form (see section 3 of [14]) ϕi(x)→Uijϕj(x), i,j ∈{ 1, 2, 3}, (3.1) with U = (Uij) a unitary 3× 3 matrix, U∈U(3). This change of basis corresponds to the following transformations of the 3× 2 matrix φ(x) and of the 3× 3 matrix K(x) deﬁned in (2.4) and (2.5), re...

work page

[5] [4]

(4.2) For the bilinears we ﬁnd from (4.1) Kα(x) Ps −→K′ α(x) =Kα(x′)

GENERALISED P AND CP TRANSFORMATIONS IN THE 3HDM The standard parity transformation, P s, reads ϕi(x) Ps −→ϕi(x′), Wi λ(x) Ps −→Wiλ(x′) Bλ(x) Ps −→Bλ(x′), (4.1) where i∈{ 1, 2, 3} and x = ( t x ) , x ′ = ( t −x ) . (4.2) For the bilinears we ﬁnd from (4.1) Kα(x) Ps −→K′ α(x) =Kα(x′). (4.3) The Lagrangian (2.2) is invariant under P s. Of course, once we in...

work page

[6] [5]

STANDARD FORMS OF GENERALISED P AND CP TRANSFORMATIONS The problem is now to ﬁnd standard forms for the matrices ˆC which satisfy our conditions (1)-(3) to which general matrices ˆC can be brought using only the ﬂavour transformations (4.17). 5.1. The equations for the matrix ˆC We start from (4.11), (4.12), and (4.13) which imply K′ a(x)K′ a(x) = ˆCabKb(...

work page

[7] [6]

The equations to be solved are the following: we have from (4.14) and (5.2) ˆC ˆC = ˆC T ˆC = 1 8

THE SOLUTIONS FOR ˆC In this section we give the solutions for the matrices ˆC (4.12). The equations to be solved are the following: we have from (4.14) and (5.2) ˆC ˆC = ˆC T ˆC = 1 8. (6.1) Using this we can write (5.6) in the form dabc ˆCbr ˆCcs = ˆCaa′da′rs. (6.2) With the help of the ﬂavour transformations, as explained in section 5, we can, without ...

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[8] [7]

Note that all parameters (2.14) of the potential, written in this form, must be real

INVARIANT POTENTIALS We consider now the potential of the 3HDM in the form (2.13) respectively (2.15). Note that all parameters (2.14) of the potential, written in this form, must be real. We consider now a generalised parity (P g) or CP transformation (CPg) satisfying the conditions (1), (2), (3) of section 4; see (4.11)-(4.14). The potential V is invari...

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