
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39232118
71222
10.1038/s41598-024-71222-8
Article
Higgs decay and CP violation phase in the CPV TNMSSM
Zhu Ning-Yu hnzhuny@163.com

Chen Hai-Xiang
Hu Huai-Cong
https://ror.org/02c9qn167 grid.256609.e 0000 0001 2254 5798 Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University, Nanning, 530004 People’s Republic of China
4 9 2024
4 9 2024
2024
14 2054225 4 2024
26 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
In this study, we calculate the Higgs mass matrix and explore the limitations of the minimum conditions of the scalar potential on parameter degrees of freedom in the CP violation TNMSSM. We discuss the contributions of some parameters to Higgs mass, and their impact on the strength of Higgs decay signals in different decay channels h→ γγ, h→ VV (V=W,Z) and h→ ff¯ (f=b,c,τ).

Keywords

Supersymmetry
Higgs mass
CP violation
Higgs decays
Subject terms

Phenomenology
Theoretical particle physics
http://dx.doi.org/10.13039/100012547 Natural Science Foundation of Guangxi Zhuang Autonomous Region 2022GXNSFDA035068 issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

The Standard Model (SM) of electroweak and strong interactions of elementary particles has achieved great success in the last century1,2. The Higgs particle was observed in 2012, which means that all particles in the SM have been discovered3,4. On the other hand, the SM also has some disadvantages, for example, its inability to explain neutrino oscillations5,6 and the asymmetry of matter and antimatter in the universe, as well as can not provide the candidates for dark matter. Therefore, some extended models of SM have been proposed in an attempt to explain these issues.

The Minimal Supersymmetric Standard Model (MSSM) is a well-known new physics model7, which has been extensively studied by physicists in the past few decades. However, the MSSM also has some unexplained problems8, such as the hierarchical and μ term problems, and physicists have proposed some extended models of the MSSM. The extension of the MSSM by addition a gauge singlet which is coupled to Higgs doublets (NMSSM) has been proposed to solve the μ term problem9. The TNMSSM introduce a forbidden bare μ term, and an effective μ term is generated by the vacuum expectation value of the singlet. However, all the couplings in the NMSSM being perturbative up to the GUT scale also does not polish up the little gauge hierarchy problem10–15. Fortunately, the next-to-minimal supersymmetric standard model with triplets (TNMSSM) is proposed by Kaustubh Agashe et al. , which combines the advantages of NMSSM and TMSSM16,17. Compared with the MSSM, the TNMSSM introduces a gauge singlet and two SU(2)L triplets with hypercharge Y=0, which can validly solve the little gauge hierarchy and μ term problems in MSSM. In addition, the TNMSSM can give neutrinos a small mass by introducing Majorana mass term without the need to introduce right-handed neutrinos16,18, which is cannot in the MSSM and NMSSM.

The violation of CP was first observed in neutral kaon decay experiments and strongly validated in B-meson decay experiments19,20. In addition to particle physics, CP violation (CPV) also provides a possible explanation for the asymmetry of matter and antimatter in the universe21. The Higgs interactions play an important role in mediating CPV, for example, CP is broken explicitly in the SM by complex Yukawa couplings of the Higgs boson to quarks. There are literatures indicating, in models that expand the Higgs sector, such as SUSY and MSSM, CP symmetries of those theories are broken spontaneously22,23.

The existence of Higgs boson has been confirmed, but further research is needed on its properties. In this paper, we consider the contributions of one-loop effective potential24 and two-loop leading-log radiative correction25,26 to Higgs mass, and calculate the mass matrix of Higgs boson in CPV TNMSSM and explore the reduction of parameter degrees of freedom by the minimum conditions of the scalar potential in Chapter II. The concrete theoretical expressions of Higgs decay are presented in Chapter III. It is found in the calculation that CPV appears in the tree level Higgs mass matrix in the CPV TNMSSM, which is absent in MSSM27. In numerical analysis, we find that the appearance of CPV in the tree level Higgs mass matrix has negative contributions to Higgs boson mass and greatly limits the range of parameter selection. Then we conduct a numerical analysis of the signal strength of Higgs decay in different decay channels in Chapter IV. We observe the effects of tanβ, Mλ (modulus of λ), tanβ′ and MλT (modulus of λT) on signal strengths when λT and χd were selected as different phase angles and Mχd (modulus of Mχd) and ReAu (real part of Au) were selected as different values. We note that because triplets have no coupling with fermions in the Lagrangian , the signal strengths of h→ ff¯ (f=b,c,τ) are almost unaffected when the phase angle of λT changes. In addition, in Chapter IV, we also calculate the electric dipole moments of neutrons and electrons, the contribution of doubly charged particles to Higgs decay, and the mass spectrum of doubly charged particles. In Chapter V, we discuss the results of the article.

THE Higgs sector in the CPV TNMSSM

The TNMSSM

Some literature has investigated B meson rare decays28, Higgs boson decays h → MZ29, and transition magnetic moment of Majorana neutrinos18 in the TNMSSM. These studies indicate that the TNMSSM is capable of making good predictions for B meson decay and neutrino transition magnetic moment, and provides valuable information for the experimental exploration of rare Higgs decays. In addition, M.A. Ouahid et al.studied the phenomenology of neutrinos and doubly charged Higgs in flavored-TNMSSM30.

Compared to the MSSM, the TNMSSM has an additional gauge singlet S and two SU(2)L triplets T and T¯. The superpotential of Higgs sector and Yukawa sector in TNMSSM are given as follows, respectively1 WHiggs=λS^H^uT·H^d+λTS^tr(T¯^T^)+k3S^3+χuH^uT·T¯^H^u+χdH^dT·T^H^d,

2 WYukawa=huQ^T·H^uU^-hdQ^T·H^dD^-hlL^T·H^dE^,

where λ, λT, κ, χu and χd are dimensionless. The field contents of the TNMSSM are given in Table 1. The singlet S, doublets Hu, Hd and triplets T, T¯ are given respectively by3 Hu=(Hu+,Hu0),Hd=(Hd0,Hd-),T=T-/2-T--T0-T-/2,

4 T¯=T¯+/2-T¯0T¯++-T¯+/2,Hu0=(vu+ϕu+iau)/2,Hd0=(vd+ϕd+iad)/2,T0=(vt+ϕt+iat)/2,S=(vs+ϕs+ias)/2,T¯0=(vt¯+ϕt¯+iat¯)/2,

where vu,vd,vs,vt and vT¯ are vacuum expectation value (VEV) of singlet S, doublets Hu and Hd, triplets T and T¯, respectively. Traditionally, tanβ and tanβ′ are can defined by tanβ≡vu/vd and tanβ′≡vt/vt¯. There is a relationship between VEV in SM and VEV in TNMSSM16v2=vu2+vd2+4vt2+4vt¯2.

The soft supersymmetry-breaking Lagrangian reads5 -Lsoft=mHu2Hu2+mHd2Hd2+mS2|S|2+mT2tr(|T|2)+mT¯2tr|T¯|2+mQ2|Q~|2+mU2|U~|2+mD2|D~|2+mL2|L~|2+mE2|E~|2+(AhuQ~·HuU~-AhdQ~·HdD~-AhlL~·HdE~+ASHu·Hd+ATStr(TT¯)+Ak3S3+AuHu·T¯Hu+AdHd·THd+h.c.)+12(M1λB~B~+M2λW~W~+M3λg~g~+h.c.).

Table 1 The field content of the TNMSSM.

Superfields	Bosons	Fermions	
Q^	Q~T=(u~,d~)L	QT=(u,d)L	
L^	L~T=(νl~,l~)L	LT=(νl,l)L	
U^	U~=u~R∗	(uR)C=(uC)L	
D^	D~=d~R∗	(dR)C=(dC)L	
E^	E~=l~R∗	(lR)C=(lC)L	
Hu^	HuT=(Hu+,Hu0)	H~uT=(H~u+,H~u0)	
Hd^	HdT=(Hd0,Hd-)	H~dT=(H~d0,H~u-)	
S^	S	S~	
T^	T~+/2-T~++T~0-T~+/2	T+/2-T++T0-T+/2	
T¯^	T¯~-/2-T¯~0T¯~---T¯~-/2	T¯-/2-T¯0T¯---T¯-/2	

The potential and Higgs mass

In the supersymmetry theory, scalar supersymmetry potentials are given in the following wayV0=12DaDa+Fi⋆Fi,

F and D are two auxiliary functionsFi=∂W/∂Ai,Da=gAi⋆TijaAj.

In the MS¯ scheme, the CPV effective potential is determined by27Veff=V0+V1,

V0 is the tree level potential, V1 is the one-loop level effective potential with246 V1=332π2∑q=t,b,c[∑i=1,2mq~i4(lnmq~i2Q2-32)-2mq4(lnmq2Q2-32)],

where mq is the q quark mass, and mq~ is the q~ squark mass and Q is the renormalization scale at TeV order.The mass matrix of squarks is given bymq~2=mq~Lq~L∗mq~Rq~L∗†mq~Lq~R∗mq~Rq~R∗,

where the matrix elements for up-type squarks aremu~Lu~L∗=112(g12-3g22)(Hu0,∗Hu0-Hd0,∗Hd0+2HT0,∗HT0-2HT¯0,∗HT¯0)+12(2mQ2+hu†huHu0,∗Hu0),mu~Lu~R∗=AhuHu0+hu(2Hd0,∗HT¯0,∗χu∗-Hd0,∗Hs0,∗λ∗),mu~Ru~R∗=-13g12(Hu0,∗Hu0-Hd0,∗Hd0+2HT0,∗HT0-2HT¯0,∗HT¯0)+12(2mu2+hu†huHu0,∗Hu0),

the matrix elements for down-type squarks aremd~Ld~L∗=112(g12+3g22)(Hu0,∗Hu0-Hd0,∗Hd0+2HT0,∗HT0-2HT¯0,∗HT¯0)+12(2mQ2+hd†hdHd0,∗Hd0),md~Ld~R∗=AhdHd0+hd(2Hu0,∗HT0,∗χd∗-Hu0,∗Hs0,∗λ∗),md~Rd~R∗=16g12(Hu0,∗Hu0-Hd0,∗Hd0+2HT0,∗HT0-2HT¯0,∗HT¯0)+12(2md2+hd†hdHd0,∗Hd0).

The mass of squarks can be obtained throughmu~1,22=12[mu~Lu~L+mu~Ru~R±(mu~Lu~L-mu~Ru~R)2-4mu~Lu~Rmu~Ru~L],md~1,22=12[md~Ld~L+md~Rd~R±(md~Ld~L-md~Rd~R)2-4md~Ld~Rmd~Rd~L].

The elements of Higgs mass matrix can be calculated by the following formula on the basis (ϕd,ϕu,ϕs,ϕt,ϕt¯,ad,au,as,at,at¯)mhij2=Veffϕiϕj∣ϕi,j=ϕd,ϕu,ϕs,ϕt,ϕt¯,ad,au,as,at,at¯.

It is found in the calculation that CPV appears in the tree level Higgs mass matrix in the CPV TNMSSM, namely mhijandmhji≠0(ϕi=ϕd,ϕu,ϕs,ϕt,ϕt¯andϕj=ad,au,as,at,at¯), which is absent in MSSM. In this paper, we consider the contributions of quarks (t,b,c) and squarks (stop, sbottom, scharm) to Higgs mass.

In addition, the Goldstone can be analytically obtained through Z†MhZ , where Z is the unitary matrix defined as7 Z=1vv5×5000000vdvu02vt-2vt¯0-vuvd02vt¯2vt000v000-2vt-2vt¯0vd-vu02vt¯-2vt0vuvd,

v5×5=v00000v00000v00000v00000v.

Moreover, we also consider the contribution of two-loop radiative correction to the Higgs boson mass25,26,318 Δmh2=3mt44π2v2[116π2(3mt22v2-32πα3)(t~2+X~tt~)]t~=LogMS2mt2,X~t=2A~t2MS(1-A~t212MS2)

where α3 is the strong coupling constant, MS =mt~1mt~2, A~t=Ahu-μ cotβ, mt~1 and mt~2 denote the stop masses and Ahu is the trilinear Higgs stop coupling.

Here we provide the mass matrices of doubly charged particles in TNMSSM. The mass matrix of the doubly charged Higgs mh±± is9 mh±±2=mT--T--,∗mT++,∗T--,∗∗mT--T++mT++,∗T++.

wheremT--T--,∗=12vs2|λT|2-14(g12-g22)(vu2-vd2+2vt2-2vt¯2)+mt¯2,mT--T++=12(λT(vuvdλ∗+vtvt¯λT)-(vsλTk∗+2AT)),mT++,∗T++=12vs2|λT|2-14(g12-g22)(vu2-vd2+2vt2-2vt¯2)+mt¯2.

The mass matrix of the doubly chargino mχ±± is10 mχ±±χ±±=12vs|λT|.

CP violation and phase angle

Consider AeiθABeiθBCeiθC is a term in Lagrangian, A, B, C are parameters or fields in Laragian, because ABCei(θA+θB+θC)=AeiθABeiθBCeiθC, we can absorb all the phase angles of the fields into the coupling coefficient through redefining eiθi=eiθAeiθBeiθC. Because all phase angles are undetermined, it is sufficient and reasonable to only add phase angles to the coupling coefficient. In this paper, we use MAieiθi to represent complex parameters Ai, MAi is modulus of complex parameters Ai.

The total tadpoles are given by:11 Tϕi=Veffϕi=Tϕi0+Tϕi1,ϕi=ϕd,ϕu,ϕs,ϕt,ϕt¯,ad,au,as,at,at¯,

whereTϕi0=V0ϕi∣ϕi=ϕd,ϕu,ϕs,ϕt,ϕt¯,ad,au,as,at,at¯Tϕi1=V1ϕi∣ϕi=ϕd,ϕu,ϕs,ϕt,ϕt¯,ad,au,as,at,at¯,

Tϕi0 and Tϕi1 are given in the appendix A. Due to the requirement of vanishing tadpole equations, the degrees of freedom of the parameters can be reduced, i.e. mi2 (i=Hu,Hd,S,T and T¯) disappear in the main diagonal elements of the Higgs mass matrix, and the imaginary parts of the five parameters Ai (Ai=Au, Ad, Ak, AT, Ahu) will be reduced. The reduced imaginary parts of those parameters are present in the appendix A (A1–A5).

THE 125 Higgs decays

The Higgs boson is a mixed state of CP-even and CP-odd components in CPV theory, differentiate from the SM Higgs boson, both of the two final states appear and could be distinguished from each other by detecting photon polarization32.

The Higgs are mainly produced by gluon pairs fusion in LHC experiments33,34. The one loop diagrams with virtual top quarks have the most significant contribution to the leading order (LO). In new physics (NP), one-loop diagrams containing virtual squarks also contribute to the LO. In this section, we use H to denote CP-even Higgs and A to denote CP-odd Higgs. The decay widths of CP-even Higgs to gluon pairs H→gg and CP-odd Higgs to gluon pairs A→gg are given respectively by35–40:12 ΓNP(H→gg)=GFαs2mH3642π3|∑qgHqqA1/2H(xq)+∑q~gHq~q~mZ2mq~2A0H(xq~)|2,

13 ΓNP(A→gg)=GFαs2mA3642π3|∑qgAqqA1/2A(xq)|2

The LO contributions to the decay Higgs to diphoton come from the one-loop diagrams. In the NP, all of the fermions, W boson and these supersymmetric partners contribute to Higgs to diphoton decay. The partial widths of CP-even Higgs and CP-odd Higgs bosons decay into diphoton are given by33,41–46:14 ΓNP(H→γγ)=GFαs2mH31282π3|∑fNcQf2gHffA1/2H(xf)+∑f~NcQf~2gHf~f~mZ2mf~2A0H(xf~)2+∑i=13gHHi+Hi-mZ2mHi±2A0H(xHi±)+∑j=12QHj++2gHHj++Hj--mZ2mHj±±2A0H(xHj±±)+∑k=13gHχk+χk-mWmχk±A1/2H(xχk±)+Qχ++2gHχ++χ--mWmχ±±A1/2H(xχ±±)+gHWWA1H(xW)|2,

15 ΓNP(A→γγ)=GFαs2mA31282π3|∑fNcQf2gAffA1/2A(xf)+∑k=13gAχk+χk-mW2mχk±2A1/2A(xχk±)+Qχ++2gAχ++χ--mW2mχ±±2A1/2A(xχ±±)|2,

where xi=mh2/(4mi2), A1H(x), A12H(x), A0H(x) and g(x) are given by4716 A1H(x)=-[2x2+3x+3(2x-1)g(x)]/x2,A12H(x)=2[x+(x-1)g(x)]/x2,A0H(x)=-(x-g(x))/x2,A1/2A(x)=2g(x)/x,g(x)=arcsin2x,x≤1-14[ln1+1-1/x1---1/x-iπ]2,x>1

The partial widths of CP-even Higgs bosons decay into vector boson pairs H → VV (V=W,Z) are given by48–55:17 ΓNP(H→WW)=3e4mH512π3sW4gHWW2F(mWmH),

18 ΓNP(H→ZZ)=e4mH2048π3sW4cW4(7-403sW2+1609sW4)×gHZZ2F(mZmH),

withF(x)=-(1-x2)(472x2-132+1x2)-3(1-6x2+4x4)lnx+3(1-8x2+20x4)4x2-1arccos(3x2-12x3)

CP-odd Higgs A is uncoupled with vector bosons in tree-level.

In the Born approximation, the partial decay widths of CP-even and CP-odd Higgs bosons decay into fermion pairs are given by35,49:19 ΓNP(H→ff¯)=NcGFmf2mH42πgHff2βf3,

20 ΓNP(A→ff¯)=NcGFmf2mA42πgAff2βf,βf=(1-4mf2mH,A2)12

In the CPV TNMSSM, the concrete expressions for gHff, gHf~f~, gHH+H-, gHH++H--, gHχ+χ-, gHχ++χ--, gHVV, gAff, gAχ+χ- and gAχ++χ-- appeared in Eqs. (6–9, 11–14) are given in the appendix B.

The signal strengths for the Higgs decay channels are5621 μγγ,VVggF=σNP(ggF)σSM(ggF)BRNP(h→γγ,VV)BRSM(h→γγ,VV),(V=W,Z)μff¯VBF=σNP(VBF)σSM(VBF)BRNP(h→ff¯)BRSM(h→ff¯),(f=b,c,τ)

The Higgs production cross sections can be simplified throughσNP(ggF)σSM(ggF)≈ΓNP(h→gg)ΓSM(h→gg),σNP(VBF)σSM(VBF)≈ΓNP(h→VV)ΓSM(h→VV),

the ratios of the signal strengths from the Higgs decay channels can be reduced as22 μγγggF=ΓSMΓNPΓNP(h→gg)ΓSM(h→gg)ΓNP(h→γγ)ΓSM(h→γγ)μVVggF=ΓSMΓNPΓNP(h→gg)ΓSM(h→gg)ΓNP(h→VV)ΓSM(h→VV)μff¯VBF=ΓSMΓNPΓNP(h→VV)ΓSM(h→VV)ΓNP(h→ff¯)ΓSM(h→ff¯),

where ΓNP = ∑f ΓNP(h → ff¯) + ∑V ΓNP(h →VV) + ΓNP(h → gg) + ΓNP(h → γ γ), represents the NP total decay width of physical Higgs.

Numerical analysis

We study the mass and decay of the lightest Higgs boson h0 in CPV TNMSSM in this section. After considering the experimental limitations, the scalar lepton masses larger than 700 GeV, and chargino masses larger than 1100 GeV57, the parameters in CPV TNMSSM were selected as23 vs=1TeV,M1=1TeV,M2=1TeV,Q=1TeV,Mk=0.9,Mχu=0.3,MA=0.8TeV,θλ=0.02,θk=0.05,θχu=π/3,θA=0.01,ReAd=0.5TeV,ReAk=-0.5TeV,ReAT=0.5TeV,Ahd=Ahl=ReAhu=0.2TeV,mQ2=mu¯2=md¯2=mL2=me¯2=1.8TeV2.

where ReAi are the real parts of Ai (Ai=Au, Ad, Ak, AT, Ahu), MAj are the modulus of Aj (Aj=λT, k, χu, χd, A), and θAk are the CP phase of Ak (Ak=λ, k, χu, A), respectively.

Due to the CP violation appears in the tree-level Higgs mass matrix which leads to a negative contribution to Higgs mass and the value of vs is large, the selection ranges of phase angles for λ, k and A are strictly limited.Fig. 1 (a) The variation of the mass of h0 with tanβ when θλT is selected as different phase angles. (b) The variation of the mass of h0 with Mλ when θχd is selected as different phase angles.

From the superpotential equations (1) and (2), it can be simply inferred that due to the vacuum expectation value vt and vt¯ are very small compared to the vacuum expectation value vu, vd and vs, the parameters that have no coupling with S, Hu and Hd are not sensitive to the influence of higgs mass and decay signal strengths. Moreover, because of the value of vs is very large, some parameters such as χu, k, A and Ak are too sensitive to higgs mass, resulting in the inability to observe those impacts on the decay signal strengths within suitable parameter ranges. After considering the above factors, we select the parameters that are sensitive to signal strengths and analyse the results.Fig. 2 (a) The variation of the mass of h0 with tanβ′ when Mχd is selected as different values. (b) The variation of the mass of h0 with MλT when ReAu is selected as different values.

We adopt the parameters as follow24 6≤tanβ≤50,0.2≤Mλ≤0.6,0.3≤tanβ′≤6,0.35≤MλT≤1,θλT=0,π/4,π/3andπ/2,θχd=0,π/3,π/2,π,Mχd=0.5,0.6,0.7,0.8,ReAu=-400,-500,-600,-700,

We explore the effects of these parameters on the Higgs mass and consider the limitations of experiments on Higgs mass, further explore their effects on Higgs decay signal strengths. In Figs. 1, 3 and 5, we adopt tanβ′ = 1.5, MλT = 0.9, Mχd = 0.8 and ReAu = − 500, in Figs. 2, 4 and 6, we adopt tanβ = 10, θλT = π/6, Mλ = 0.4 and θχd = 0.8.

In Fig. 1a and b, we select Mλ = 0.4, θχd=π/3 when observing the effects of tanβ and θλT on h0 mass, select tanβ = 10, θλT=π/6 when observing the effects of Mλ and θχd on h0 mass. In Fig. 2a and b, we select MλT = 0.9, ReAu=-500 when observing the effects of tanβ′ and Mχd on h0 mass, select tanβ′ = 1.5, Mχd=0.8 when observing the effects of MλT and ReAu on h0 mass.

We take the parameter ranges to accept 124 GeV ≤mh0≤ 126.5 GeV and study the impacts of these two sets of parameters on signal strengths.After considering the experimental data of Higgs mass, these parameter spaces are further limited25 8≤tanβ≤50,0.35≤Mλ≤0.55,0.5≤tanβ′≤5,0.45≤MλT≤0.8,

We analyse signal strengths within the new parameter ranges.Fig. 3 (a) The variation of neutron EDM with θχd and θλT. (b) The variation of neutron EDM with Mλ when θλT is selected as different values. (c) the variation of neutron EDM with tanβ′ when Mχd is selected as different values. (d) The variation of neutron EDM with MλT when ReAu is selected as different values.

Fig. 4 (a) The variation of electron EDM with θχd and θλT. (b) The variation of electron EDM with Mλ when θλT is selected as different values. (c) The variation of electron EDM with tanβ′ when Mχd is selected as different values. (d) The variation of electron EDM with MλT when ReAu is selected as different values.

In addition, the selection of complex parameters, especially the phase angle, is strongly limited by the electric dipole moments (EDM) of electrons and neutrons. To further verify the rationality of these parameters, we calculated the electric dipole moments of electrons and neutrons using some unique parameters of TNMSSM. These results are presented in Figs. 3 and 4.

The effective Lagrangian for spin-12 particles EDMs can be written asLIEDM=-i2dfψ¯σμνγ5ψFμν

Similarly, the effective Lagrangian for spin-12 particles EDMs can be written asLICEDM=-i2d~qCq¯σμνγ5TaqGμνa.

The concrete formulas of EDMs and CEDMs can be seen in58–62. The research by T.Ibrahim et al. shows that the main factors affecting fermion EDM are the phase of the fermion field , vector Superfields, Higgs field, and the coupling parameters between Higgs fields58. Due to the fact that this article only considers the phase of parameters related to Higgs decay, and λ and χu have already been selected, only the phase of χd and λT needs to be considered.

The experimental limitations for neutron and electron EDM are dn<0.18×1025 and de<0.11×1028, respectively, which are denoted by blue dashed lines in the Figs. 3 and 4. Due to the influence of λ and χu not being equal to 0, the EDM of electrons and neutrons is not equal to 0 when χd and λT are equal to 0. When λT changes, the EDM of electrons and neutrons does not change. This is because singlet and triplets are not coupled with fermions. The calculation results are in good agreement with experimental limits, leading us to believe that these parameters are appropriate.Fig. 5 (a, b) The variation of the signal strength μγγggF and μVVggF with tanβ when θλT is selected as different phase angles. (c, d) The variation of the signal strength μγγggF and μVVggF with Mλ when θχd is selected as different phase angles.

Fig. 6 (a, b) The variation of the signal strength μγγggF and μVVggF with tanβ′ when Mχd is selected as different values. (c, d) The variation of the signal strength μγγggF and μVVggF with MλT when ReAu is selected as different values.

Fig. 7 (a, b) The variation of the signal strength μbbVBF, μττVBF and μccVBF with tanβ when θλT is selected as different phase angles. (c, d) The variation of the signal strength μbbVBF, μττVBF and μccVBF with Mλ when θχd is selected as different phase angles.

Fig. 8 (a, b) The variation of the signal strength μbbVBF, μττVBF and μccVBF with tanβ′ when Mχd is selected as different values. (c, d) The variation of the signal strength μbbVBF, μττVBF and μccVBF with MλT when ReAu is selected as different values.

When different phase angles for λT and χd are selected, μγγggF in Fig. 5a and b first increases and then decreases, and reaches its maximum at tanβ = 12 and Mλ = 0.47, μVVggF in Fig. 5a and d first increases and then decreases, and reaches its maximum at tanβ = 14 and Mλ = 0.46, respectively. According to the datas provided in PDG57, μγγexp = 1.10 ± 0.0763–66, μWWexp = 1.19 ± 0.1265,66, μZZexp = 1.01 ± 0.0765,67,68, the maximum error between the experimental data and the theoretical prediction of μγγ is about 1σ and of μWW is less than 1σ and of μZZ is approaching 2.5σ.

When different vlaues for Mχd and ReAu are selected, μγγggF gradually increasing in Fig. 6a and gradually decreasing in Fig. 6b, μVVggF gradually increasing in Fig. 6c and gradually decreasing in Fig. 6d, respectively. Because the value of vd is relatively small compared to vu when tanβ=10, the effects of Mχd on signal strengths are insensitive, so that μγγggF and μVVggF remain almost unchange when Mχd changes. According to the datas provided in PDG, the maximum error between the experimental data and the theoretical prediction of μγγ is about 1.1σ and of μWW is less than 0.8σ and of μZZ is approaching 2.4σ.

When different phase angles for λT and χd are selected, μbbVBF and μττVBFin Fig. 7a and b gradually decreasing, μccVBF in Fig. 7c and d gradually increasing. According to the datas provided in PDG57, μbbexp = 0.98 ± 0.1265,66,69,70, μccexp = 37 ± 2071,72, and μττexp = 1.15-0.15+0.1665,66,73, the maximum error between the experimental data and the theoretical prediction of μbb is about 1.6σ, of μcc is less than 1.5σ and of μττ is much less than 1σ. When different phase angles of λT are selected, μbbVBF, μττVBFand μccVBF remain almost unchanged, because the Higgs singlet and triplets are uncoupled with fermions in the Lagrangian.

When different vlaues for Mχd and ReAu are selected, μγγggF almost unchange in Fig. 8a and b, μVVggF almost unchange in Fig. 8d and only slightly reduce in Fig. 8c. This meets our expectation because these parameters are uncoupled with fermions in the Lagrangian.

We calculated ΓNP(h→γγ) after ignoring the contribution of doubly charged particles ΓNPno doubly(h→γγ) and compared it with ΓNP(h→γγ) to quantify the contribution of two charged particles to the decay h→γγ.These results are presented in Fig. 9. The Cdoubly in Fig. 9 is defined byCdoubly=ΓNP(h→γγ)-ΓNPno doubly(h→γγ)ΓNP(h→γγ).

Fig. 9 (a) The variation of Cdoubly with tanβ when θλT is selected as different phase angles. (b) The variation of Cdoubly with Mλ when θχd is selected as different phase angles. (c) The variation of Cdoubly with tanβ′ when Mχd is selected as different values. (d) The variation of Cdoubly with MλT when ReAu is selected as different values.

Due to the large mass of the doubly charged Higgs, its contribution to the decay width is small. For the doubly charged chargino, because the vacuum expectation values of triplets are very small, its contribution to the decay width is also small.

We calculate the masses of the lightest doubly charged Higgs boson h++ and doubly charged chargino χ++, and compared them with experimental results. The analytical expressions for the mass of doubly charged Higgs boson and doubly charged chargino are given in Eqs. (9) and (10), and the numerical results are shown in Figs. 10 and 11.Fig. 10 (a) The variation of the mass of h++ with tanβ when θλT is selected as different phase angles. (b) The variation of the mass of h++ with Mλ when θχd is selected as different phase angles. (c) The variation of the mass of h++ with tanβ′ when Mχd is selected as different values. (d) The variation of the mass of h++ with MλT when ReAu is selected as different values.

Fig. 11 The variation of the mass of χ++ with MλT.

The limitation of the experiment is that mh++ > 1080 GeV74–76, we mark it with a blue dashed line in the Fig. 10. From the Eq. (10), it can be seen that the mass of mχ++ depends only on λT.

Conclusion

As an extended model of MSSM, the TNMSSM introduces a new gauge singlet S and two SU(2)L triplets T, T¯. The neutral parts of Higgs singlet, two Higgs doublets(Hd and Hu) and two Higgs singlets mix together, which constitute a 10×10 mass squared matrix of Higgs boson after considering CP violation. After considering the contributions of one-loop effective potential and two-loop radiative correction, we obtain complete Higgs mass matrix and the lightest Higgs h0 with a mass mh0 near 125 GeV in Chapter II. We show concrete theoretical expressions of the Higgs decay in Chapter III, and provide the concrete expressions of gHff, gHf~f~, gHH+H-, gHH++H--, gHχ+χ-, gHχ++χ--, gHVV, gAff, gAχ+χ- and gAχ++χ-- in the appendix B.

We explore the limitations of the minimum conditions of the scalar potential on the degrees of freedom of parameters and present the relevant results in the appendix A. After considing the minimum conditions of the scalar potential in the CPV TNMSSM model, we reduce five parameters mi2 (i=Hu, Hd, S, T and T¯) and the imaginary parts of the five parameters Ai (Ai = Au, Ad, Ak, AT, Ahu) . In Chapter IV, after considering limitations of the minimum conditions of the scalar potential and Higgs mass, we obtain a set of parameters and calculate the signal strengths of Higgs decay in different decay channels h→ γγ, h→ VV (V=W,Z) and h→ ff¯ (f=b,c,τ) , and compare them with experimental results in PDG, where μγγVBF, μWWVBF and μττVBF match the experimental data very well. Compared to the SM, the theoretical calculations of new physics better conform to the experimental results. In addition, the calculation result of the signal strengths μffVBF of h→ff¯ are almost the same when θλT takes different values, we analyze that this is because the triplets are uncoupled with fermions in the Lagrangian. Moreover, we explore the effects of some parameters that beyond the MSSM on Higgs mass and decay signal strengths, such as tanβ′, θχd, θλT, Mχd and MλT. From the Chapter IV, it can be seen that θχd and MλT are sensitive to Higgs decay signal strengths. In addition, in Chapter IV, we calculate the EDM of electrons and neutrons, as well as the mass spectrum of doubly charged particles, which well meet the experimental limitations. We also discuss the contribution of doubly charged particles to Higgs decay in the “Numerical analysis” Section. Due to the large mass of h++ and small vt, the contribution of doubly charged particles to Higgs decay is relatively small. We look forward to more experimental measurements of these decays in the future, which will be beneficial for understanding more about the properties of Higgs boson.

Supplementary Information

Supplementary Information.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-71222-8.

Acknowledgements

The work has been supported by Natural Science Foundation of Guangxi Autonomous Region with Grant No. 2022GXNSFDA035068.

Author contributions

N.Z. wrote the main manuscript text, H.C. and H.H. prepared figures 1–2. All authors reviewed the manuscript.

Data availibility

All data generated or analysed during this study are included in this published article.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Weinberg S A model of leptons Phys. Rev. Lett. 1967 19 1264 10.1103/PhysRevLett.19.1264
Weinberg, S. A model of leptons. Phys. Rev. Lett. 19, 1264. 10.1103/PhysRevLett.19.1264 (1967).10.1103/PhysRevLett.19.1264
2. Glashow SL Partial symmetries of weak interactions Nucl. Phys. 1961 22 579 10.1016/0029-5582(61)90469-2
Glashow, S. L. Partial symmetries of weak interactions. Nucl. Phys. 22, 579. 10.1016/0029-5582(61)90469-2 (1961).10.1016/0029-5582(61)90469-2
3. Aad G Observation of a new particle in the search for the Standard Model Higgs Boson with the ATLAS detector at the LHC Phys. Lett. B 2012 716 1 10.1016/j.physletb.2012.08.020
Aad, G. et al. Observation of a new particle in the search for the Standard Model Higgs Boson with the ATLAS detector at the LHC. Phys. Lett. B 716, 1. 10.1016/j.physletb.2012.08.020 (2012).10.1016/j.physletb.2012.08.020
4. Chatrchyan S Observation of a New Boson at a mass of 125 GeV with the CMS experiment at the LHC Phys. Lett. B 2012 716 30 10.1016/j.physletb.2012.08.021
Chatrchyan, S. et al. Observation of a New Boson at a mass of 125 GeV with the CMS experiment at the LHC. Phys. Lett. B 716, 30. 10.1016/j.physletb.2012.08.021 (2012).10.1016/j.physletb.2012.08.021
5. Abe K Indication of electron neutrino appearance from an accelerator-produced off-axis muon neutrino beam Phys. Rev. Lett. 2011 107 041801 10.1103/PhysRevLett.107.041801 21866992
Abe, K. et al. Indication of electron neutrino appearance from an accelerator-produced off-axis muon neutrino beam. Phys. Rev. Lett. 107, 041801. 10.1103/PhysRevLett.107.041801 (2011).21866992 10.1103/PhysRevLett.107.041801
6. Adamson P Improved search for muon-neutrino to electron-neutrino oscillations in MINOS Phys. Rev. Lett. 2011 107 181802 10.1103/PhysRevLett.107.181802 22107623
Adamson, P. et al. Improved search for muon-neutrino to electron-neutrino oscillations in MINOS. Phys. Rev. Lett. 107, 181802. 10.1103/PhysRevLett.107.181802 (2011).22107623 10.1103/PhysRevLett.107.181802
7. Rosiek J Complete set of Feynman rules for the minimal supersymmetric extension of the standard model Phys. Rev. D 1990 41 3464 10.1103/PhysRevD.41.3464
Rosiek, J. Complete set of Feynman rules for the minimal supersymmetric extension of the standard model. Phys. Rev. D 41, 3464. 10.1103/PhysRevD.41.3464 (1990).10.1103/PhysRevD.41.3464
8. Kim JE Nilles HP The mu problem and the strong CP problem Phys. Lett. B 1984 138 150 10.1016/0370-2693(84)91890-2
Kim, J. E. & Nilles, H. P. The mu problem and the strong CP problem. Phys. Lett. B 138, 150. 10.1016/0370-2693(84)91890-2 (1984).10.1016/0370-2693(84)91890-2
9. Ellwanger U Hugonie C Teixeira AM The next-to-minimal supersymmetric standard model Phys. Rept. 2010 496 1 10.1016/j.physrep.2010.07.001
Ellwanger, U., Hugonie, C. & Teixeira, A. M. The next-to-minimal supersymmetric standard model. Phys. Rept. 496, 1. 10.1016/j.physrep.2010.07.001 (2010).10.1016/j.physrep.2010.07.001
10. Mason JD Gauge Mediation with a small mu term and light Squarks Phys. Rev. D 2009 80 015026 10.1103/PhysRevD
Mason, J. D. Gauge Mediation with a small mu term and light Squarks. Phys. Rev. D 80, 015026. 10.1103/PhysRevD (2009).10.1103/PhysRevD
11. Ellwanger U Hugonie C The Upper bound on the lightest Higgs mass in the NMSSM revisited Mod. Phys. Lett. A 2007 22 1581 10.1142/S0217732307023870
Ellwanger, U. & Hugonie, C. The Upper bound on the lightest Higgs mass in the NMSSM revisited. Mod. Phys. Lett. A 22, 1581. 10.1142/S0217732307023870 (2007).10.1142/S0217732307023870
12. Ananthanarayan B Pandita PN Particle spectrum in the nonminimal supersymmetric standard model with tan beta approximately = m(t)/m(b) Phys. Lett. B 1996 371 245 10.1016/0370-2693(96)00010-X
Ananthanarayan, B. & Pandita, P. N. Particle spectrum in the nonminimal supersymmetric standard model with tan beta approximately = m(t)/m(b). Phys. Lett. B 371, 245 (1996).10.1016/0370-2693(96)00010-X
13. Ananthanarayan B Pandita PN The Nonminimal supersymmetric standard model at large tan beta Int. J. Mod. Phys. A 1997 12 2321 10.1142/S0217751X97001353
Ananthanarayan, B. & Pandita, P. N. The Nonminimal supersymmetric standard model at large tan beta. Int. J. Mod. Phys. A 12, 2321. 10.1142/S0217751X97001353 (1997).10.1142/S0217751X97001353
14. Ellwanger U Hugonie C Yukawa induced radiative corrections to the lightest Higgs boson mass in the NMSSM Phys. Lett. B 2005 623 93 10.1016/j.physletb.2005.07.039
Ellwanger, U. & Hugonie, C. Yukawa induced radiative corrections to the lightest Higgs boson mass in the NMSSM. Phys. Lett. B 623, 93. 10.1016/j.physletb.2005.07.039 (2005).10.1016/j.physletb.2005.07.039
15. Degrassi G Slavich P On the radiative corrections to the neutral Higgs boson masses in the NMSSM Nucl. Phys. B 2010 825 119 10.1016/j.nuclphysb.2009.09.018
Degrassi, G. & Slavich, P. On the radiative corrections to the neutral Higgs boson masses in the NMSSM. Nucl. Phys. B 825, 119. 10.1016/j.nuclphysb.2009.09.018 (2010).10.1016/j.nuclphysb.2009.09.018
16. Agashe K Azatov A Katz A Kim D Improving the tunings of the MSSM by adding triplets and singlet Phys. Rev. D 2011 84 115024 10.1103/PhysRevD.84.115024
Agashe, K., Azatov, A., Katz, A. & Kim, D. Improving the tunings of the MSSM by adding triplets and singlet. Phys. Rev. D 84, 115024. 10.1103/PhysRevD.84.115024 (2011).10.1103/PhysRevD.84.115024
17. Espinosa JR Quiros M Higgs triplets in the supersymmetric standard model Nucl. Phys. B 1992 384 113 10.1016/0550-3213(92)90464-M
Espinosa, J. R. & Quiros, M. Higgs triplets in the supersymmetric standard model. Nucl. Phys. B 384, 113. 10.1016/0550-3213(92)90464-M (1992).10.1016/0550-3213(92)90464-M
18. Zhang, Z.-Y. Yang, J.-L. Zhang, H.-B. & Feng, T.-F. Transition magnetic moment of Majorana neutrinos in the triplets next-to-minimal MSSM. arXiv:2406.18323 [hep-ph] (2024).
19. Christenson JH Cronin JW Fitch VL Turlay R Evidence for the 2π Decay of the K20 Meson Phys. Rev. Lett. 1964 13 138 10.1103/PhysRevLett.13.138
Christenson, J. H., Cronin, J. W., Fitch, V. L. & Turlay, R. Evidence for the Decay of the Meson. Phys. Rev. Lett. 13, 138. 10.1103/PhysRevLett.13.138 (1964).10.1103/PhysRevLett.13.138
20. Neubert M B decays and CP violation Int. J. Mod. Phys. A 1996 11 4173 10.1142/S0217751X96001966
Neubert, M. B decays and CP violation. Int. J. Mod. Phys. A 11, 4173. 10.1142/S0217751X96001966 (1996).10.1142/S0217751X96001966
21. Sakharov AD Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe Pisma Zh. Eksp. Teor. Fiz. 1967 5 32 10.1070/PU1991v034n05ABEH002497
Sakharov, A. D. Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe. Pisma Zh. Eksp. Teor. Fiz. 5, 32. 10.1070/PU1991v034n05ABEH002497 (1967).10.1070/PU1991v034n05ABEH002497
22. Lee TD A theory of spontaneous T violation Phys. Rev. D 1973 8 1226 10.1103/PhysRevD.8.1226
Lee, T. D. A theory of spontaneous T violation. Phys. Rev. D 8, 1226. 10.1103/PhysRevD.8.1226 (1973).10.1103/PhysRevD.8.1226
23. Weinberg S Gauge theory of CP violation Phys. Rev. Lett. 1976 37 657 10.1103/PhysRevLett.37.657
Weinberg, S. Gauge theory of CP violation. Phys. Rev. Lett. 37, 657. 10.1103/PhysRevLett.37.657 (1976).10.1103/PhysRevLett.37.657
24. Coleman SR Weinberg EJ Radiative corrections as the origin of spontaneous symmetry breaking Phys. Rev. D 1973 7 1888 10.1103/PhysRevD.7.1888
Coleman, S. R. & Weinberg, E. J. Radiative corrections as the origin of spontaneous symmetry breaking. Phys. Rev. D 7, 1888. 10.1103/PhysRevD.7.1888 (1973).10.1103/PhysRevD.7.1888
25. Carena M Espinosa JR Quiros M Wagner CEM Analytical expressions for radiatively corrected Higgs masses and couplings in the MSSM Phys. Lett. B 1995 355 209 10.1016/0370-2693(95)00694-G
Carena, M., Espinosa, J. R., Quiros, M. & Wagner, C. E. M. Analytical expressions for radiatively corrected Higgs masses and couplings in the MSSM. Phys. Lett. B 355, 209. 10.1016/0370-2693(95)00694-G (1995).10.1016/0370-2693(95)00694-G
26. Carena M Quiros M Wagner CEM Effective potential methods and the Higgs mass spectrum in the MSSM Nucl. Phys. B 1996 461 407 10.1016/0550-3213(95)00665-6
Carena, M., Quiros, M. & Wagner, C. E. M. Effective potential methods and the Higgs mass spectrum in the MSSM. Nucl. Phys. B 461, 407. 10.1016/0550-3213(95)00665-6 (1996).10.1016/0550-3213(95)00665-6
27. Carena M Ellis JR Pilaftsis A Wagner CEM Renormalization group improved effective potential for the MSSM Higgs sector with explicit CP violation Nucl. Phys. B 2000 586 92 10.1016/S0550-3213(00)00358-8
Carena, M., Ellis, J. R., Pilaftsis, A. & Wagner, C. E. M. Renormalization group improved effective potential for the MSSM Higgs sector with explicit CP violation. Nucl. Phys. B 586, 92. 10.1016/S0550-3213(00)00358-8 (2000).10.1016/S0550-3213(00)00358-8
28. Chen H-X Cui S-K Zhu N-Y Zhang Z-Y Hu H-C B meson rare decays in the TNMSSM* Chin. Phys. C 2024 48 053104 10.1088/1674-1137/ad2a62
Chen, H.-X., Cui, S.-K., Zhu, N.-Y., Zhang, Z.-Y. & Hu, H.-C. B meson rare decays in the TNMSSM*. Chin. Phys. C 48, 053104. 10.1088/1674-1137/ad2a62 (2024).10.1088/1674-1137/ad2a62
29. Hu, H.-C. Zhang, Z.-Y. Zhu, N.-Y. & Chen, H.-X. Higgs boson decays in the TNMSSM. arXiv:2406.00946 [hep-ph] (2024).
30. Ouahid MA Ahl Laamara R Neutrino and doubly charged Higgs boson phenomenology in flavored-TNMSSM Nucl. Phys. B 2022 974 115640 10.1016/j.nuclphysb.2021.115640
Ouahid, M. A. & Ahl Laamara, R. Neutrino and doubly charged Higgs boson phenomenology in flavored-TNMSSM. Nucl. Phys. B 974, 115640. 10.1016/j.nuclphysb.2021.115640 (2022).10.1016/j.nuclphysb.2021.115640
31. Carena M Gori S Shah NR Wagner CEM A 125 GeV SM-like Higgs in the MSSM and the γγ rate J. High Energy Phys. 2012 03 014 10.1007/JHEP03(2012)014
Carena, M., Gori, S., Shah, N. R. & Wagner, C. E. M. A 125 GeV SM-like Higgs in the MSSM and the rate. J. High Energy Phys. 03, 014 (2012).10.1007/JHEP03(2012)014
32. Oshimo N Coexistence of CP eigenstates in Higgs boson decay Prog. Theor. Exp. Phys. 2013 2013 083B04 10.1093/ptep/ptt062
Oshimo, N. Coexistence of CP eigenstates in Higgs boson decay. Prog. Theor. Exp. Phys. 2013, 083B04. 10.1093/ptep/ptt062 (2013).10.1093/ptep/ptt062
33. Ellis JR Gaillard MK Nanopoulos DV A phenomenological profile of the Higgs Boson Nucl. Phys. B 1976 106 292 10.1016/0550-3213(76)90382-5
Ellis, J. R., Gaillard, M. K. & Nanopoulos, D. V. A phenomenological profile of the Higgs Boson. Nucl. Phys. B 106, 292. 10.1016/0550-3213(76)90382-5 (1976).10.1016/0550-3213(76)90382-5
34. Djouadi A The Anatomy of electro-weak symmetry breaking. II. The Higgs bosons in the minimal supersymmetric model Phys. Rept. 2008 459 1 10.1016/j.physrep.2007.10.005
Djouadi, A. The Anatomy of electro-weak symmetry breaking. II. The Higgs bosons in the minimal supersymmetric model. Phys. Rept. 459, 1. 10.1016/j.physrep.2007.10.005 (2008).10.1016/j.physrep.2007.10.005
35. Gunion, J. F. & Haber, H. E. Errata for Higgs bosons in supersymmetric models: 1, 2 and 3. arXiv:hep-ph/9301205 (1992).
36. Wilczek F Decays of heavy vector mesons into Higgs particles Phys. Rev. Lett. 1977 39 1304 10.1103/PhysRevLett.39.1304
Wilczek, F. Decays of heavy vector mesons into Higgs particles. Phys. Rev. Lett. 39, 1304. 10.1103/PhysRevLett.39.1304 (1977).10.1103/PhysRevLett.39.1304
37. Georgi HM Glashow SL Machacek ME Nanopoulos DV Higgs bosons from two gluon annihilation in proton proton collisions Phys. Rev. Lett. 1978 40 692 10.1103/PhysRevLett.40.692
Georgi, H. M., Glashow, S. L., Machacek, M. E. & Nanopoulos, D. V. Higgs bosons from two gluon annihilation in proton proton collisions. Phys. Rev. Lett. 40, 692. 10.1103/PhysRevLett.40.692 (1978).10.1103/PhysRevLett.40.692
38. Kileng B Effects of scalar mixing in g g–> Higgs –> gamma gamma Z. Phys. C 1994 63 87 10.1007/BF01577547
Kileng, B. Effects of scalar mixing in g g– Higgs – gamma gamma. Z. Phys. C 63, 87. 10.1007/BF01577547 (1994).10.1007/BF01577547
39. Djouadi A Squark effects on Higgs boson production and decay at the LHC Phys. Lett. B 1998 435 101 10.1016/S0370-2693(98)00784-9
Djouadi, A. Squark effects on Higgs boson production and decay at the LHC. Phys. Lett. B 435, 101. 10.1016/S0370-2693(98)00784-9 (1998).10.1016/S0370-2693(98)00784-9
40. Belanger G Boudjema F Donato F Godbole R Rosier-Lees S SUSY Higgs at the LHC: Effects of light charginos and neutralinos Nucl. Phys. B 2000 581 3 10.1016/S0550-3213(00)00243-1
Belanger, G., Boudjema, F., Donato, F., Godbole, R. & Rosier-Lees, S. SUSY Higgs at the LHC: Effects of light charginos and neutralinos. Nucl. Phys. B 581, 3. 10.1016/S0550-3213(00)00243-1 (2000).10.1016/S0550-3213(00)00243-1
41. Shifman MA Vainshtein AI Voloshin MB Zakharov VI Low-energy theorems for Higgs boson couplings to photons Sov. J. Nucl. Phys. 1979 30 711
Shifman, M. A., Vainshtein, A. I., Voloshin, M. B. & Zakharov, V. I. Low-energy theorems for Higgs boson couplings to photons. Sov. J. Nucl. Phys. 30, 711 (1979).
42. Gavela MB Girardi G Malleville C Sorba P A nonlinear R(xi) gauge condition for the electroweak SU(2) X U(1) model Nucl. Phys. B 1981 193 257 10.1016/0550-3213(81)90529-0
Gavela, M. B., Girardi, G., Malleville, C. & Sorba, P. A nonlinear R(xi) gauge condition for the electroweak SU(2) X U(1) model. Nucl. Phys. B 193, 257. 10.1016/0550-3213(81)90529-0 (1981).10.1016/0550-3213(81)90529-0
43. Kalyniak P Bates R Ng JN Two photon decays of scalar and pseudoscalar bosons in supersymmetry Phys. Rev. D 1986 33 755 10.1103/PhysRevD.33.755
Kalyniak, P., Bates, R. & Ng, J. N. Two photon decays of scalar and pseudoscalar bosons in supersymmetry. Phys. Rev. D 33, 755. 10.1103/PhysRevD.33.755 (1986).10.1103/PhysRevD.33.755
44. Bates R Ng JN Kalyniak P Two photon decay widths of Higgs bosons in minimal broken supersymmetry Phys. Rev. D 1986 34 172 10.1103/PhysRevD.34.172
Bates, R., Ng, J. N. & Kalyniak, P. Two photon decay widths of Higgs bosons in minimal broken supersymmetry. Phys. Rev. D 34, 172. 10.1103/PhysRevD.34.172 (1986).10.1103/PhysRevD.34.172
45. Kane GL Kribs GD Martin SP Wells JD Two photon decays of the lightest Higgs boson of supersymmetry at the LHC Phys. Rev. D 1996 53 213 10.1103/PhysRevD.53.213
Kane, G. L., Kribs, G. D., Martin, S. P. & Wells, J. D. Two photon decays of the lightest Higgs boson of supersymmetry at the LHC. Phys. Rev. D 53, 213. 10.1103/PhysRevD.53.213 (1996).10.1103/PhysRevD.53.213
46. Djouadi A Driesen V Hollik W Illana JI The Coupling of the lightest SUSY Higgs boson to two photons in the decoupling regime Eur. Phys. J. C 1998 1 149 10.1007/BF01245805
Djouadi, A., Driesen, V., Hollik, W. & Illana, J. I. The Coupling of the lightest SUSY Higgs boson to two photons in the decoupling regime. Eur. Phys. J. C 1, 149. 10.1007/BF01245805 (1998).10.1007/BF01245805
47. Gunion JF Haber HE Kane GL Dawson S The Higgs Hunter’s Guide 2000 CRC Press
Gunion, J. F., Haber, H. E., Kane, G. L. & Dawson, S. The Higgs Hunter’s Guide (CRC Press, 2000).
48. Lee BW Quigg C Thacker HB Weak interactions at very high-energies: The role of the Higgs boson mass Phys. Rev. D 1977 16 1519 10.1103/PhysRevD.16.1519
Lee, B. W., Quigg, C. & Thacker, H. B. Weak interactions at very high-energies: The role of the Higgs boson mass. Phys. Rev. D 16, 1519. 10.1103/PhysRevD.16.1519 (1977).10.1103/PhysRevD.16.1519
49. Resnick L Sundaresan MK Watson PJS Is there a light scalar boson? Phys. Rev. D 1973 8 172 10.1103/PhysRevD.8.172
Resnick, L., Sundaresan, M. K. & Watson, P. J. S. Is there a light scalar boson?. Phys. Rev. D 8, 172. 10.1103/PhysRevD.8.172 (1973).10.1103/PhysRevD.8.172
50. Rizzo TG Decays of heavy Higgs bosons Phys. Rev. D 1980 22 722 10.1103/PhysRevD.22.722
Rizzo, T. G. Decays of heavy Higgs bosons. Phys. Rev. D 22, 722. 10.1103/PhysRevD.22.722 (1980).10.1103/PhysRevD.22.722
51. Pocsik G Torma T On the decays of heavy Higgs bosons Z. Phys. C 1980 6 1 10.1007/BF01427913
Pocsik, G. & Torma, T. On the decays of heavy Higgs bosons. Z. Phys. C 6, 1. 10.1007/BF01427913 (1980).10.1007/BF01427913
52. Keung W-Y Marciano WJ Higgs scalar decays: H –> W+- X Phys. Rev. D 1984 30 248 10.1103/PhysRevD.30.248
Keung, W.-Y. & Marciano, W. J. Higgs scalar decays: H – W+- X. Phys. Rev. D 30, 248. 10.1103/PhysRevD.30.248 (1984).10.1103/PhysRevD.30.248
53. Cahn RN The Higgs boson Rept. Prog. Phys. 1989 52 389 10.1088/0034-4885/52/4/001
Cahn, R. N. The Higgs boson. Rept. Prog. Phys. 52, 389. 10.1088/0034-4885/52/4/001 (1989).10.1088/0034-4885/52/4/001
54. Grau A Panchieri G Phillips RJN Contributions of off-shell top quarks to decay processes Phys. Lett. B 1990 251 293 10.1016/0370-2693(90)90939-4
Grau, A., Panchieri, G. & Phillips, R. J. N. Contributions of off-shell top quarks to decay processes. Phys. Lett. B 251, 293. 10.1016/0370-2693(90)90939-4 (1990).10.1016/0370-2693(90)90939-4
55. Kniehl BA The Higgs boson decay H→Z gg Phys. Lett. B 1990 244 537 10.1016/0370-2693(90)90360-I
Kniehl, B. A. The Higgs boson decay HZ . Phys. Lett. B 244, 537. 10.1016/0370-2693(90)90360-I (1990).10.1016/0370-2693(90)90360-I
56. Arbey A Deandrea A Mahmoudi F Tarhini A Anomaly mediated supersymmetric models and Higgs data from the LHC Phys. Rev. D 2013 87 115020 10.1103/PhysRevD.87.115020
Arbey, A., Deandrea, A., Mahmoudi, F. & Tarhini, A. Anomaly mediated supersymmetric models and Higgs data from the LHC. Phys. Rev. D 87, 115020. 10.1103/PhysRevD.87.115020 (2013).10.1103/PhysRevD.87.115020
57. Workman RL Particle data group Rev. Part. Phys. 2022 2022 083C01
Workman, R. L. et al. Particle data group. Rev. Part. Phys. 2022, 083C01 (2022).
58. Ibrahim, T. & Nath, P. The Neutron and the lepton EDMs in MSSM, large CP violating phases, and the cancellation mechanism. Phys. Rev. D 58, 111301. 10.1103/PhysRevD.58.111301 (1998). Note [Erratum: Phys. Rev. D 60, 099902 (1999)].
59. Ibrahim T Itani A Nath P Electron electric dipole moment as a sensitive probe of PeV scale physics Phys. Rev. D 2014 90 055006 10.1103/PhysRevD.90.055006
Ibrahim, T., Itani, A. & Nath, P. Electron electric dipole moment as a sensitive probe of PeV scale physics. Phys. Rev. D 90, 055006. 10.1103/PhysRevD.90.055006 (2014).10.1103/PhysRevD.90.055006
60. Aboubrahim A Ibrahim T Nath P Zorik A Chromoelectric dipole moments of quarks in MSSM extensions Phys. Rev. D 2015 92 035013 10.1103/PhysRevD.92.035013
Aboubrahim, A., Ibrahim, T., Nath, P. & Zorik, A. Chromoelectric dipole moments of quarks in MSSM extensions. Phys. Rev. D 92, 035013. 10.1103/PhysRevD.92.035013 (2015).10.1103/PhysRevD.92.035013
61. Yang J-L Feng T-F Zhang H-B Electroweak baryogenesis and electron EDM in the B-LSSM Eur. Phys. J. C 2020 80 210 10.1140/epjc/s10052-020-7753-9
Yang, J.-L., Feng, T.-F. & Zhang, H.-B. Electroweak baryogenesis and electron EDM in the B-LSSM. Eur. Phys. J. C 80, 210. 10.1140/epjc/s10052-020-7753-9 (2020).10.1140/epjc/s10052-020-7753-9
62. Yang J-L Electric dipole moments of neutron and heavy quarks in the B-LSSM J. High Energy Phys. 2020 04 013 10.1007/JHEP04(2020)013
Yang, J.-L. et al. Electric dipole moments of neutron and heavy quarks in the B-LSSM. J. High Energy Phys. 04, 013. 10.1007/JHEP04(2020)013 (2020).10.1007/JHEP04(2020)013
63. Sirunyan AM Measurements of Higgs boson properties in the diphoton decay channel in proton-proton collisions at s= 13 TeV J. High Energy Phys. 2018 11 185 10.1007/JHEP11(2018)185
Sirunyan, A. M. et al. Measurements of Higgs boson properties in the diphoton decay channel in proton-proton collisions at 13 TeV. J. High Energy Phys. 11, 185. 10.1007/JHEP11(2018)185 (2018).10.1007/JHEP11(2018)185
64. Aaboud M Measurements of Higgs boson properties in the diphoton decay channel with 36 fb-1 of pp collision data at s=13 TeV with the ATLAS detector Phys. Rev. D 2018 98 052005 10.1103/PhysRevD.98.052005
Aaboud, M. et al. Measurements of Higgs boson properties in the diphoton decay channel with 36 fb of collision data at TeV with the ATLAS detector. Phys. Rev. D 98, 052005. 10.1103/PhysRevD.98.052005 (2018).10.1103/PhysRevD.98.052005
65. Aad G Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at s=7 and 8 TeV J. High Energy Phys. 2016 08 045 10.1007/JHEP08(2016)045
Aad, G. et al. Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at and 8 TeV. J. High Energy Phys. 08, 045. 10.1007/JHEP08(2016)045 (2016).10.1007/JHEP08(2016)045
66. Aaltonen T Higgs boson studies at the Tevatron Phys. Rev. D 2013 88 052014 10.1103/PhysRevD.88.052014
Aaltonen, T. et al. Higgs boson studies at the Tevatron. Phys. Rev. D 88, 052014. 10.1103/PhysRevD.88.052014 (2013).10.1103/PhysRevD.88.052014
67. Aaboud M Measurement of the Higgs boson coupling properties in the H→ZZ∗→4ℓ decay channel at s = 13 TeV with the ATLAS detector J. High Energy Phys. 2018 03 095 10.1007/JHEP03(2018)095
Aaboud, M. et al. Measurement of the Higgs boson coupling properties in the decay channel at = 13 TeV with the ATLAS detector. J. High Energy Phys. 03, 095. 10.1007/JHEP03(2018)095 (2018).10.1007/JHEP03(2018)095
68. Sirunyan AM Measurements of properties of the Higgs boson decaying into the four-lepton final state in pp collisions at s=13 TeV J. High Energy Phys. 2017 11 047 10.1007/JHEP11(2017)047
Sirunyan, A. M. et al. Measurements of properties of the Higgs boson decaying into the four-lepton final state in pp collisions at TeV. J. High Energy Phys. 11, 047. 10.1007/JHEP11(2017)047 (2017).10.1007/JHEP11(2017)047
69. Sirunyan AM Observation of Higgs boson decay to bottom quarks Phys. Rev. Lett. 2018 121 121801 10.1103/PhysRevLett.121.121801 30296133
Sirunyan, A. M. et al. Observation of Higgs boson decay to bottom quarks. Phys. Rev. Lett. 121, 121801. 10.1103/PhysRevLett.121.121801 (2018).30296133 10.1103/PhysRevLett.121.121801
70. Aaboud M Observation of H→bb¯ decays and VH production with the ATLAS detector Phys. Lett. B 2018 786 59 10.1016/j.physletb.2018.09.013
Aaboud, M. et al. Observation of decays and production with the ATLAS detector. Phys. Lett. B 786, 59. 10.1016/j.physletb.2018.09.013 (2018).10.1016/j.physletb.2018.09.013
71. Tumasyan A Search for Higgs boson decay to a charm quark-antiquark pair in proton-proton collisions at s = 13 TeV Phys. Rev. Lett. 2023 131 061801 10.1103/PhysRevLett.131.061801 37625071
Tumasyan, A. et al. Search for Higgs boson decay to a charm quark-antiquark pair in proton-proton collisions at s = 13 TeV. Phys. Rev. Lett. 131, 061801. 10.1103/PhysRevLett.131.061801 (2023).37625071 10.1103/PhysRevLett.131.061801
72. Tumasyan A Search for Higgs boson and observation of Z boson through their decay into a charm quark-antiquark pair in boosted topologies in proton-proton collisions at s = 13 TeV Phys. Rev. Lett. 2023 131 041801 10.1103/PhysRevLett.131.041801 37566854
Tumasyan, A. et al. Search for Higgs boson and observation of Z boson through their decay into a charm quark-antiquark pair in boosted topologies in proton-proton collisions at s = 13 TeV. Phys. Rev. Lett. 131, 041801. 10.1103/PhysRevLett.131.041801 (2023).37566854 10.1103/PhysRevLett.131.041801
73. Sirunyan AM Observation of the Higgs boson decay to a pair of τ leptons with the CMS detector Phys. Lett. B 2018 779 283 10.1016/j.physletb.2018.02.004
Sirunyan, A. M. et al. Observation of the Higgs boson decay to a pair of leptons with the CMS detector. Phys. Lett. B 779, 283. 10.1016/j.physletb.2018.02.004 (2018).10.1016/j.physletb.2018.02.004
74. Novak, T. Searches for singly- and doubly-charged Higgs bosons with the ATLAS detector. 10.22323/1.449.0433 ( 2024)
75. Aad G Search for doubly charged Higgs boson production in multi-lepton final states using 139 fb-1 of proton-proton collisions at s = 13 TeV with the ATLAS detector Eur. Phys. J. C 2023 83 605 10.1140/epjc/s10052-023-11578-9
Aad, G. et al. Search for doubly charged Higgs boson production in multi-lepton final states using 139 fb of proton-proton collisions at = 13 TeV with the ATLAS detector. Eur. Phys. J. C 83, 605. 10.1140/epjc/s10052-023-11578-9 (2023).10.1140/epjc/s10052-023-11578-9
76. Leban, B. Search for doubly charged Higgs boson production in multi-lepton final states using 139 fb of proton-proton collisions at = 13 TeV with the ATLAS detector. 10.22323/1.414.1081 ( 2022)
