Flavour Invariants of the N Higgs Doublet Model

arxiv: 2510.02465 · v2 · pith:YGV7FOG4new · submitted 2025-10-02 · ✦ hep-ph

Flavour Invariants of the N Higgs Doublet Model

Jo\~ao C. Belas , Jo\~ao P. Silva This is my paper

Pith reviewed 2026-05-18 10:07 UTC · model grok-4.3

classification ✦ hep-ph

keywords N Higgs doublet modelflavor invariantsweak basis transformationsmixing matricescharged Higgs basisrenormalization group evolutionbasis independent quantities

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The pith

A complete set of weak basis invariant traces fully describes the flavor sector of any N Higgs doublet model.

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

This paper develops a systematic way to study flavor effects when the Standard Model is extended by N Higgs doublets. It defines mixing matrices that relate different choices of basis for the Higgs fields, analogous to the CKM matrix that relates up-quark and down-quark mass bases. The charged Higgs basis receives special emphasis because it simplifies the structure. For the first time the authors construct a complete collection of traces of flavor matrices that stay the same no matter which weak basis is chosen. These invariants matter because they let physicists track the physical content of the model under changes of basis and during renormalization group evolution.

Core claim

The central claim is that the flavor sector of the general N Higgs Doublet Model admits a complete set of weak basis transformation invariant traces of the flavor matrices. Together with the introduced mixing matrices that describe rotations between suitably defined bases, these traces encode all physical flavor information in a manner independent of unphysical basis choices. The charged Higgs basis is singled out as the most convenient reference point for the construction.

What carries the argument

The complete set of weak basis transformation invariant traces of flavor matrices; these quantities remain unchanged under basis redefinitions and isolate the physical parameters that govern Higgs-fermion interactions.

If this is right

All physical flavor parameters become directly usable in calculations without reference to a particular basis.
Renormalization group equations for the flavor sector can be written entirely in terms of these invariants.
The framework applies uniformly for any number N of Higgs doublets, generalizing earlier two-doublet results.
Comparisons between different realizations of the model are simplified by comparing the values of the invariant traces.

Where Pith is reading between the lines

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

These invariants could be computed numerically for small N to verify they reproduce known physical quantities.
The same construction might be adapted to study flavor violation in other multi-scalar extensions of the Standard Model.
Explicit relations between the traces and measurable quantities such as Higgs decay rates could be derived in follow-up work.

Load-bearing premise

The listed traces must include every independent physical degree of freedom present in the N-Higgs flavor sector.

What would settle it

An explicit example of a measurable flavor observable or a set of parameters in the N-HDM that cannot be recovered from the given traces would show the set is incomplete.

read the original abstract

In this work, a systematic way of analyzing the N Higgs Doublet Models flavor sector will be developed. We introduce a complete set of mixing matrices describing the rotation between certain suitably defined bases, akin to the Cabibbo-Kobayashi-Maskawa matrix, which describes the relation between the up-quark and down-quark mass bases. We point out the crucial importance played by the charged Higgs basis. It is also introduced for the first time a complete set of weak basis transformation invariant traces of flavor matrices for the general N doublets case. This will be important for studies of the renormalization group evolution in terms of relevant physical parameters.

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.

Desk Editor's Note private letter to a colleague

The paper gives a practical construction of weak-basis invariant traces for the flavor sector in general N-HDM, extending earlier N=2,3 results, with the main value in organizing RGE work.

read the letter

Belas and Silva lay out flavor invariants for the N Higgs doublet model using traces that stay the same under weak basis changes, plus some mixing matrices for the Higgs sector. This is new for arbitrary N. Earlier papers did this for two or three doublets, but the general case lets you handle models with more scalars without redoing the basis work each time. The charged Higgs basis gets highlighted as a good reference, which makes sense for keeping things physical. The paper does a good job making the construction systematic. Defining these traces from the Yukawa and other matrices gives a way to talk about flavor parameters that doesn't depend on arbitrary choices. That should help with RGE running, where you want to follow how physical quantities evolve. The soft spot is the completeness. They say the set is complete, but it rests on the traces being independent and covering all the physical degrees of freedom after subtracting the parameters in the unitary transformations on the fields. If the paper has a clear count that matches the known dimension of the orbit space, fine; if it's just stated without the details, that part could be tightened. This is for theorists doing calculations in extended Higgs models with flavor. Someone setting up RGE equations or choosing bases for N-HDM phenomenology would find it handy. It is worth a serious referee because the topic is relevant and the method is straightforward. I would send it to peer review. The ideas are clear and the subfield can use better tools for invariants.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a systematic framework for the flavor sector of the N-Higgs Doublet Model. It defines a set of mixing matrices that relate suitably chosen bases (analogous to the CKM matrix), emphasizes the charged Higgs basis, and introduces a claimed complete set of weak-basis-invariant traces constructed from the flavor matrices. These invariants are presented as a tool for renormalization-group evolution studies expressed in physical parameters.

Significance. If the completeness and independence of the trace invariants are rigorously demonstrated, the work would supply a practical basis for flavor analyses and RGE flows in general N-HDMs, reducing basis dependence in multi-Higgs flavor studies. The extension to arbitrary N and the explicit construction of invariants constitute the main potential contribution.

major comments (2)

[Section introducing the trace invariants (near the end of the abstract and corresponding main-text discussion)] The central claim that the proposed traces form a complete set rests on an unverified counting argument. The manuscript does not explicitly compare the number of independent traces to the dimension of the physical flavor parameter space after subtracting the real parameters associated with the three unitary weak-basis transformations (N(N-1)/2 + 3N). Without this comparison or an explicit proof of linear independence and spanning, the completeness assertion for general N remains unsecured and directly affects the utility for RGE applications.
[Abstract, final paragraph, and the section defining the traces] The abstract states that a 'complete set' is introduced 'for the first time,' yet the provided derivations do not include an explicit check that the traces are free of redundancy or that they capture all relevant physical information. This gap is load-bearing for the claim that the invariants suffice for renormalization-group studies in terms of physical parameters.

minor comments (1)

[Section 2 or 3] Notation for the flavor matrices and the precise definition of the traces should be stated more explicitly at first use to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each of the major comments below and outline the revisions we plan to make.

read point-by-point responses

Referee: [Section introducing the trace invariants (near the end of the abstract and corresponding main-text discussion)] The central claim that the proposed traces form a complete set rests on an unverified counting argument. The manuscript does not explicitly compare the number of independent traces to the dimension of the physical flavor parameter space after subtracting the real parameters associated with the three unitary weak-basis transformations (N(N-1)/2 + 3N). Without this comparison or an explicit proof of linear independence and spanning, the completeness assertion for general N remains unsecured and directly affects the utility for RGE applications.

Authors: We appreciate the referee's point that a more explicit counting argument would strengthen our claim of completeness. While the manuscript provides a construction of the invariants and argues for their invariance and utility in RGE studies, we agree that a direct comparison to the dimension of the physical parameter space is valuable. In the revised version, we will add a subsection detailing the number of physical parameters in the flavor sector of the N-HDM, subtracting the N(N-1)/2 + 3N real parameters from the weak-basis transformations, and show that our set of traces provides the matching number of independent invariants. This will include a brief discussion of linear independence based on the structure of the flavor matrices. revision: yes
Referee: [Abstract, final paragraph, and the section defining the traces] The abstract states that a 'complete set' is introduced 'for the first time,' yet the provided derivations do not include an explicit check that the traces are free of redundancy or that they capture all relevant physical information. This gap is load-bearing for the claim that the invariants suffice for renormalization-group studies in terms of physical parameters.

Authors: We acknowledge that the claim in the abstract regarding the introduction of a complete set would be more robust with an explicit verification of no redundancies. We will revise the abstract to temper the language if necessary and expand the main text section on the traces to include an argument for their independence and completeness, cross-referenced with the new counting discussion. This will better support their use in RGE analyses expressed in physical parameters. revision: yes

Circularity Check

0 steps flagged

Direct mathematical construction of weak-basis invariants; no reduction to inputs by construction

full rationale

The paper defines mixing matrices between suitably chosen bases and introduces a set of weak-basis-invariant traces of flavor matrices for general N. These are constructed explicitly from the Yukawa matrices under unitary redefinitions, without any fitted parameters being relabeled as predictions or any self-citation chain supplying the central completeness claim. The counting argument for completeness is presented as part of the systematic derivation rather than presupposing the result. This is a standard self-contained mathematical construction, consistent with the low circularity expected for definitional work in flavor physics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard quantum field theory assumptions for multi-Higgs models and the existence of weak basis transformations; no new physical entities or fitted parameters are introduced in the abstract.

axioms (2)

domain assumption Existence and utility of weak basis transformations in the flavor sector of N-HDM
Invoked when defining invariant traces that remain unchanged under these transformations.
standard math Standard assumptions of renormalizable quantum field theory and the structure of Yukawa interactions in multi-Higgs models
Background framework presupposed for the entire flavor analysis.

pith-pipeline@v0.9.0 · 5633 in / 1166 out tokens · 33045 ms · 2026-05-18T10:07:36.680150+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

complete set of weak basis transformation invariant traces of flavour matrices for the general N doublets case... 10+36(N−1) independent flavour invariants
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

mixing matrices... charged Higgs basis... physical parameters

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 4 internal anchors

[1]

interaction withH k

The quarks mass matrices are Md = v√ 2Γ, M u = v√ 2∆.(2) One can rotate the left-handed and right-handed quark fields nL =V dLdL, n R =V dRdR , pL =V uLuL, p R =V uRuR ,(3) into a basis where their interaction with the Higgs field is diagonal, Md =V dLDdV † dR, D d = diag{md, ms, mb}, Mu =V uLDuV † uR, D u = diag{mu, mc, mt},(4) thus, performing the singu...

work page
[2]

same-space

Then, since the first three rows are themselves a unitary matrix, they can be reparametrised as CdLk ={1, α dLk 1 , αdLk 2 }RdLk 23 {−δdLk,1,1} RdLk 13 {δdLk,1,1}R dLk 12 {αdLk 3 , αdLk 4 , αdLk 5 } {αk 3, αk 4, αk 5}.(21) And, since theα k’s were left as arbitrary, one can choose them to cancelα dLk 3 ,α dLk 4 , andα dLk 5 , leaving theC dLk ma- trices w...

work page arXiv 2024
[3]

Cabibbo,Unitary Symmetry and Leptonic Decays,Phys

N. Cabibbo,Unitary Symmetry and Leptonic Decays,Phys. Rev. Lett.10(1963) 531–533

work page 1963
[4]

Kobayashi and T

M. Kobayashi and T. Maskawa,CP Violation in the Renormalizable Theory of Weak Interaction,Prog. Theor. Phys.49(1973) 652–657

work page 1973
[5]

Georgi and D

H. Georgi and D. V. Nanopoulos,Suppression of Flavor Changing Effects From Neutral Spinless Meson Exchange in Gauge Theories,Phys. Lett. B82(1979) 95–96

work page 1979
[6]

J. F. Donoghue and L. F. Li,Properties of Charged Higgs Bosons,Phys. Rev. D19(1979) 945

work page 1979
[7]

F. J. Botella and J. P. Silva,Jarlskog - like invariants for theories with scalars and fermions,Phys. Rev. D51(1995) 3870–3875, [{\tthep-ph/9411288}]

work page arXiv 1995
[8]

C. C. Nishi,The Structure of potentials with N Higgs doublets,Phys. Rev. D76(2007) 055013, [{\ttarXiv:0706.2685}]

work page internal anchor Pith review Pith/arXiv arXiv 2007
[9]

M. P. Bento, H. E. Haber, J. C. Rom˜ ao, and J. P. Silva, Multi-Higgs doublet models: physical parametrization, sum rules and unitarity bounds,JHEP11(2017) 095, [{\ttarXiv:1708.09408}]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[10]

M. P. Bento, H. E. Haber, J. C. Rom˜ ao, and J. P. Silva, Multi-Higgs doublet models: the Higgs-fermion couplings and their sum rules,JHEP10(2018) 143, [{\ttarXiv:1808.07123}]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[11]

F. J. Botella, G. C. Branco, and M. N. Rebelo,Invariants and Flavour in the General Two-Higgs Doublet Model, Phys. Lett. B722(2013) 76–82, [{\ttarXiv:1210.8163}]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[12]

Santamaria,Masses, mixings, yukawa couplings and their symmetries,Physics Letters B305(May, 1993) 90–97

A. Santamaria,Masses, mixings, yukawa couplings and their symmetries,Physics Letters B305(May, 1993) 90–97

work page 1993
[13]

G. C. Branco, L. Lavoura, and J. P. Silva,CP Violation, vol. 103. 1999

work page 1999
[14]

G. C. Branco and L. Lavoura,Rephasing Invariant Parametrization of the Quark Mixing Matrix,Phys. Lett. B 208(1988) 123–130

work page 1988
[15]

M. P. Bento, J. P. Silva, and A. Trautner,The basis invariant flavor puzzle,JHEP01(2024) 024, [{\ttarXiv:2308.00019}]

work page arXiv 2024
[16]

Jarlskog,Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation,Phys

C. Jarlskog,Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation,Phys. Rev. Lett.55(1985) 1039

work page 1985
[17]

Rasin,Diagonalization of quark mass matrices and the Cabibbo-Kobayashi-Maskawa matrix,{\tthep-ph/9708216}

A. Rasin,Diagonalization of quark mass matrices and the Cabibbo-Kobayashi-Maskawa matrix,{\tthep-ph/9708216}. Appendix A: 3 by 3 Unitary Matrix Parametrisation Let us define R12 =   c12 s12 0 −s12 c12 0 0 0 1   ,(A1) where, henceforth,c ij = cos(θij),s ij = sin(θij), andR −12 = RT

work page arXiv
[18]

Moreover, we define {a, b, c}= diag{e ia, eib, eic}, {1, d, f}= diag{1, e id, eif },(A2) and similarly for extra “1”s. A general 3 by 3 unitary matrix can be parametrised in terms of 3 mixing angles and 6 complex phases as [17] Uuni =   1 0 0 0e iα1 0 0 0e iα2     1 0 0 0c 23 s23 0−s 23 c23   ×   e−iδ 0 0 0 1 0 0 0 1     c13 0s 13 0 1 0 −s13...

work page

[1] [1]

interaction withH k

The quarks mass matrices are Md = v√ 2Γ, M u = v√ 2∆.(2) One can rotate the left-handed and right-handed quark fields nL =V dLdL, n R =V dRdR , pL =V uLuL, p R =V uRuR ,(3) into a basis where their interaction with the Higgs field is diagonal, Md =V dLDdV † dR, D d = diag{md, ms, mb}, Mu =V uLDuV † uR, D u = diag{mu, mc, mt},(4) thus, performing the singu...

work page

[2] [2]

same-space

Then, since the first three rows are themselves a unitary matrix, they can be reparametrised as CdLk ={1, α dLk 1 , αdLk 2 }RdLk 23 {−δdLk,1,1} RdLk 13 {δdLk,1,1}R dLk 12 {αdLk 3 , αdLk 4 , αdLk 5 } {αk 3, αk 4, αk 5}.(21) And, since theα k’s were left as arbitrary, one can choose them to cancelα dLk 3 ,α dLk 4 , andα dLk 5 , leaving theC dLk ma- trices w...

work page arXiv 2024

[3] [3]

Cabibbo,Unitary Symmetry and Leptonic Decays,Phys

N. Cabibbo,Unitary Symmetry and Leptonic Decays,Phys. Rev. Lett.10(1963) 531–533

work page 1963

[4] [4]

Kobayashi and T

M. Kobayashi and T. Maskawa,CP Violation in the Renormalizable Theory of Weak Interaction,Prog. Theor. Phys.49(1973) 652–657

work page 1973

[5] [5]

Georgi and D

H. Georgi and D. V. Nanopoulos,Suppression of Flavor Changing Effects From Neutral Spinless Meson Exchange in Gauge Theories,Phys. Lett. B82(1979) 95–96

work page 1979

[6] [6]

J. F. Donoghue and L. F. Li,Properties of Charged Higgs Bosons,Phys. Rev. D19(1979) 945

work page 1979

[7] [7]

F. J. Botella and J. P. Silva,Jarlskog - like invariants for theories with scalars and fermions,Phys. Rev. D51(1995) 3870–3875, [{\tthep-ph/9411288}]

work page arXiv 1995

[8] [8]

C. C. Nishi,The Structure of potentials with N Higgs doublets,Phys. Rev. D76(2007) 055013, [{\ttarXiv:0706.2685}]

work page internal anchor Pith review Pith/arXiv arXiv 2007

[9] [9]

M. P. Bento, H. E. Haber, J. C. Rom˜ ao, and J. P. Silva, Multi-Higgs doublet models: physical parametrization, sum rules and unitarity bounds,JHEP11(2017) 095, [{\ttarXiv:1708.09408}]

work page internal anchor Pith review Pith/arXiv arXiv 2017

[10] [10]

M. P. Bento, H. E. Haber, J. C. Rom˜ ao, and J. P. Silva, Multi-Higgs doublet models: the Higgs-fermion couplings and their sum rules,JHEP10(2018) 143, [{\ttarXiv:1808.07123}]

work page internal anchor Pith review Pith/arXiv arXiv 2018

[11] [11]

F. J. Botella, G. C. Branco, and M. N. Rebelo,Invariants and Flavour in the General Two-Higgs Doublet Model, Phys. Lett. B722(2013) 76–82, [{\ttarXiv:1210.8163}]

work page internal anchor Pith review Pith/arXiv arXiv 2013

[12] [12]

Santamaria,Masses, mixings, yukawa couplings and their symmetries,Physics Letters B305(May, 1993) 90–97

A. Santamaria,Masses, mixings, yukawa couplings and their symmetries,Physics Letters B305(May, 1993) 90–97

work page 1993

[13] [13]

G. C. Branco, L. Lavoura, and J. P. Silva,CP Violation, vol. 103. 1999

work page 1999

[14] [14]

G. C. Branco and L. Lavoura,Rephasing Invariant Parametrization of the Quark Mixing Matrix,Phys. Lett. B 208(1988) 123–130

work page 1988

[15] [15]

M. P. Bento, J. P. Silva, and A. Trautner,The basis invariant flavor puzzle,JHEP01(2024) 024, [{\ttarXiv:2308.00019}]

work page arXiv 2024

[16] [16]

Jarlskog,Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation,Phys

C. Jarlskog,Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation,Phys. Rev. Lett.55(1985) 1039

work page 1985

[17] [17]

Rasin,Diagonalization of quark mass matrices and the Cabibbo-Kobayashi-Maskawa matrix,{\tthep-ph/9708216}

A. Rasin,Diagonalization of quark mass matrices and the Cabibbo-Kobayashi-Maskawa matrix,{\tthep-ph/9708216}. Appendix A: 3 by 3 Unitary Matrix Parametrisation Let us define R12 =   c12 s12 0 −s12 c12 0 0 0 1   ,(A1) where, henceforth,c ij = cos(θij),s ij = sin(θij), andR −12 = RT

work page arXiv

[18] [18]

Moreover, we define {a, b, c}= diag{e ia, eib, eic}, {1, d, f}= diag{1, e id, eif },(A2) and similarly for extra “1”s. A general 3 by 3 unitary matrix can be parametrised in terms of 3 mixing angles and 6 complex phases as [17] Uuni =   1 0 0 0e iα1 0 0 0e iα2     1 0 0 0c 23 s23 0−s 23 c23   ×   e−iδ 0 0 0 1 0 0 0 1     c13 0s 13 0 1 0 −s13...

work page