Baryogenesis from mixed particle decays

E. Roulet; L. Covi

REVIEW

Baryogenesis from mixed particle decays

L. Covi, E. Roulet

High Energy Physics - Phenomenology · Submitted 26 Nov 1996 · 20 references

Pith's verdict: A clean, self-contained extension of the Liu–Segrè mixing formalism to arbitrary mass splittings, with the peak at splitting ~ width derived rather than assumed; the leptogenesis application is honest about its initial-state caveat.

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arXiv IDhep-ph/9611425 (version 1)

Pith IDHNB4TA55, Pith's permanent record of this paper and its review

CitationE. Roulet, L. Covi. "Baryogenesis from mixed particle decays." arXiv:hep-ph/9611425 (1996). https://pith.science/paper/HNB4TA55

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Editor's note

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This is the paper that puts the wave-function CP asymmetry on a footing valid for any mass splitting, not just the hierarchical limit. The core result is Eq. (21): a closed expression for the integrated asymmetry function F1, from which the peak condition M2² − M1² ~ Γ² (Eq. 23) and the peak height drop out. That peak condition is the part worth remembering — it is derived, not fitted, and it is the statement that resonant-ish degeneracy is where the wave contribution wins. The sneutrino application (Eqs. 27, 29) is a straightforward substitution, and the ratio to the vertex asymmetry in Fig. 2 is a useful, checkable number. Two consistency checks are done properly: the large-splitting limit recovers their own published result (Eq. 30 vs. ref. [7]), and the small-splitting limit is compared against ref. [8] with an explicit note of where the two differ and why (the surviving 1/(M1(Γ1²+Γ2²)) term). That kind of honesty about a mismatch is rarer than it should be, and it is the reason I trust the algebra more than I otherwise would in a paper this compressed. The derivation is analytic throughout, so anyone can redo it in an afternoon; no code needed, no parameters fitted to anything. The soft spots are real but bounded. The biggest is the one the authors themselves flag: the asymmetry depends on the production basis, and they assume M²₁₂ = 0 with no prescription for a general production process. They say so plainly, so it is a limitation of the application, not a hidden error — but it does mean the quoted enhancement is conditional, and a reader who plugs these numbers into a model with flavour-superposed production will get a different answer. Second, the enhancement requires M2² − M1² ~ Γ², which for leptogenesis widths is a tuning of order 10⁻¹⁰ in mass squared; the paper cites ref. [19] for degenerate spectra but does not quantify the tuning, and a referee would reasonably ask for that. Third, the disagreement with ref. [18] over H12H21 versus (H12+H21)² is asserted in a footnote rather than argued; the paper's own Eq. (14) and Eq. (12) are internally consistent with H12H21, so I side with the authors, but the dispute is left open. None of this touches the central formula. This is for people working on leptogenesis with near-degenerate right-handed sneutrinos, and for anyone who needs the finite-splitting wave contribution as an input. It deserves a referee, and the referee should focus on the tuning question and the ref. [18] discrepancy, not on the derivation.

Referee report

Major objections (3)

[Eq. (22)] Eq. (22) is dimensionally inconsistent: from Eq. (20), F_a must have dimensions of inverse mass cubed for ǫ^wf_a to be dimensionless, whereas 2γ_a/(M2²−M1²) has dimensions of inverse mass. The correct limit of Eq. (21) in the regime M2² ≫ M1² ≫ |Γ²_ab| is 2/(γ_a(M2²−M1²)), which does reproduce Eq. (30). Please correct Eq. (22) and check that the text's claim of recovering the large-splitting result is consistent with the corrected form.
[Eqs. (21) and (24)] The last term of Eq. (21) as printed is proportional to (m1−m2) and therefore vanishes when M1²→M2², but the text below Eq. (24) states that a non-vanishing term survives in this limit because the propagation eigenstates are not degenerate when Γ²_11≠Γ²_22. Please check the sign and structure of this term (it may need to be proportional to m1+m2) and verify that Eq. (21) and Eq. (24) are consistent at M1²=M2².
[Initial-state specification, before Eq. (20) and in the sneutrino section] The paper correctly notes that the asymmetry depends on the basis in which the initial state is specified, but the leptogenesis application then assumes M²_12=0 without justifying this choice for the relevant production mechanisms (e.g., inflaton decay or thermal production). Since the numerical enhancement depends on this assumption, a more explicit discussion of how the result changes for a general initial-state density matrix would strengthen the application. The authors acknowledge the issue, so this is a scope limitation rather than an error.

Minor objections (6)

[Eq. (32)] The ratio ǫ^w_1/ǫ^v_1 should be dimensionless, but the printed expression −Γ²_11(Γ²_11+Γ²_22) ln2 has dimensions of mass⁴; please check for a missing M1⁴ in the denominator.
[Eq. (24)] The dimensional consistency of the two terms in Eq. (24) is not transparent; please check the powers of masses.
[Discussion of ref. [18]] The discrepancy with ref. [18] is stated but not explained; it would help to identify whether it arises from a different treatment of the initial state, the absorptive part, or the off-shell momentum.
[Fig. 2] The caption should state the values of Γ²_11, Γ²_22, and |Γ²_12| used, and clarify the meaning of the dotted line.
[Vertex contribution] The paper ignores vertex CP violation; while this is safe at the peak where ǫ_w≫ǫ_v, a sentence quantifying the error for moderate splittings would be useful.
[Naturalness of the degenerate spectrum] The required mass splitting M2²−M1² ~ Γ² is very small; a brief comment on the fine-tuning and on the plausibility of such degenerate spectra (beyond the citation of ref. [19]) would help.

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The paper

The paper claims

The paper derives the integrated CP asymmetry from mixing of two heavy scalars at arbitrary mass splitting and shows it peaks when the squared mass difference is of order the squared decay widths, with an application to sneutrino leptogénes

The reading

The paper asks how much CP violation can come from mixing between two unstable heavy scalar particles when their masses are close, rather than widely split. The authors extend a known formalism in which the heavy states propagate through a non-Hermitian effective mass matrix $H^2=M^2-iGamma^2$; the imaginary part comes from absorptive loops and is what makes a decay-rate difference possible. They derive the integrated asymmetry for arbitrary mass splitting and show that the wave-function contribution is not monotonic: it peaks when $M_2^2-M_1^2$ is of order the squared widths, and falls again as the states become exactly degenerate. Applied to heavy scalar-neutrino decays in leptogenesis, the peak can exceed the standard vertex asymmetry by a large factor, so the observed baryon asymmetry might be generated with smaller couplings or phases than a large-splitting estimate would suggest.

What carries the argument

The mechanism is the wave-function contribution: a loop of light particles mixes the decaying heavy state with a second heavy state before it decays. The load-bearing object is the non-Hermitian effective squared-mass matrix $H^2=M^2-iGamma^2$, whose imaginary part $Gamma^2$ comes from absorptive loops. It is diagonalized by a non-unitary matrix $V$, so the propagation eigenstates are not orthonormal; the mismatch between production and propagation bases generates the asymmetry. The integrated result is controlled by $F_1$ in Eq. (21), which encodes the competition between the mass splitting and the widths and produces the peak at $M_2^2-M_1^2simGamma^2$.

Extended reading notes

Core claim

The central claim is that the mixing, or wave-function, CP asymmetry in the decay of a heavy scalar is governed by Eq. (20) with the function $F_1$ of Eq. (21), and that for arbitrary mass splittings this function has a maximum at $M_2^2-M_1^2simeq Gamma_1^2+Gamma_2^2$ when $|Gamma_{12}^2|$ is small compared with $|Gamma_{22}^2-Gamma_{11}^2|$, with peak value $F_1simeq M_1/[Gamma_1^2(Gamma_1^2+Gamma_2^2)]$. In the large-splitting limit $F_1$ reduces to the earlier result, while for the sneutrino leptogenesis application the summed asymmetry is given by Eq. (29). The authors emphasize that the result depends on the initial state: they take the produced state to be a mass eigenstate, $M_{12}^2=0$, and warn that ignoring mixing at production gives incorrect asymmetries.

Load-bearing premise

The leptogenesis application assumes that the heavy scalar is produced as an eigenstate of the renormalized mass matrix, $M_{12}^2=0$, and the paper itself warns that a different production basis changes the asymmetry without supplying a general production prescription.

Editorial extensions

If this is right

Near-degenerate heavy scalars with a mass splitting of order their widths can generate a wave-function CP asymmetry much larger than the large-splitting estimate, so leptogenesis may require smaller Yukawa couplings or CP phases.
The enhancement is resonant in the mass splitting, peaking at $M_2^2-M_1^2simeq Gamma_1^2+Gamma_2^2$ when the off-diagonal width is small, which makes the degeneracy pattern of heavy states a relevant input for baryogenesis.
In the large-splitting limit the new expression reduces to the earlier result, and in the authors' example the wave contribution approaches twice the vertex contribution.
For the equal-width case, the propagation eigenstates become degenerate at $|M_2^2-M_1^2|=2|Gamma_{12}^2|$, where the asymmetry vanishes.

Reading between the lines

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

The same non-Hermitian effective-mass mechanism should apply to any unstable mixed scalar, not only sneutrinos; the resonant peak is a general feature of unstable mixing when width and mass splittings are comparable.
Because the asymmetry depends on the production basis, a full quantum-kinetic treatment of sneutrino production is needed before the peak can be used as a quantitative leptogenesis prediction; this is the main gap between the derivation and an observable.
If the enhancement survives, it loosens the usual tension between leptogenesis and low-scale supersymmetry or lepton-flavour constraints by allowing smaller couplings, but it also makes the baryon asymmetry sensitive to the detailed spectrum of the two lightest heavy sneutrinos.
The paper's comparison with the vertex contribution assumes the same Yukawa combination; with three families and independent phases, the third heavy state can still dominate, so a realistic estimate needs the full three-flavour interference.

Original abstract

We consider the $CP$ violating asymmetries produced in the decay of heavy particles, studying the effects of heavy particle mixing for arbitrary mass splittings. A considerable enhancement of the asymmetries is achieved when the masses of the mixed states are comparable, and the enhancement is maximal for mass splittings of the order of the widths of the decaying particles. We apply the results to the particular case of heavy scalar neutrino decays relevant for leptogenesis scenarios.

The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,952 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.19 · compares with prior results 0.11 · 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 consider the $CP$ violating asymmetries produced in the decay of heavy particles, studying the effects of heavy particle mixing for arbitrary mass splittings. A considerable enhancement of the asymmetries is achieved when the masses of the mixed states are comparable, and the enhancement is maximal for mass splittings of the order of the widths of the decaying particles. We apply the results to the particular case of heavy scalar neutrino decays relevant for leptogenesis scenarios.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider the $CP$ violating asymmetries produced in the decay of heavy particles, studying the effects of heavy particle mixing for arbitrary mass splittings.	contribution 0.99	0.85	0.07	unclear 0.60
s1	A considerable enhancement of the asymmetries is achieved when the masses of the mixed states are comparable, and the enhancement is maximal for mass splittings of the order of the widths of the decaying particles.	result 1.00	0.85	0.75	asserted 1.00
s2	We apply the results to the particular case of heavy scalar neutrino decays relevant for leptogenesis scenarios.	contribution 0.94	0.85	0.07	asserted 0.95

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
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P abstract

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R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Diagrams which interfere to produce the CP violation in the heavy particle decay. Fig. 1b gives the so called vertex contribution while Fig. 1c gives the wave function one

**Figure 2.** Wave function contribution to the CP asymmetry, normalised to the vertex one, as a function of M2/M1, assuming M2 12 = 0 and taking Γ2 22 = M2 1 /10, Γ2 11 = |Γ 2 12| = Γ 2 22/10. 13

Figures reproduced from arXiv:hep-ph/9611425 by the authors.

References

Works cited (20)

[7]

L. Covi, E. Roulet and F. Vissani, Phys. Lett B384 (1996) 169

View on Pith 1996
[8]

J. Liu and G. Segr` e, Phys. Rev. D49 (1994) 1342

View on Pith 1994
[19]

J. A. Harvey, D. B. Reiss and P. Ramond, Nucl. Phys. B199 (1982) 223; S. Di- mopoulos, L. Hall and S. Raby, Phys. Rev. D47 (1993) R3697; Y. Achiman and Th. Greiner, Nucl. Phys. B443 (1995) 3

View on Pith 1982
[18]

M. Flanz, E. A. Paschos, U. Sarkar and J. Weiss, hep-ph/9607 310

View on Pith
[1]

A. D. Sakharov, JETP Lett. 5 (1967) 24

View on Pith 1967
[2]

J. A. Harvey, E. W. Kolb, D. B. Reiss and S. Wolfram, Nucl. Phys. B201 (1982) 16

View on Pith 1982
[3]

A. Yu. Ignatiev, V. A. Kuzmin and M. E. Shaposhnikov, JETP Lett . 30 (1979) 688

View on Pith 1979
[4]

F. J. Botella and J. Roldan, Phys. Rev. D44 (1991) 966

View on Pith 1991
[5]

J. Liu and G. Segr` e, Phys. Rev. D48 (1993) 4609

View on Pith 1993
[6]

M. Flanz, E. A. Paschos and U. Sarkar, Phys. Lett. B345 (1995) 248; Erratum: ibidem B382 (1996) 447

View on Pith 1995
[9]

See e.g. I. Dunietz and J. L. Rosner, Phys. Rev. D34 (1986) 1404

View on Pith 1986
[10]

M. Fukugita and T. Yanagida, Phys. Lett. B174 (1986) 45

View on Pith 1986
[11]

M. A. Luty, Phys. Rev. D45 (1992) 455

View on Pith 1992
[12]

C. E. Vayonakis, Phys. Lett. B286 (1992) 92

View on Pith 1992
[13]

B. A. Campbell, S. Davidson and K. A. Olive, Nucl. Phys. B399 (1993) 111

View on Pith 1993
[14]

H. Murayama, H. Suzuki, T. Yanagida and J. Yokoyama, Phys. R ev. Lett. 70 (1993) 1912. 11

View on Pith 1993
[15]

M. Pl¨ umacher, hep–ph/9604229

View on Pith
[16]

S. Khlebnikov and M. Shaposhnikov, Nucl. Phys. B308 (1988) 883; J. A. Harvey and M. S. Turner, Phys. Rev. D42 (1990) 3344

View on Pith 1988
[17]

N. S. Manton, Phys. Rev. D28 (1983) 2212; V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B155 (1985) 36

View on Pith 1983
[20]

A. Acker, H. Kikuchi, E. Ma and U. Sarkar, Phys. Rev. D48 (1993) 5006. 12 Figure captions: Figure 1: Diagrams which interfere to produce the CP violation in the heavy particle decay. Fig. 1b gives the so called vertex contribution while Fig. 1c give s the wave function one. Figure 2: Wave function contribution to the CP asymmetry, normalised to the vertex...

View on Pith 1993

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How the literature uses it

32 arXiv papers cite this in 47 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 · 30 sentences without a confident label

Uses its method or tool 2 papers

This method was used in [ref] to calculate the CP asymmetry in the decay of heavy particles with arbitrary mass splittings that mix, which elucidated various issues.

CP violation in mixing and oscillations for leptogenesis with quasi-degenerate neutrinos · Introduction · 0.84
The first part of our discussion will follow closely [ref] , where leptogenesis from mixed particle decays was considered.

Probing Resonant Leptogenesis at the LHC · Oscillation derivation · 0.94

Compares with it 1 paper

This is different from flavored-scalar leptogenesis [ref] , where more delicate collider confirmation is needed.

Forbidden neutrinogenesis · Phenomenology · 0.97

Cites it as background 10 papers

The mass scale of leptogenesis can be significantly lowered if two HNLs are nearly degenerate in mass, this phenomenon was dubbed resonant leptogenesis [ref] .

Uniting low-scale leptogeneses · Introduction · 0.80
In this scenario, a lepton asymmetry is produced through out-of-equilibrium and $CP$-violating decays of the RHNs [ref] .

Hot Leptogenesis · Introduction · 0.91
In its simplest realisation, the lepton asymmetry is generated via out-of-equilibrium decays of heavy Majorana neutrinos [ref] .

Three-Flavoured Non-Resonant Leptogenesis at Intermediate Scales · Introduction · 0.96

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.

[1] [7]

L. Covi, E. Roulet and F. Vissani, Phys. Lett B384 (1996) 169

View on Pith 1996

[2] [8]

J. Liu and G. Segr` e, Phys. Rev. D49 (1994) 1342

View on Pith 1994

[3] [19]

J. A. Harvey, D. B. Reiss and P. Ramond, Nucl. Phys. B199 (1982) 223; S. Di- mopoulos, L. Hall and S. Raby, Phys. Rev. D47 (1993) R3697; Y. Achiman and Th. Greiner, Nucl. Phys. B443 (1995) 3

View on Pith 1982

[4] [18]

M. Flanz, E. A. Paschos, U. Sarkar and J. Weiss, hep-ph/9607 310

View on Pith

[5] [1]

A. D. Sakharov, JETP Lett. 5 (1967) 24

View on Pith 1967

[6] [2]

J. A. Harvey, E. W. Kolb, D. B. Reiss and S. Wolfram, Nucl. Phys. B201 (1982) 16

View on Pith 1982

[7] [3]

A. Yu. Ignatiev, V. A. Kuzmin and M. E. Shaposhnikov, JETP Lett . 30 (1979) 688

View on Pith 1979

[8] [4]

F. J. Botella and J. Roldan, Phys. Rev. D44 (1991) 966

View on Pith 1991

[9] [5]

J. Liu and G. Segr` e, Phys. Rev. D48 (1993) 4609

View on Pith 1993

[10] [6]

M. Flanz, E. A. Paschos and U. Sarkar, Phys. Lett. B345 (1995) 248; Erratum: ibidem B382 (1996) 447

View on Pith 1995

[11] [9]

See e.g. I. Dunietz and J. L. Rosner, Phys. Rev. D34 (1986) 1404

View on Pith 1986

[12] [10]

M. Fukugita and T. Yanagida, Phys. Lett. B174 (1986) 45

View on Pith 1986

[13] [11]

M. A. Luty, Phys. Rev. D45 (1992) 455

View on Pith 1992

[14] [12]

C. E. Vayonakis, Phys. Lett. B286 (1992) 92

View on Pith 1992

[15] [13]

B. A. Campbell, S. Davidson and K. A. Olive, Nucl. Phys. B399 (1993) 111

View on Pith 1993

[16] [14]

H. Murayama, H. Suzuki, T. Yanagida and J. Yokoyama, Phys. R ev. Lett. 70 (1993) 1912. 11

View on Pith 1993

[17] [15]

M. Pl¨ umacher, hep–ph/9604229

View on Pith

[18] [16]

S. Khlebnikov and M. Shaposhnikov, Nucl. Phys. B308 (1988) 883; J. A. Harvey and M. S. Turner, Phys. Rev. D42 (1990) 3344

View on Pith 1988

[19] [17]

N. S. Manton, Phys. Rev. D28 (1983) 2212; V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B155 (1985) 36

View on Pith 1983

[20] [20]

A. Acker, H. Kikuchi, E. Ma and U. Sarkar, Phys. Rev. D48 (1993) 5006. 12 Figure captions: Figure 1: Diagrams which interfere to produce the CP violation in the heavy particle decay. Fig. 1b gives the so called vertex contribution while Fig. 1c give s the wave function one. Figure 2: Wave function contribution to the CP asymmetry, normalised to the vertex...

View on Pith 1993