We discuss the three different classes of CP-symmetries that can be realized
in a two-Higgs-doublet model, CP1, CP2 and CP3. We express conditions for
realizing these symmetries in terms of masses and couplings of the model,
thereby providing a way of verifying which, if any, of these symmetries is
realized by nature.Comment: Contribution to the conference proceedings of DISCRETE 1

Grzadkowski, Bohdan

Ogreid, Odd Magne

Osland, Per

Journal of Physics Conference Series

English

arXiv

Ogreid, O M

Osland, P

Grządkowski, Bohdan

REIN UW  Institutional Repository of the University of Warsaw

The CP-symmetries of the 2HDM

We discuss the three different classes of CP-symmetries that can be realized in a two-Higgs-doublet model, CP1, CP2 and CP3. We express conditions for realizing these symmetries in terms of masses and couplings of the model, thereby providing a way of verifying which, if any, of these symmetries is realized by nature.publishedVersio

Øgreid, Odd Magne

HVL Open (Høgskulen på Vestlandet)

Journal of Physics: Conference SeriesPAPER • OPEN ACCESSThe CP-symmetries of the 2HDMTo cite this article: B Grzadkowski et al 2020 J. Phys.: Conf. Ser. 1586 012046 View the article online for updates and enhancements.This content was downloaded from IP address 85.166.128.71 on 03/02/2021 at 17:43Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd6th Symposium on Prospects in the Physics of Discrete Symmetries - DISCRETE 2018Journal of Physics: Conference Series 1586 (2020) 012046IOP Publishingdoi:10.1088/1742-6596/1586/1/0120461The CP-symmetries of the 2HDMB Grzadkowski1, O M Ogreid2 and P Osland31 Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland2 Western Norway University of Applied Sciences, Postboks 7030, N-5020 Bergen, Norway3 Department of Physics, University of Bergen, Postboks 7803, N-5020 Bergen, NorwayE-mail: bohdan.grzadkowski@fuw.edu.pl, omo@hvl.no, per.osland@uib.noAbstract. We discuss the three different classes of CP-symmetries that can be realized ina two-Higgs-doublet model, CP1, CP2 and CP3. We express conditions for realizing thesesymmetries in terms of masses and couplings of the model, thereby providing a way of verifyingwhich, if any, of these symmetries is realized by nature.1. IntroductionMulti-Higgs-doublet models provide attractive extensions of the Standard Model (SM) dueto their ability to accommodate Dark Matter and/or CP-violation [1]. The two-Higgs-doublet model (2HDM) is the simplest of these extensions. Its CP-properties are well knownand have been studied and expressed in many different ways, using different formalisms[2, 3, 4, 5, 6, 7, 8, 9, 10]. In the 2HDM Lagrangian, interactions involving scalars are expressedusing the two doublets Φ1 and Φ2. It is also possible to form combinations of these as Φi → UijΦ′j ,where Uij is any U(2)-matrix, and express the Lagrangian in terms of the transformed doublets.This is referred to as a change of basis. Clearly, physics should not depend upon our choice ofbasis, and this motivates basis-invariant formulations of physical conditions. In our formalism,we focus on expressing the CP-properties in terms of physical observable quantities only, i.e.only masses and physical couplings of the model. All masses and measurable couplings involvingthe scalars are naturally basis-invariant. For this purpose we introduce the physical parametersetP ≡ {M2H± ,M21 ,M22 ,M23 , e1, e2, e3, q1, q2, q3, q}, (1)containing the squared masses of the charged scalar as well as those of the tree neutral ones.Furthermore, ei parametrizes the HiV V couplings V = (W,Z), qi parametrizes the HiH+H−couplings and q parametrizes the H+H+H−H− coupling. See [9, 10, 11, 12] for more details.All physical observables arising from the bosonic sector of the 2HDM can be expressed interms of the 11 parameters of P. In order to make the translation from the parameters ofthe potential and the vacuum to the physical parameter set we can either translate explicitlybasis-invariant quantities, or we can translate a combination of non-invariant constraints thattogether constitutes a basis-invariant constraint. The method for translating from parametersof the potential to the physical parameter set is presented in detail in [11].6th Symposium on Prospects in the Physics of Discrete Symmetries - DISCRETE 2018Journal of Physics: Conference Series 1586 (2020) 012046IOP Publishingdoi:10.1088/1742-6596/1586/1/01204622. Three classes of CP-symmetriesIn the 2HDM, the potential is CP-invariant provided it is symmetric under the followingtransformation of the Higgs-doublets,Φi → XijΦ∗j , (2)where Xij is any U(2)-matrix. In [13], three different classes of CP-symmetries were presented,according to the form of the Xij . These three symmetries were denoted CP1, CP2 and CP3;• CP1: Xij =(1 00 1).• CP2: Xij =(0 1−1 0).• CP3: Xij =(cos θ sin θ− sin θ cos θ), 0 < θ < π/2.CP1 is the “ordinary” CP-symmetry of the potential, which can be shown to be equivalent tothe possibility of having a real potential [5]. There is still the possibility for a non-real vacuum.CP2 and CP3 denote even higher degrees of symmetry in the potential. Whenever the potentialpossesses the CP2 or CP3 symmetry, it also possesses the CP1 symmetry. Basis-independentformulations of conditions for invariance of the 2HDM-potential under CP1, CP2 and CP3 werepresented in [14] in terms of properties of vectors ξ, η and a matrix E. For a detailed study ofvacua within models invariant under these symmetries see [15].3. Physical conditions for CP1, CP2 and CP3The conditions for CP1 in terms of the physical parameter set P are well known from previousworks [9, 10]. In the following three sub-sections we are going to express the conditions for CP2and CP3 derived in [14] in terms of physical parameters using the technique outlined in [11].Hereafter we will refer to CP1 either as CP1 or just CP.3.1. Physical conditions for CP1Here, we just repeat the results found in [9, 10]. The “ordinary” CP-symmetry, CP1, can berealized with four different physical configurations.• Case A: M1 = M2 = M3. (Full mass degeneracy.)• Case B: Mi = Mj , (ejqi − eiqj) = 0. (Partial mass degeneracy.)• Case C: ek = qk = 0. (No mass degeneracy.)• Case D: M2H± =v22D[e1q1M22M23 + e2q2M21M23 + e3q3M21M22 −M21M22M23],q =12D[(e2q3 − e3q2)2M21 + (e3q1 − e1q3)2M22 + (e1q2 − e2q1)2M23 +M21M22M23].where D = e21M22M23 +e22M21M23 +e23M21M22 . The conditions for cases A, B and C guarantee thatboth the potential and the vacuum conserve CP, while case D only guarantees a CP conservingpotential and will lead to spontaneous CP-violation (SCPV) unless the constraints of either CaseA, B or C are satisfied in addition to the condition of Case D. We will hereafter refer to Case Das the SCPV-condition.In [9] we identified several physical processes involving decaying Z-bosons where theamplitude contains CP-odd form factors, thereby facilitating CP-violating processes. In [16]we studied the effects of one-loop generated effective ZZZ and ZWW vertices along with aset of CP-sensitive asymmetries for ZZ and W+W− production at e+e−colliders that directlydepends on a CP-odd basis-invariant.6th Symposium on Prospects in the Physics of Discrete Symmetries - DISCRETE 2018Journal of Physics: Conference Series 1586 (2020) 012046IOP Publishingdoi:10.1088/1742-6596/1586/1/01204633.2. Physical conditions for CP2A CP2-symmetric 2HDM will necessarily have both a CP-conserving potential and vacuum, sothe physical conditions for CP2 will be sub-cases of Cases A, B and C above. We find that thepotential is CP2-invariant in the following physical configurations:• Case ABBBD: M1 = M2 = M3, (e2q1 − e1q2) = (e3q1 − e1q3) = (e2q3 − e3q2) = 0,M2H± =12(e1q1 + e2q2 + e3q3 −M21 ), q =M212v2.• Case BCD: Mi = Mj , (ejqi − eiqj) = 0, ek = qk = 0,M2H± =12(eiqi + ejqj −M2i ), q =M2i2v2.• Case CCD: ej = ek = qj = qk = 0, M2H± =12(eiqi −M2i ), q =M2i2v2.The logic behind this labeling is that the physical configurations present here can be interpretedas sub-cases of multiple of the four different cases (A, B, C and D) of CP1. A repeated letterindicates that two (or more) separate constraints (with different indices) of said case is neededto get CP2. In each of these physical configurations, the SCPV-condition (Case D) is presentas constraints on M2H± and q in combination with at least two separate conditions for CP-invariance (Cases A, B or C) in addition. In some crude sense we can say that these conditions(symmetries) are stacked on top of each other.3.3. Physical conditions for CP3A CP3-symmetric 2HDM will also necessarily have both a CP-conserving potential and vacuum,so the physical conditions for CP3 will be sub-cases of the cases for CP1. More than this, anymodel with CP3 symmetry will also have the CP2 symmetry. Thus, the cases of CP3 will besub-cases of the cases we found for CP2. We find that the potential is CP3-invariant in thefollowing physical configurations:• Case ABBBD: As in the previous sub-section.• Case BC0D: As Case BCD with the additional constraint Mk = 0.• Case B0CD: As Case BCD with the additional constraint Mi = 0.• Case BCCD: As Case CCD with the additional constraint Mj = Mk.• Case CC0D: As Case CCD with the additional constraint Mj = 0 (or Mk = 0).The subscript 0 attached to a constraint tells us that the mass corresponding to the scalar(s)involved in the constraint vanishes1. Comparing to the conditions for CP2, we see that theconditions for CP3 are identical with the exception that in some cases we need additional massdegeneracy, or some mass(es) must vanish in order to get CP3.4. SummaryWe have expressed conditions for the CP1, CP2 and CP3 symmetries of the 2HDM in termsof observable parameters, i.e. masses and physical couplings. Experimentally, we know that ifnature has in fact realized a 2HDM in which the Higgs-boson discovered at the LHC is one of thethree neutral scalars of the model, then we are close to the so-called alignment limit defined bythe requirement that one of the three neutral scalars (H1) exactly mimics the SM-Higgs boson.1 If we have labeling B0, where B refers to the constraints Mi = Mj , ejqi − eiqj = 0, then the subscript 0 tellsus that Mi = Mj = 0. If we have labeling C0, where C is the constraint ek = qk = 0, then the subscript 0 tellsus that Mk = 0.6th Symposium on Prospects in the Physics of Discrete Symmetries - DISCRETE 2018Journal of Physics: Conference Series 1586 (2020) 012046IOP Publishingdoi:10.1088/1742-6596/1586/1/0120464In our formalism, alignment can be expressed as e1 = v, e2 = e3 = 0. We would like to stressthe fact that in a CP2-symmetric model, alignment is a natural consequence of the symmetryprovided the neutral masses are non-degenerate. For a discussion of natural alignment in the2HDM see [17]. The conditions for the CP1, CP2 and CP3 symmetries of the 2HDM presentedhere provide a direct way to verify if nature has realized these symmetries in the 2HDM.References[1] Branco G C, Ferreira P M, Lavoura L, Rebelo M N, Sher M and Silva J P, Phys. Rept. 516 (2012) 1[2] Lavoura L and Silva J P, Phys. Rev. D 50 (1994) 4619[3] Botella F J and Silva J P, Phys. Rev. D 51 (1995) 3870[4] Branco G C, Rebelo M N and Silva-Marcos J I, Phys. Lett. B 614 (2005) 187[5] Gunion J F and Haber H E, Phys. Rev. D 72 (2005) 095002[6] Ivanov I P, Phys. Lett. B 632 (2006) 360[7] Nishi C C, Phys. Rev. D 74 (2006) 036003 Erratum: [Phys. Rev. D 76 (2007) 119901][8] Maniatis M, von Manteuffel A and Nachtmann O, Eur. Phys. J. C 57 (2008) 719[9] Grzadkowski B, Ogreid O M and Osland P, JHEP 1411 (2014) 084[10] Grzadkowski B, Ogreid O M and Osland P, Phys. Rev. D 94 (2016) no.11, 115002[11] Ogreid O M, PoS CORFU 2017 (2018) 065[12] Grzadkowski B, Haber H E, Ogreid O M and Osland P, JHEP 1812 (2018) 056[13] Ferreira P M, Haber H E and Silva J P, Phys. Rev. D 79 (2009) 116004[14] Ferreira P M, Haber H E, Maniatis M, Nachtmann O and Silva J P, Int. J. Mod. Phys. A 26 (2011) 769[15] Battye R A, Brawn G D and Pilaftsis A, JHEP 1108 (2011) 020[16] Grzadkowski B, Ogreid O M and Osland P, JHEP 1605 (2016) 025 Erratum: [JHEP 1711 (2017) 002][17] Dev P S B and Pilaftsis A, JHEP 1412 (2014) 024 Erratum: [JHEP 1511 (2015) 147]

We discuss the three different classes of CP-symmetries that can be realized in a two-Higgs-doublet model, CP1, CP2 and CP3. We express conditions for realizing these symmetries in terms of masses and couplings of the model, thereby providing a way of verifying which, if any, of these symmetries is realized by nature

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The CP-symmetries of the 2HDM

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