Resonant CP violation induced by particle mixing in transition amplitudes

Apostolos Pilaftsis (MPI; Munich)

REVIEW

Resonant CP violation induced by particle mixing in transition amplitudes

Apostolos Pilaftsis (MPI, Munich)

High Energy Physics - Phenomenology · Submitted 21 Feb 1997 · 67 references

Pith's verdict: A solid, careful paper whose real contribution is the Jordan-form maximal-mixing limit and the gauge-invariant resummation; the 'order one' collider numbers are a stated ceiling, not a prediction, and the paper mostly says so.

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CitationApostolos Pilaftsis (MPI, Munich). "Resonant CP violation induced by particle mixing in transition amplitudes." arXiv:hep-ph/9702393 (1997). https://pith.science/paper/4O6KFYIB

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Worth your time, mainly for Sections 3–5. The genuinely new thing here is the observation that maximal mixing-induced CP violation corresponds to a non-diagonalizable (Jordan-form) effective Hamiltonian, where the similarity transformation X goes singular but the scattering amplitude stays perfectly well defined. That is a clean, correct point, and it explains why the propagator-resummation approach is the right tool rather than a stylistic preference. The pinch-technique treatment unifying the bosonic and fermionic cases is also done properly, and the CPT section is honest: it shows explicitly that a_CP stays below unity even when |q/p| hits its extreme, via the toy Hamiltonian (4.15)–(4.17). The appendices on mixing renormalization are standard but competently assembled and useful to have in one place. The analytic core — propagator inversion, Eq. (3.5), the CPT relations (4.9)–(4.14), the Lee-Wolfenstein saturation argument — checks out on inspection. I found no error in it. On the stress-test note: the claim that the paper 'never scans xA' does not hold up. Figs. 5b and 6b are precisely plots of A_CP against xA, and the text says outright that the asymmetry becomes large 'if the parameter xA was tuned.' The paper is not hiding the tuning. What is genuinely missing is a naturalness argument — nothing shows that a realistic two-Higgs or mirror-fermion parameter point lands near xA ~ 1, and the EDM bound (7.14) only permits it. That is a real gap, but a modest one, and the paper's own LHC sentence ('chances ... appear to be quite limited') is more measured than the abstract's framing. The fermionic section is the thinner half: the factorization in Eq. (9.11) is asserted rather than justified, and the cancellation argument for unpolarized a_CP is sketched. The CPT cross-check of Eq. (7.8) against (7.9) is not shown explicitly, though it should follow by construction. None of this is load-bearing. Send it out; the Jordan-form section alone will be cited by people working on degenerate unstable systems.

Referee report

Major objections (1)

[Section 7, Eq. (7.8) and Figs. 5–6] The headline numerical claim that CP asymmetries can be 'of order one' at colliders rests on tuning the ratio xA = Pi_HA / Im(Pi_HH - Pi_AA) to specific values chosen by hand (xA = 1 for process (a), xA = 3 for process (b), and xA = 0.07 or 3 for the electron-collider case). The paper is transparent that these are tuned values, and the neutron-EDM bound (Eq. 7.14) does not forbid xA ~ O(1) for narrow widths, so the tuning is not unphysical. However, no scan over xA or over the mass splitting M_H - M_A (which must also be within ~8 GeV, Section 7) is presented to show that a natural region of the model parameter space populates the tuning. Consequently the observability claim in the introduction ('may hence be observed at future high-energy pp, e+e- or mu+mu- machines') is a statement about the ceiling of the mechanism, not about a typical or even reasonably probable point. I recommend either adding a modest scan over xA and the mass splitting to establish that xA ~ O(1) is reachable in a natural region, or explicitly framing the 'order one' result as a ceiling and toning down the observability language in the introduction and conclusions.

Minor objections (6)

[Section 7, Eq. (7.8)] The distribution of x_f and r_d factors in the denominator does not follow directly from Eqs. (7.6)–(7.7) unless additional assumptions about the mu+mu- production vertex (e.g., r_mu = 1) are made. As written, the x_f r_d^2 multiplying (s - M_A^2)^2 and the x_f multiplying Pi_AH^2 differ from what one obtains by factoring |M(A* -> mu)|^2 |M(A* -> f fbar)|^2 |Delta_AH|^2 out of Eqs. (7.6)–(7.7). Please clarify the assumptions or correct the expression.
[Section 7, Eq. (7.8) and Eq. (4.14)] The CPT sum rule (4.14) is demonstrated explicitly only for the toy Hamiltonian (4.15)–(4.17). It would strengthen the paper to verify explicitly that Eq. (7.8) satisfies the CPT constraint (7.9) for the full two-Higgs model, rather than relying on the general argument of Section 4.
[Section 5] The paper claims that the resummation approach is free from the pathological singularities of the effective Hamiltonian in the Jordan limit. It would be helpful to show explicitly how the propagator Delta(s) behaves in this limit, e.g., by giving the explicit form of the transition amplitude for the Hamiltonian (5.7) and demonstrating that it remains finite and well defined.
[Section 3, Eq. (3.2) and Eq. (3.5)] The sign convention for the i Pi_AH term in Eq. (3.2) differs from what one obtains from the basis transformation (3.1) with the standard definition of the self-energy; the two are related by a convention for the sign of Pi_AH. Please state this convention explicitly so that the reader can verify the consistency of Eqs. (3.2) and (3.5) without ambiguity.
[Section 9, Eq. (9.22)] The derivation of Eq. (9.22) uses the approximation A_ij^2 << 1. Please state the parametric regime in which this approximation is valid, and comment on how the result changes if A_ij is not small.
[Notation] The use of both H(s) and H~ for the effective Hamiltonian in different bases could be confusing. A brief reminder of the basis at each use, or a table of notation, would help the reader.

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The paper

The paper claims

CP violation peaks when two mixed particles nearly coincide

The reading

The paper asks when the mixing of two particles with opposite CP parities can produce a CP asymmetry far larger than the usual small effects, and answers: when the two particles are nearly degenerate, with a mass difference comparable to their decay widths. Working with scattering amplitudes rather than a diagonalized effective Hamiltonian, the author derives the conditions under which the mixing parameter $|q/p|$ departs strongly from 1, then checks that CPT invariance does not forbid the effect but does cap it. Two concrete settings are worked out: mixing of a CP-even Higgs $H$ with a CP-odd Higgs $A$ in two-Higgs doublet models, and mixing of the top quark with a new up-type quark $t'$ in mirror-fermion models. The payoff, if the reasoning holds, is that CP asymmetries of order one could sit on a Higgs resonance line or in polarized top decays, reachable at a muon collider, the LHC, or a linear $e^+e^-$ machine.

What carries the argument

The load-bearing object is the resummed inverse propagator matrix $s - H(s)$ for the mixed pair, together with the ratio $|q/p| = |tilde H_{21}/tilde H_{12}|^{1/2}$ that measures CP-asymmetric mixing in the $K^0$ basis. Gauge invariance of the resummation is secured by the pinch technique, whose Ward identities convert $ZH$ and $ZZ$ strings into would-be-Goldstone $G^0H$ and $G^0G^0$ ones before resummation, so that only $H$ and $G^0$ mix and no CP-odd gauge artifacts appear off mass shell. The machinery does the work of separating two things that the effective-Hamiltonian approach conflates: the orthogonal, order-by-order renormalized mass eigenstates of the Hermitian Lagrangian, and the non-orthogonal eigenvectors of the non-Hermitian effective Hamiltonian. That separation is what lets the author claim a physically meaningful maximal-CP-violation limit at degeneracy.

Extended reading notes

Core claim

The central claim is that mixing-induced CP violation in a transition amplitude is resonantly amplified when two intermediate particles of opposite CP parity have a mass splitting of the order of their widths, and that the effect reaches its maximum allowed value when the two non-free particles are exactly degenerate. The author expresses the transition amplitude through a resummed $2$-by-$2$ propagator matrix $[s-H(s)]^{-1}$ built from $HH$, $AA$ and $HA$ self-energies, and shows that in a $K^0$-like basis the CP-violating parameter obeys $|q/p|^2 = |tilde H_{21}/tilde H_{12}|$, which departs from 1 only when the off-diagonal mixing competes with the difference of absorptive self-energies. Two conditions are stated: for a dispersive mixing term, $Im(hatPi_{HH}-hatPi_{AA}) sim pm 2 Re,hatPi_{HA}$; for a mass-basis mixing term, $M_H^2 - M_A^2 sim pm 2 Im,hatPi_{HA}$. At exact degeneracy the effective Hamiltonian becomes non-diagonalizable, taking the Jordan form, so that one of $tilde H_{12}, tilde H_{21}$ vanishes and CP violation through mixing is maximal, saturating the Lee-Wolfenstein unitarity bound $|varepsilon|^2 le 1$. The author stresses that this singular limit breaks the similarity transformation used to diagonalize the effective Hamiltonian, while the scattering-amplitude formulation remains well defined, and that CPT invariance enforces a compensating reduction: summed over all final states, the CP-violating differences cancel, so the observable asymmetry stays below unity even when $|q/p|$ is extreme.

Load-bearing premise

The headline claim that CP asymmetries can reach order one rests on tuning the ratio of the $HA$ mixing term to the difference of absorptive self-energies to particular values chosen by hand for each process, and on there being a natural reason for two particles to sit within a width of each other in mass.

Editorial extensions

If this is right

A CP asymmetry of order one can appear on a Higgs resonance line when $M_H \approx M_A$ to within a width, giving a distinctive peak-shaped signal in $\mu^+\mu^- \to H^*,A^* \to f\bar f$ or $e^+e^- \to H^*,A^* \to f_L\bar f_L X$ rather than a flat small asymmetry.
The same mechanism in the top sector predicts an order-one asymmetry in polarized top decays, $A_{CP}^{pol}$, for $10^{-4} < (m_t^2-m_{t'}^2)/m_t^2 < 10^{-1}$, potentially probed with the $10^6$–$10^7$ top decays expected at the LHC.
Neutron and electron electric dipole moment bounds constrain how large the $HA$ mixing term can be, giving $\hat\Pi_{HA}/M^2 < 0.10$ to $0.24$ for $M = 200$ to $600$ GeV, so the resonant region survives existing EDM limits only for a sufficiently small $H$–$A$ mass splitting.
Because CPT forces the sum of all partial CP-violating differences to vanish, a measured asymmetry in one channel implies compensating asymmetries elsewhere, which is a checkable pattern rather than a free parameter.
The gauge-invariant resummation removes CP-odd off-shell transitions between states of different CP parity that a naive $R_\xi$-gauge calculation would produce, so predictions for $HZ$ mixing are correspondingly small.

Reading between the lines

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

The paper demonstrates a ceiling, not a typical size: the order-one asymmetries shown in the figures require tuning $x_A = \hat\Pi_{HA}/\Im m(\hat\Pi_{HH}-\hat\Pi_{AA})$ to values near 1, 3, or 0.07. A natural follow-up would ask what distribution of $x_A$ a generic soft-CP-violating two-Higgs model predicts, since that, not the ceiling, decides whether the effect is discoverable.
The Jordan-form limit is mathematically striking but physically delicate: at exact degeneracy the mass eigenstates lose meaning, so any experimental statement about 'the two particles' becomes a statement about a single resonant structure with an internal two-state character. This suggests looking for the effect as a lineshape distortion rather than as a pair of separated peaks.
The same criterion — mass splitting comparable to width, mixing term comparable to the absorptive difference — should transfer to any two nearly degenerate unstable states with opposite discrete quantum numbers, including neutral-meson-like systems in other sectors and possibly scalar dark sectors with near-degenerate CP-even and CP-odd states.
If the CPT cancellation is as rigid as claimed, the largest observable asymmetry should appear in channels that exclude the initial state from the final-state sum, which is a concrete selection rule for experimental analyses.

Original abstract

We analyze CP violation in resonant transitions involving scalar as well as fermionic intermediate states, using a gauge-invariant resummation approach implemented by the pinch technique. We derive the necessary conditions for resonantly enhanced CP violation induced by particle mixing, by paying special attention to CPT invariance, and apply the results of our analysis to two representative new-physics scenarios: (i) the indirect mixing of a CP-even Higgs scalar with a CP-odd Higgs particle in two-Higgs doublet models and (ii) the CP-asymmetric mixing between the top quark and a new heavy up-type quark, $t'$, in mirror fermion models. Furthermore, we explicitly demonstrate the equivalence of our scattering-amplitude formalism with that based on the effective Hamiltonian and discuss the possibility of maximal CP violation in the limiting case of two degenerate particles.

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We analyze CP violation in resonant transitions involving scalar as well as fermionic intermediate states, using a gauge-invariant resummation approach implemented by the pinch technique. We derive the necessary conditions for resonantly enhanced CP violation induced by particle mixing, by paying special attention to CPT invariance, and apply the results of our analysis to two representative new-physics scenarios: (i) the indirect mixing of a CP-even Higgs scalar with a CP-odd Higgs particle in two-Higgs doublet models and (ii) the CP-asymmetric mixing between the top quark and a new heavy up-type quark, $t'$, in mirror fermion models. Furthermore, we explicitly demonstrate the equivalence of our scattering-amplitude formalism with that based on the effective Hamiltonian and discuss the possibility of maximal CP violation in the limiting case of two degenerate particles.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We analyze CP violation in resonant transitions involving scalar as well as fermionic intermediate states, using a gauge-invariant resummation approach implemented by the pinch technique.	method 0.58	0.94	0.07	asserted 0.95
s1	We derive the necessary conditions for resonantly enhanced CP violation induced by particle mixing, by paying special attention to CPT invariance, and apply the results of our analysis to two representative new-physics scenarios: (i) the indirect mixing of a CP-even Higgs scalar with a CP-odd Higgs particle in two-Higgs doublet models and (ii) the CP-asymmetric mixing between the top quark and a new heavy up-type quark, $t'$, in mirror fermion models.	contribution 0.92	0.96	0.51	asserted 1.00
s2	Furthermore, we explicitly demonstrate the equivalence of our scattering-amplitude formalism with that based on the effective Hamiltonian and discuss the possibility of maximal CP violation in the limiting case of two degenerate particles.	result 0.57	0.94	0.20	asserted 0.92

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Figures

**Figure 1.** Resonant CP violation induced by HA mixing. Let us consider the resonant prototype process ab → H∗ , A∗ → cd in

**Figure 2.** PT WI’s involving the HZ mixing. We will now discuss the mixing of a CP-even (Higgs) scalar, H, with a massive (gauge) vector particle, e.g., the Z boson. It is obvious that the scalar, H, can only couple to the longitudinal component of the gauge particle due to angular momentum conservation. In spontaneous symmetry breaking (SSB) theories, the longitudinal component of a gauge boson, Z, may be represented equally well by the respective would-be Goldstone boson, G0 , which is a CP-odd scalar. The advantages of our gauge-invariant resummation approach may be seen in the description…

**Figure 3.** CPT invariance in the squared transition amplitudes. 13

**Figure 4.** HA mixing induced by heavy Majorana neutrinos. As shown in

**Figure 5.** (a) Numerical estimates of production cross-sections and CP violation for µ − L,R µ + L,R → h ∗ (H∗ ), A∗ → f ¯f as a function of c.m. energy √ s. (b) CP asymmetry versus xA = ΠSA/ℑm(Πb SS − Πb AA), with S = h, H

**Figure 7.** Diagrams contributing to the EDM of a fermion. At the LHC, the respective CP asymmetry A (p) CP (ˆs) may be obtained from Eq. (7.11), for Higgs particles that have a production mechanism similar to that at the NLC. If the Higgs is produced via gluon fusion, one has to use the analytic expression of A (µ) CP (ˆs) and replace rd with ru in Eq. (7.8). Unless CP violation is resonantly amplified, i.e., of order one, the chances to detect CP-violating phenomena on the Higgs-resonance line after removing the contributing background appear to be quite limited at the LHC. It is therefore…

**Figure 8.** The two-loop Barr-Zee mechanism for generating EDM. The presence of a HA operator may also contribute to other low-energy CP-violating observables. CP-violating quantities sensitive to HA terms, for which there exist quite strict experimental upper bounds, are the EDM’s of the neutron, electron and muon. From the one-loop flavour diagrams shown in

**Figure 9.** Resonant CP-violating tt′ transitions. It is now worth emphasizing that we consider CP violation that originates mainly from the mixing of the intermediate up-type fermionic states t and t ′ . However, one should bear in mind that the tree-level production and decay vertices V d,s i , e.g., W uidj , contain extra flavour rotations which result from diagonalizing the mass matrix of the up-quark family by means of bi-unitary transformations. In the SM, these rotations are parameterized by the well-known Cabbibo-Kobayashi-Maskawa (CKM) mixing matrix in the flavour basis where the d

**Figure 10.** tt′ transitions in models with sequential doublets. In models with additional sequential doublets, one can generally induce a tt′ transition amplitude as shown in

Showing 9 of the paper's 10 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/9702393 by the authors.

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Forward citations

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How later papers used it

71 later papers cite it

Evidence: substantial. 59 of the citing papers share no author with it, and 10 of those carry a confident label from Pith's citation classifier.

Pith read 12 of the 117 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

1 paper by the same authors is listed below: 1 built on it.

Builds on it 1 of 2 papers with this label · 1 by the same authors

By the same authors

Reviewed and then extended the former formalism for unstable particle mixing presented in this paper.

CP Violation in Correlated Production and Decay of Unstable Particles (2011) (same authors) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

68 arXiv papers cite this in 118 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 · 90 sentences without a confident label

Contests it 1 paper

In the derivation discussed in reference [ref] , the tree-level propagator $i / (p - m_N_β)$ was replaced by $i / [p - m_N_β + Σ_ββ (p)]$, which implies that the resummation had not been fully taken.

Generation of quasiparticles by flavor mixing and CP violation: heavy Majorana neutrinos, part I · Introduction · contest 0.82

Builds on it 2 papers

In this section, we will first review and then extend the former formalism for unstable particle mixing, presented in [ref] .

CP Violation in Correlated Production and Decay of Unstable Particles · Formalism for Scalar Particle Mixing · 0.98
This formalism can be easily extended to fermionic resonances as discussed in [ref] and applied to heavy neutrino mixing in [ref] .

Nearby resonances beyond the Breit-Wigner approximation · Conclusions · 0.86

Uses its method or tool 6 papers

The scheme proposed in [ref] introduces a shift in $α$, the angle from the rotation matrix $R_α$, [equation] and by relating it to the field renormalization matrix constant leads to the following counterterm for $α$,

One-loop Corrections to the Higgs Boson Invisible Decay in a Complex Singlet Extension of the SM · Renormalization of the Mixing Angle $α$ · 0.95
This procedure is described in detail in [ref] .

One-loop Corrections to the Higgs Boson Invisible Decay in the Dark Doublet Phase of the N2HDM · Mixing angle $α$ · 0.88
It is useful to decompose the CP asymmetry in $N_i$ decays as the sum of a Vertex contribution ($V_j$) and of a Self-energy contribution ($S_j$) [ref] .

TeV scale double seesaw in left-right symmetric theories · Leptogenesis · 0.89

Cites it as support 2 papers

Whereas the standard mixing contributions [ref] are associated with the mass shells $ω_i$ ($i=1,2$) of the corresponding quasi-particles, the oscillation contribution is associated with an intermediate shell at $ω̅=(ω_1+ω_2)/2$, which we will refer to as the oscillation shell in the remainder of this paper.

Lepton asymmetry from mixing and oscillations · Shell structure for two-particle mixing in the Heisenberg picture · 0.97
In [ref] , it was shown that such critical non-oscillatory two-level quantum systems maximise the phenomenon of resonant CP violation through mixing of states.

Critical Unstable Qubits: an Application to $B^0\bar{B}^0$-Meson System · Critical Evolution of a Pure State Qubit · 0.97

Cites it as background 16 papers

It is well known that mixing of states with equal conserved quantum numbers is strongly enhanced when these states are nearly degenerate, i.e., when their mass difference is of the order of their widths [ref] .

Disentangling CP phases in nearly degenerate resonances: neutralino production via Higgs at a muon collider · Introduction · 0.80
The PT resummation approach to unstable particles [ref] was extensively studied in the literature originally within the context of the SM [ref] and more recently in two Higgs-doublet models [ref] .

Gauge-Independent Approach to Resonant Dark Matter Annihilation · Introduction · 0.88
This mechanism is called resonant leptogenesis [ref] , and this scenario is particularly interesting since such relatively light RH Majorana neutrinos can be discovered in the particle colliders currently available.

Generation of quasiparticles by flavor mixing and CP violation: heavy Majorana neutrinos, part I · Introduction · 0.85

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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