Constraints on CP violation in the Higgs sector from the $\rho$ parameter

Alex Pomarol; Roberto Vega

REVIEW

Constraints on CP violation in the Higgs sector from the $\rho$ parameter

Alex Pomarol, Roberto Vega

High Energy Physics - Phenomenology · Submitted 15 May 1993 · 29 references

Pith's verdict: The custodial-CP theorem is Weinberg's and they say so; the new part is the ρ/T application, which is real but softer than the abstract implies.

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arXiv IDhep-ph/9305272 (version 1)

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CitationAlex Pomarol, Roberto Vega. "Constraints on CP violation in the Higgs sector from the $\rho$ parameter." arXiv:hep-ph/9305272 (1993). https://pith.science/paper/ZMRRHPYT

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Editor's note

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Bottom line first: the theorem in the title-adjacent claim is not theirs. Footnote 1 credits Weinberg for the observation that a custodial-invariant Higgs potential conserves CP, and Section 3 plus the appendix is a clean re-derivation of it in the two-doublet language. What is new is the use of that theorem: converting the experimental bound on T into a constraint on CP mixing in the neutral Higgs sector. That is a legitimate, modest contribution, and the paper is honest about its own softness in a way most submissions are not. I would send it out, expecting minor revision, not a fight. The appendix is the strongest part. Eqs. (A.1)–(A.4) are explicit and checkable, and the statement that custodial invariance forces A0 to be a mass eigenstate degenerate with H+ is exactly the kind of thing a reader can verify in an afternoon. The case II analysis is the most interesting thing here and gets only a few sentences: defining the custodial symmetry via M21 rather than Mi allows an approximate custodial invariance in the potential alongside maximal CP violation in the Yukawa sector, at the price of tan β ≃ 1 and m_H0 ≃ m_H+. That is a real observation and deserves more space than it gets. The soft spot is Section 4, and the stress-test note has it right. Eq. (27) shows T depends on both O_31 and the splitting Δm²_{+3}; the tight linear bound in Fig. 2a requires the hand-chosen relation Eq. (29) plus O_32 = O_31 and a fixed neutral spectrum. Fig. 2b, where the splitting is taken to zero, gives a much weaker quadratic dependence. The authors say plainly that the results are 'very sensitive to the different parameters' and invoke naturalness to argue O_32 should not be large while the rest is suppressed. So Fig. 2a is a benchmark, not a bound, and the paper would be stronger if it said so in those words. The top-mass cancellation in Eq. (14) is a second acknowledged softness, and the conclusion's claim to give 'the only bounds' on Higgs-sector CP violation is a bit strong, though defensible once you restrict to heavy Higgs masses where EDM constraints weaken. Who should read it: anyone working on 2HDM CP violation or on custodial symmetry as a organizing principle, and anyone who wants a clean worked example of how a precision observable constrains a scalar sector. It is a useful reference paper, not a landmark. Recommendation: accept for review; the referee's main job is to make them relabel Fig. 2a as illustrative and expand the case II discussion.

Referee report

Significance

The paper makes a clean, checkable structural point that is worth having in the literature: the custodial-CP connection is derived rather than assumed, and the appendix gives the explicit parameter restrictions (Eqs. A.2 and A.4) that a reader can verify line by line. The resulting statement that CP violation in the Higgs-gauge sector forces a quadratically growing contribution to T, and that the constraint strengthens with increasing Higgs mass (the opposite of the electric-dipole-moment situation), is a falsifiable expectation that can be tested once the top mass is known. The paper also states a concrete future benchmark, Delta T = +/- 0.1 at the 'Ultimate LEP' configuration (Eq. 15), at which the constraint would become meaningful. The case-II loophole of Eq. (25) is a genuinely useful caveat for model builders, and the authors correctly credit Weinberg (Ref. [2], footnote 1) for the earlier observation of the CP-custodial link.

Major objections (1)

[Abstract, sentence 2; Section 5, first and second paragraphs; Section 4, Eq. (29) and Figs. 2a-2b] The abstract states without qualification that the authors 'exploit these facts to constrain CP violation using the experimental bounds on rho-1', and Section 5 states that these are 'the only bounds on CP nonconservation in the Higgs sector'. Section 4 itself shows that the extracted bound is not a derived constraint but a naturalness-dependent expectation: Eq. (27) makes T a function of both O_31 and the custodial-breaking splitting Delta m^2_{+3}, and the numerical bound in Fig. 2a follows only after imposing the hand-chosen relation Eq. (29), O_32 = O_31 and m_H3 = (1 + O_31/(2 O_31^max)) m_H+, together with m_H1 = m_H2/2 = 3 m_H+/2. Fig. 2b, where Delta m^2_{+3} is taken to zero, shows the bound weakening substantially because T then depends quadratically on O_31 (Eq. 28). The authors acknowledge this ('very sensitive to the different parameters', and the remark that it 'seems unnatural' for O_32 to be large while the other custodial-breaking parameters are suppressed), so the issue is not concealment but calibration: the abstract and conclusions should say that the T bound constrains custodial breaking, and that CP violation is bounded only under the assumption that the CP-violating source dominates that breaking. A CP-conserving but custodially breaking sector (Delta m^2_{+3} nonzero with O_31 = 0) also contributes to T, so the one-way implication proved in Section 3 does not by itself convert a T bound into a CP-violation bound. The fix is textual and does not require new calculation, but it is load-bearing for how strongly the paper's stated contribution can be read.

Minor objections (7)

[Appendix, Eqs. (A.1) and (A.2)] The identification lambda = lambda_4 = lambda_5 in Eq. (A.2) should be checked against the normalization of the Tr^2{M_1^dagger M_2} operator in Eq. (A.1) and the explicit 1/2 prefactor on the lambda_5 term in Eq. (2). Depending on whether Tr^2 denotes (Tr)^2 or |Tr|^2, a relative factor of two appears between lambda and lambda_4 or lambda_5. The conclusion that all parameters are real is unaffected, but the stated correspondence should be made convention-explicit so that a reader can reproduce it.
[Section 3, paragraph beginning 'In the THDM there are two possible ways of defining the global SU(2)_L x SU(2)_R Symm et] The exhaustiveness of the two definitions (cases I and II) is asserted rather than argued, and it is load-bearing for the generality of the central theorem. Ref. [26] is cited at that sentence; please state explicitly that Ref. [26] establishes that no third inequivalent definition exists, so that the theorem's scope is clear to a reader who does not consult that reference.
[Section 2, Eq. (11) and the sentence immediately following it] The claim that when the charged Higgs is degenerate with two neutral Higgs bosons 'it can be easily shown' that Eq. (11) also holds is given without the one-line argument. Please supply it, since Eq. (11) is the criterion used throughout Section 4.
[Section 2, Eq. (5)] The matrix element O_i1 is used repeatedly (Eqs. 7, 11, 12, 26-29) but never defined in words. Please state that O_i1 is the h0 component of the mass eigenstate H_i, and note that O is orthogonal. In the present text the matrix is rendered as 'OOO' throughout, which should be corrected in the typeset version.
[Fig. 1 caption and Eq. (14)] The caption should record the neutral-Higgs mass sets used in panels (a) and (b) and the value of m_Href, and Eq. (14) should state explicitly the top-quark mass at which the quoted interval -0.93 < T < 0.33 applies, since the text notes that a top contribution can cancel a negative Higgs contribution.
[Abstract, final sentence] The 'possible exception' should be named in the abstract: it is case II, with the accompanying conditions tan beta approximately 1 and m_H0 approximately m_H+ of Eq. (25). As written, the reader cannot tell whether the exception is a loophole in the theorem or a separate model class.
[Whole manuscript] The preprint text contains numerous broken words from typesetting ('violatio n', 'symm etry', 'H iggs', 'cust dial', 'paramet er', 'accurac y', 'lagrang ian', 'const rained', 'repres entatio ns', 'de v oted'), and Section 2 has 'insured' for 'ensured'. These should be cleaned in the journal version.

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The paper

The paper claims

CP violation in the Higgs-gauge sector is possible only if the custodial SU(2) symmetry is broken, which ties Higgs-sector CP violation to the measured rho parameter.

The reading

The paper asks whether CP violation in the Higgs sector can be constrained by the fact that the rho parameter is measured to be essentially one. Its central claim is that CP violation in the Higgs-gauge interactions requires the custodial SU(2) symmetry — the approximate symmetry that keeps rho at one — to be broken. In the two-Higgs-doublet model the authors show that a custodial-invariant scalar potential has only real parameters and therefore conserves CP, so any CP-violating Higgs-gauge coupling must come with custodial breaking. That breaking produces a one-loop contribution to the $T$ parameter (equivalent to $rho-1$) that grows quadratically with the Higgs masses, so the experimental bound on $T$ limits how much CP violation the Higgs sector can contain. The authors also examine Higgs-fermion CP violation and identify a narrow exception in which an approximate custodial symmetry can coexist with CP violation in the Yukawa couplings.

What carries the argument

The load-bearing object is the custodial SU(2) symmetry of the two-Higgs-doublet potential, defined through matrix fields $M_i$ or $M_{21}$ that transform as $M to L M R^dagger$. The paper classifies the two ways of embedding this symmetry, derives the resulting constraints on the general potential in the appendix, and shows that in case I the custodial-invariant potential forces $m_{12}^2, lambda_6, lambda_7$ real and $lambda_4=lambda_5=lambda$ real, so CP is conserved. The observable that carries the argument is the $T$ parameter, defined from the difference of vacuum polarizations $Pi_{11}(0)-Pi_{33}(0)$, whose Higgs contribution contains a term quadratic in the Higgs masses unless Eq. (11) holds: a neutral Higgs with $m_{H_i}=m_{H^+}$ and $O_{i1}=0$. The mixing entries $O_{i1}$ measure how much the mass eigenstates contain the CP-odd state $A^0$, so they are the quantitative handle on CP violation in the Higgs-gauge sector.

Extended reading notes

Core claim

The paper establishes a link between two symmetries of the Higgs sector: CP and the custodial SU(2) that protects the rho parameter. Working in the two-Higgs-doublet model, it shows that if the scalar potential is invariant under the custodial SU(2), all its parameters can be taken real and CP is conserved by the Higgs-gauge sector; conversely, CP violation in that sector requires custodial breaking. The practical consequence is Eq. (11): the one-loop contribution to $T$ that grows quadratically with the Higgs masses vanishes only when some neutral Higgs has mass equal to the charged Higgs and has vanishing mixing with the CP-odd state, which is the CP-conserving limit. Because the experimental value of $T$ is near zero, the size of the custodial-breaking mixing parameter is bounded, and the bound becomes stronger as the Higgs masses grow. For Higgs-fermion interactions the authors find a possible exception, a custodial-invariant potential combined with CP-violating Yukawa couplings, but it requires $tan beta$ close to one and the neutral and charged Higgs masses nearly degenerate.

Load-bearing premise

The numerical constraint on CP violation assumes that the custodial-breaking pattern can be represented by a single mixing parameter with $O_{32}=O_{31}$ and a specific relation between $m_{H_3}$ and $m_{H^+}$, rather than following from the symmetry argument alone.

Editorial extensions

If this is right

If the custodial SU(2) is exact in the Higgs potential, the Higgs-gauge sector conserves CP and the quadratically growing contribution to $T$ is absent.
Any CP-violating Higgs-gauge coupling implies custodial breaking and therefore a one-loop correction to $rho-1$; for heavy Higgs masses this correction grows with the mass scale, so precision electroweak data become a direct constraint on Higgs-sector CP violation.
In the two-Higgs-doublet model, the quadratic $T$ contribution cancels only in the CP-conserving limit $m_{H_i}=m_{H^+}$ and $O_{i1}=0$, so a future measurement of $T$ near zero with $Delta T simeq pm0.1$ would squeeze the allowed CP-violating mixing.
CP violation in Higgs-fermion couplings can survive an approximate custodial symmetry only in the special case $tan beta simeq 1$ and $m_{H^0} simeq m_{H^+}$, which makes that exception sharply testable.
If the discrete symmetries used to suppress flavor-changing neutral currents are exact, the minimal CP-violating three-doublet model has a custodial-invariant potential with real parameters, so CP is conserved by the full Lagrangian.

Reading between the lines

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

The argument turns electroweak precision into a CP-violation sieve: any scalar sector in which CP violation requires custodial breaking will leave a correlated footprint in $rho$, so a future global fit can test CP-violating Higgs models even without observing the CP-odd states directly.
If a CP-violating Higgs-gauge signal is established while $T$ remains at its Standard Model value, the most economical reading within this paper is the fermion-sector exception or a fine-tuned cancellation of custodial-breaking contributions, not a custodial-invariant CP-violating gauge sector.
The numerical bound is contingent on a naturalness assumption about how the custodial limit is approached; a model that arranges cancellations among the custodial-breaking parameters could evade the bound without preserving custodial symmetry, so the constraint should be read as applying to natural CP violation rather than as a theorem forbidding all CP violation.
Because the correction grows with Higgs mass, the constraint is complementary to electric-dipole-moment bounds, which weaken at large Higgs masses; a heavy Higgs sector could therefore be bounded more sharply by $rho$ than by dipole experiments.

Original abstract

We discuss the relation between the CP symmetry and the custodial $SU(2)$ symmetry in the Higgs sector. In particular, we show that CP violation in the Higgs-gauge sector is allowed only if the custodial $SU(2)$ symmetry is broken. We exploit these facts to constrain CP violation using the experimental bounds on $\rho-1$. CP nonconservation in the Higgs-fermion interactions can also be constrained in a similar way although a possible exception is pointed out.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,543 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.82 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.07 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We discuss the relation between the CP symmetry and the custodial $SU(2)$ symmetry in the Higgs sector. In particular, we show that CP violation in the Higgs-gauge sector is allowed only if the custodial $SU(2)$ symmetry is broken. We exploit these facts to constrain CP violation using the experimental bounds on $\rho-1$. CP nonconservation in the Higgs-fermion interactions can also be constrained in a similar way although a possible exception is pointed out.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We discuss the relation between the CP symmetry and the custodial $SU(2)$ symmetry in the Higgs sector.	background 0.71	0.43	0.06	unclear 0.72
s1	In particular, we show that CP violation in the Higgs-gauge sector is allowed only if the custodial $SU(2)$ symmetry is broken.	result 0.94	0.84	0.55	asserted 1.00
s2	We exploit these facts to constrain CP violation using the experimental bounds on $\rho-1$.	method 0.69	0.86	0.28	asserted 0.98
s3	CP nonconservation in the Higgs-fermion interactions can also be constrained in a similar way although a possible exception is pointed out.	limitation 0.45	0.73	0.08	hedged possibility 0.53

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.54.

What was examined, and what was not

sources read

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References

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How the literature uses it

40 arXiv papers cite this in 60 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 · 41 sentences without a confident label

Compares with it 3 papers

In the custodial limit basindcust applies, and it follows that m^2_H^±=m^2_A^0= Y_2 +Z_3 v^2 , in agreement with the results of [ref] .

Basis-independent methods for the two-Higgs-doublet model III: The CP-conserving limit, custodial symmetry, and the oblique parameters S, T, U · Higgs mass degeneracy in the custodial limit · 0.88
which agrees with earlier results [ref] .

Democratic three Higgs-doublet models: the custodial limit and wrong-sign Yukawa · Examples with 2HDMs · 0.97
The former is called the custodial $SU(2)_C$ symmetry in Refs. [ref] and is completely the same as Eq. [ref], i.e., the custodial symmetry of the case II studied by Pomarol and Vega [ref] .

Stable $Z$-strings with topological polarization in two Higgs doublet model · Difference of conventions from previous works · 0.88

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At the one loop level, the $T$ parameter is not affected by the effects from the $Im[λ_7]$ term [ref] .

Electroweak baryogenesis via top-charm mixing · Constraints on the model · 0.96
These choices are well-motivated, because only in these two cases [ref] can the $H$ or the $A$ have a sizeable mass splitting from the rest of the non-SM Higgses without being in conflict with EW precision measurements.

Searching for heavy Higgs bosons in the $t \bar t Z$ and $t b W$ final states · Heavy Higgs interactions and decays · 0.99
Thus, the one-loop corrections to the $ρ$ parameter from the scalar boson loops almost completely vanish due to the approximate custodial symmetry [ref] . ].

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In the 2HDM, the oblique parameters have been extensively studied [ref] .

CDF $W$ boson mass and muon $g-2$ in type-X two-Higgs-doublet model with a Higgs-phobic light pseudoscalar · Scanning strategies and the results · 0.99
Second, we discuss oblique parameters such as $S, T, U$ [ref] , especially the $T$ parameter. $λ_4 - Re[λ_5]$ and $Im[λ_7]$ terms in the potential violate the custordial symmetry [ref] .

New benchmark scenarios of electroweak baryogenesis in aligned two Higgs double models · Collider experiments · 0.89
The custodial transformation acting on the matrix $H$ [ref] is defined as the following global $SU(2)$ transformation: $H →U H U ^†$, $U ∈SU(2)_C$ [ Note that this $SU(2)_C$ transformation is different from the $U(2)$ basis transformation: $Φ_i →∑_j=1^2M_ij Φ_j$ ($i=1,2$). ].

Topological Nambu monopole in two Higgs doublet models · The model · 0.94

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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