Conditions for CP-Violation in the General Two-Higgs-Doublet Model

Howard E. Haber; John F. Gunion

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Conditions for CP-Violation in the General Two-Higgs-Doublet Model

John F. Gunion, Howard E. Haber

High Energy Physics - Phenomenology · Submitted 23 Jun 2005 · 22 references

Pith's verdict: A genuine classification theorem for 2HDM CP violation, proved rather than asserted; the weak part is the invariant survey in Section IV, and the authors say so themselves.

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arXiv IDhep-ph/0506227 (version 2)

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CitationHoward E. Haber, John F. Gunion. "Conditions for CP-Violation in the General Two-Higgs-Doublet Model." arXiv:hep-ph/0506227 (2005). https://pith.science/paper/NMOT6XK7

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Editor's note

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The thing worth knowing: Theorem 2 is real and it is proved here. Four basis-independent invariants, eqs. (23)-(26), and the potential is explicitly CP-conserving iff all four have vanishing imaginary part. Necessity is immediate; sufficiency is a case analysis in Section III backed by Lemmas 2 and 4 in the appendices. I went at Lemma 2 hard because it is the load-bearing step at the isolated point lambda1 = lambda2, lambda7 = -lambda6, and it holds up. G(theta,xi) is a polynomial in sin/cos, so continuity is fine, and G(0,xi) = -G(pi/2,xi) = 16 lambda6^2 |lambda5| sin(theta5) forces a sign change unless lambda5 is already real, in which case there is nothing to prove. The degenerate cases are not hand-waved either: if fa = fb = 0 then Im lambda5' vanishes identically and eq. (C8) fixes chi; if fc = fd = 0 the roles swap and eq. (C7) does the job. That is a construction, not an assertion. The stress-test note is right on this and I agree with it against the reader's weaker read of that lemma. Credit where due: the paper is explicit that two invariants come from Davidson-Haber and three were used by Branco-Rebelo-Silva-Marcos. The new content is the fourth invariant, the completeness argument, and the sufficiency proof. That is a real addition to a literature that previously had only the sufficient basis-dependent conditions of eq. (33). Where it is soft, and the authors say so: Section IV. The sixth-order Z scan covers about nine million of 12! cases, and the 2Y4Z and Y5Z conclusions rest on partial scans of roughly two million invariants (footnote 17). Eq. (54) is offered as a conjecture, and the text calls it that. None of this touches Theorem 2, which is why I would not hold the paper hostage to it. One small presentational gripe: the abstract says all four invariants are needed, while the body correctly notes that away from the isolated point only three matter. Worth tightening, not worth a fight. Who should read it: anyone doing 2HDM phenomenology, MSSM effective-potential work, or basis-independent model building. It is a reference paper, not a beach read. Send it out.

Referee report

Significance

The result is a useful reference tool for the 2HDM community: it converts the question of whether a real basis exists into an algebraic test on four basis-independent quantities that can be evaluated in any starting basis, without first having to find the special basis in which lambda7 = -lambda6. The paper is careful to separate explicit from spontaneous CP violation, and it is explicit about the limits of the Section IV survey (partial scans of sixth-order and higher invariants), correctly noting that statement 9 is a consequence of Theorem 2 rather than of the scans. The appendices contain complete proofs of the lemmas, including the continuity argument in Lemma 2 and the explicit U(2) constructions in Lemma 4, and the paper is self-contained apart from the cited results of refs. [6], [11] and [4], which are used for supporting material rather than for the central theorem.

Minor objections (8)

[Abstract and Theorem 2 (Section III)] The abstract and Theorem 2 describe the four invariants as 'independent', but Section III immediately shows that at most two need be checked in any given model, and only one at the isolated point lambda1 = lambda2, lambda7 = -lambda6. The word 'independent' is presumably meant in the technical sense of 'not expressible in terms of lower-order invariants', and it would help to say so explicitly in the abstract or in the paragraph following Eq. (27).
[Section IV.D, first sentence] 'The analyzes of Sections IV.A and IV.B permit us to conjecture...' should read 'The analyses of Sections IV.A and IV.B permit us to conjecture...'.
[Footnote 20] 'only one has an non-vanishing imaginary part' should read 'only one has a non-vanishing imaginary part'.
[Eqs. (39) and (40)] The symbol Lambda is used in Eq. (39) before its definition in Eq. (40). A forward reference such as '(Lambda is defined in Eq. (40))' at first use would help the reader.
[Section IV.C] The text notes that only about nine million of the 12! = 479,001,600 sixth-order Z-invariants were scanned. This is an honest limitation, but the surrounding discussion would be clearer if it stated explicitly that statement 9 rests on Theorem 2 and that the scan is illustrative rather than probative.
[Section IV.D, Eq. (54)] The results of Section IV.D are introduced as a conjecture, which is appropriate, but some of the surrounding prose reads as if the results were established. A consistent label ('conjecture' or 'expected') throughout Section IV.D would avoid ambiguity.
[Section III, paragraph below Eq. (22)] The parameter-counting argument (nine real parameters and two phases) is clear, but it would be worth stating explicitly that the four invariants are not all independent in the sense of parameter counting, which is precisely the point of the following paragraph. This would preempt a natural reader objection.
[Section V, Theorem 5] The statement of Theorem 5 is correct, but the text could note that the direction 'CP-invariance implies J1, J2, J3 real' is immediate from the existence of a real basis with real vevs, while the sufficiency direction is the cited result of refs. [11] and [4]. This would make the logical structure of the theorem clearer.

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The paper

The paper claims

Four basis-independent invariants decide whether the general two-Higgs-doublet potential breaks CP explicitly, and three more decide whether the vacuum breaks it spontaneously.

The reading

In the general two-Higgs-doublet model the scalar potential carries four parameters that can be complex, but a change of basis in the two-dimensional Higgs 'flavor' space shuffles those parameters around. The paper asks how to tell, without first hunting for a convenient basis, whether some basis exists in which every parameter is real. Its answer is a set of four basis-independent combinations of the mass and quartic-coupling tensors whose imaginary parts, if all zero, guarantee that such a real basis exists, and if any one is nonzero, guarantee that none does. Because the invariants are polynomials in the couplings, the test can be run directly in whatever basis a model builder happens to write down. The paper then adds three further invariants built with the vacuum expectation values so that explicit CP violation can be separated from spontaneous CP violation, where the potential is real but the vacuum is not.

What carries the argument

The machinery is the tensorial reformulation of the potential as $V = Y_{āb}Φ^{†ā}Φ^b + ½ Z_{ābc̄d}(Φ^{†ā}Φ^b)(Φ^{†c̄}Φ^d)$, in which $Y$ and $Z$ transform covariantly under the global $U(2)$ of Higgs-flavor rotations. Contracting the indices of these tensors in every allowed way produces $U(2)$-scalar invariants; the ones that can carry a phase are the I-invariants. The paper reports a systematic survey: every invariant of cubic order or less is manifestly real, all quartic invariants that can be complex reduce to $I_{Y^3Z}$ and $I_{2Y^2Z}$, and the genuinely new conditions first appear at sixth order with $I_{6Z}$ (pure $Z$) and $I_{3Y^3Z}$ (three $Y$'s and three $Z$'s). In the special $λ_7 = -λ_6$ basis these collapse to compact expressions such as $I_{Y^3Z} = -(λ_1-λ_2)^2$ Im$(Y_{12}λ_6^*)$ and $I_{6Z} = -(λ_1-λ_2)^3$ Im$(λ_6^2λ_5^*)$, which is what makes the sufficiency proof tractable.

Extended reading notes

Core claim

The central claim is Theorem 2: the general two-Higgs-doublet scalar potential is explicitly CP-conserving if and only if the imaginary parts of four specific invariant combinations vanish, namely $I_{Y^3Z}$, $I_{2Y^2Z}$, $I_{6Z}$ and $I_{3Y^3Z}$ as defined in eqs. (23)–(26). The 'only if' direction is immediate, since a nonzero imaginary part of a basis-independent quantity can never be removed by a basis change. The 'if' direction is the real work: the authors transform to a basis where $Z^{(1)}$ and $Z^{(2)}$ are simultaneously diagonal (so $λ_7 = -λ_6$) and where $λ_6$ is real, then split the analysis into $λ_1 ≠ λ_2$ and the isolated point $λ_1 = λ_2$, showing in each case that vanishing of the relevant invariants forces the remaining potentially complex parameters to be real or removable. The paper also shows that the four conditions are genuinely independent by exhibiting four models in each of which exactly one invariant is nonzero, and that no smaller set suffices in general even though any single model needs at most two of them checked. A companion set of results (Theorems 3–5) uses the three vacuum-dependent invariants $J_1$, $J_2$, $J_3$ to state necessary and sufficient conditions for spontaneous CP violation.

Load-bearing premise

The sufficiency proof at the isolated point $λ_1 = λ_2$, $λ_7 = -λ_6$ rests on Lemma 2, which argues that a basis with all $λ_i$ real always exists by noting that a certain function changes sign between $θ = 0$ and $θ = π/2$; the degenerate cases where the auxiliary quantities vanish are handled by brief assertion rather than by an explicit construction.

Editorial extensions

If this is right

A model builder can test explicit CP conservation by evaluating four polynomials in the couplings as written, with no need to search for the basis in which $λ_7 = -λ_6$.
The often-quoted conditions Im$(Y_{12}^2λ_5^*) = $ Im$(Y_{12}λ_6^*) = $ Im$(Y_{12}λ_7^*) = $ Im$(λ_5^*λ_6^2) = \cdots = 0$ are sufficient but not necessary; the paper exhibits three explicitly CP-conserving models that violate them.
Combining the four I-invariants with the three vacuum-dependent J-invariants gives a decision procedure that separates explicit from spontaneous CP violation without ever fixing a basis.
Although four invariants appear in the theorem, at most two need be checked for any particular model, because only two physical phases survive the $U(2)$ freedom.
Because the invariants are polynomial in the parameters, they can be re-expressed in terms of fundamental supersymmetric parameters once loop corrections generate phases in the MSSM Higgs sector.

Reading between the lines

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

The theorem is really a statement about the ring of $U(2)$-invariant polynomials of $Y$ and $Z$: the four imaginary parts are the odd-parity generators, and the paper's survey of orders up to six is an empirical hunt for a minimal generating set. A clean algebraic characterisation of that ring would turn the case-by-case proof into a one-line orbit-space argument.
Since the invariants are cheap to evaluate, a natural extension is a public numerical scan over 2HDM parameter space mapping the regions where each invariant is nonzero, which would tell experimenters which CP-violating patterns are reachable at all.
The observation that at most two invariants matter per model suggests syzygies among the four; writing them down explicitly would give a smaller, model-adaptive test and would clarify why the naive phase counting (two physical phases) and the invariant counting (four conditions) do not conflict.
If a future lepton collider measures Higgs mixing angles with enough precision, the I-invariants offer basis-independent coordinates in which to report the extracted CP-odd parameters, avoiding the ambiguity of quoting phases that a basis change can rotate away.

Original abstract

The most general Higgs potential of the two-Higgs-doublet model (2HDM) contains three squared-mass parameters and seven quartic self-coupling parameters. Among these, one squared-mass parameter and three quartic coupling parameters are potentially complex. The Higgs potential explicitly violates CP symmetry if and only if no choice of basis exists in the two-dimensional Higgs ``flavor'' space in which all the Higgs potential parameters are real. We exhibit four independent potentially complex invariant (basis-independent) combinations of mass and coupling parameters and show that the reality of all four invariants provides the necessary and sufficient conditions for an explicitly CP-conserving 2HDM scalar potential. Additional potentially complex invariants can be constructed that depend on the Higgs field vacuum expectation values (vevs). We demonstrate how these can be used together with the vev-independent invariants to distinguish between explicit and spontaneous CP-violation in the Higgs sector.

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The most general Higgs potential of the two-Higgs-doublet model (2HDM) contains three squared-mass parameters and seven quartic self-coupling parameters. Among these, one squared-mass parameter and three quartic coupling parameters are potentially complex. The Higgs potential explicitly violates CP symmetry if and only if no choice of basis exists in the two-dimensional Higgs ``flavor'' space in which all the Higgs potential parameters are real. We exhibit four independent potentially complex invariant (basis-independent) combinations of mass and coupling parameters and show that the reality of all four invariants provides the necessary and sufficient conditions for an explicitly CP-conserving 2HDM scalar potential. Additional potentially complex invariants can be constructed that depend on the Higgs field vacuum expectation values (vevs). We demonstrate how these can be used together with the vev-independent invariants to distinguish between explicit and spontaneous CP-violation in the Higgs sector.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	The most general Higgs potential of the two-Higgs-doublet model (2HDM) contains three squared-mass parameters and seven quartic self-coupling parameters.	background 1.00	0.13	0.02	asserted 1.00
s1	Among these, one squared-mass parameter and three quartic coupling parameters are potentially complex.	background 0.99	0.22	0.03	hedged possibility 0.78
s2	The Higgs potential explicitly violates CP symmetry if and only if no choice of basis exists in the two-dimensional Higgs ``flavor'' space in which all the Higgs potential parameters are real.	background 0.67	0.21	0.18	asserted 1.00
s3	We exhibit four independent potentially complex invariant (basis-independent) combinations of mass and coupling parameters and show that the reality of all four invariants provides the necessary and sufficient conditions for an explicitly CP-conserving 2HDM scalar potential.	result 0.63	0.96	0.83	asserted 1.00
s4	Additional potentially complex invariants can be constructed that depend on the Higgs field vacuum expectation values (vevs).	contribution 0.47	0.52	0.03	asserted 0.51
s5	We demonstrate how these can be used together with the vev-independent invariants to distinguish between explicit and spontaneous CP-violation in the Higgs sector.	contribution 0.74	0.93	0.10	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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full text
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References

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After the 125 GeV Higgs, 2HDM scalar parameter space is squeezed by stability, unitarity, and T-parameter constraints, with softly broken U(1) symmetry allowing heavier individual nonstandard scalars than exact Z2.
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A pedagogical survey of how extended Higgs sectors are built and tested, covering 2HDM, NHDM, singlet and triplet extensions, and their collider and astroparticle signatures.

How later papers used it

72 later papers cite it

Evidence: substantial. 67 of the citing papers share no author with it, and 11 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 133 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 1 used its method or tool and 1 cited it as support. 1 more paper by the same authors is listed: 1 built on it.

Builds on it 1 paper · 1 by the same authors

By the same authors

Generalized this paper, in which relations 4 and 6 were noted and discussed.

Group-theoretic Condition for Spontaneous CP Violation (2012) (same authors) · 0.98 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 of 9 papers with this label

Used the invariants J_1, J_2 and J_3 defined in this paper, finding it convenient to do so.

Natural Two-Higgs-Doublet Model (2010) · 0.93 · paraphrased: the licence does not allow quoting

Cites it as support 1 paper

This shows that the leading imaginary divergent contribution to $m_12^2$ always occurs at one higher loop order than the leading imaginary divergent contribution to $λ_5$, thereby ensuring that the phase of $λ_5$ thus generated cannot be trivially rotated away [ref] .

Is the real two-Higgs-doublet model consistent? (2024) · Type-I 2HDM · 0.95

How the literature uses it

60 arXiv papers cite this in 107 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 72 sentences without a confident label

Builds on it 1 paper

This generalizes [ref] , in which relations 4 and 6 were noted and discussed.

Group-theoretic Condition for Spontaneous CP Violation · Invariant relations · 0.98

Uses its method or tool 7 papers

In the MR basis, the most general potential $V[Φ_a]$ can be written as [ref] .

Constraining Flavour Symmetries At The EW Scale I: The A4 Higgs Potential · The $A_4$ Scalar Potential · 0.85
In [ref] it is shown how to construct basis-invariant quantities from contractions between tensor indices of $V_ab̅$, $Y_ab̅$ and $Z_ab̅cd̅$ following a certain pattern.

Diagnosing CP properties of the 2HDM · Appendix. Observable parameters of the potential · 0.90
For descriptions of this technique, confer [ref] , see also Refs. [ref] .

Complex $S_3$-symmetric 3HDM · Explicit CP violation · 0.97

Compares with it 2 papers

However, we have verified that for the model which was considered in this paper, the conditions for CP conservation obtained in [ref] coincides with those found in [ref] .

Diagnosing CP properties of the 2HDM · The model · 0.99
This result is identical to the one found previously in [ref] , which is based on the construction of all possible Jarlskog-like [ref] , Higgs-basis independent CP-odd invariants [ref] (for a recent review, see [ref] ).

Vacuum Topology of the Two Higgs Doublet Model · Neutral Vacuum Solutions from a Singular Matrix N · 0.98

Cites it as background 22 papers

While these models admit explicit CPV in the scalar potential [ref] , most studies have imposed CP conservation in the scalar sector, yielding the model known as the real 2HDM.

Is the real two-Higgs-doublet model consistent? · Introduction · 0.89
A two-Higgs-doublet model (2HDM) [ref] is not only one of the simplest extensions of the SM which may provide new sources for CPV [ref] , but also the prototype employed in numerous more elaborate new physics models [ref] .

Higgs alignment and novel $CP$-Violating observables in two-Higgs-doublet models · Introduction · 0.85
This should ensure, in models with extra Higgses[CP Violation in the 2 Higgs Doublet Model is discussed in [ref] , where $d_e$ has recently been studied in the decoupling limit [ref] .] or “leptons” (hypercharge $-1/2$ fermionic doublets[For instance, a fourth generation of “vector-like” leptons, as discussed in [ref] , or see [ref] for phenomenological constraints.]), that it is the light “standard” Higgs and leptons which participate in the Yukawa coupling with the ${e_R^i}$.

Left-Handed Physics is not right for EDMs · Vanishing one-loop edms · 0.93

Discussion

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