e + e − →hhZ in the B-L symmetric SSM
Dan-Dan Cui, Tai-Fu Feng, Yu-Li Yan, Hai-Bin Zhang, Guo-Zhu Ning, Jin-Lei Yang
e + e − → hhZ in the B − L symmetric SSM Dan-Dan Cui,
Tai-Fu Feng, † Yu-Li Yan, ‡ Hai-Bin Zhang ,
Guo-Zhu Ning, and Jin-Lei Yang ∥ Department of Physics, Hebei University, Baoding 071002, China Key Laboratory of High-precision Computation and Applicationof Quantum Field Theory of Hebei Province, Baoding 071002, China Department of Physics, Chongqing University, Chongqing 400044, China CAS Key Laboratory of Theoretical Physics, School of Physical Sciences,University of Chinese Academy of Sciences, Beijing 100049, China (Received 11 January 2020; accepted 14 September 2020; published 5 October 2020)The double Higgs boson production through e þ e − → hhZ is analyzed in the minimal super-symmetric extension of the Standard Model (SM) with the local gauge symmetry U ð Þ B − L , where h denotes the lightest Higgs boson with 125 GeV. Considering the constraints from the updatedprediction data, we find that the production cross section of this process in the model depends on someparameters strongly. DOI: 10.1103/PhysRevD.102.075002
I. INTRODUCTION
Among the particles predicted by the StandardModel (SM), the Higgs boson was the last particlediscovered. Its discovery proves that correctness of theparticles mass produced by the spontaneous symmetrybreaking. Nevertheless, some other questions also arise. Forexample, is the discovered particle really the Higgs bosonpredicted by the SM? Are there other neutral or chargedscalar fields? Whether or not the coupling of Higgs withmatter fields and gauge particles meets the theoreticalpredictions of the SM? In addition, the SM can not providethe candidates for dark matter, can not explain the asym-metry of matter and antimatter in the universe, etc., so theserequire us to look for new physics beyond the SM. The B − L Symmetric SM (B-LSSM) is one of the simplestextension models of the Minimal Supersymmetric StandardModel (MSSM), which is based on the gauge symmetrygroup SU ð Þ C ⊗ SU ð Þ L ⊗ U ð Þ Y ⊗ U ð Þ B − L , where B stands for the baryon number and L stands for the leptonnumber, respectively.The B-LSSM alleviates the hierarchy problem arisen inthe MSSM, because the exotic singlet Higgs boson andright-handed (s)neutrinos [1 –
7] can alleviate the constraints from the experimental data of LHC, Tevatron and LEP.Furthermore, U ð Þ B − L gauge group can help to understandthe possible broken ways of R parity in the supersymmetricmodels [8 – – – – e þ e − → hhZ is oneof the main processes [22] for studying the self-coupling ofHiggs boson.In this work, we analyze the cross section of e þ e − → hhZ in the B-LSSM. The cross section and angulardistribution of this process can be used to determine theself-coupling of the Higgs at future collider experiments[23]. The process has been studied in the frameworks of theSM, the two-Higgs-doublet model [13,23 – E cm ¼ GeV, 0.12 fb for E cm ¼ GeV. Iffuture experimental observations are much larger than thetheoretical predictions of the SM, which can be consideredas an evidence of new physics beyond the SM. Regarding thecross section, there are no experimental observations, but theproduction cross section for the e þ e − → hhZ process istypically of the order of 0.1 fb at the collision energy justabove the threshold at about 400 GeV, and at theinternational linear collider with a center-of-mass energyof 500 GeV, the trilinear Higgs boson coupling can bemeasured via this process [26]. In addition, if the exper-imental measurement value deviates obviously from the SMvalue in the future, the model can be considered as anexplanation to account for the deviations. If the experimentalmeasured values are consistent with the SM, it will constrainour parameter space. * [email protected] † [email protected] ‡ [email protected] § [email protected] ∥ [email protected] Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article ’ s title, journal citation,and DOI. Funded by SCOAP . PHYSICAL REVIEW D = = = he paper is organized as follows. In Sec. II, weintroduce the B-LSSM in detail. In Sec. III, we analyzethe dependence of the cross section and angular distributionof the final state particles on the parameters in this model.The numerical results are given in Sec. IV, and someconclusions are summarized in Sec. V. II. INTRODUCTION OF THE MODEL
In this section, we briefly introduce the basic character-istics of the B-LSSM. About the B-LSSM, there are severaldifferent versions. Here, we apply the version describedin Refs. [27 – – SU ð Þ C ⊗ SU ð Þ L ⊗ U ð Þ Y ⊗ U ð Þ B − L , where the U ð Þ B − L is an additional gaugesymmetry.With the local gauge group SU ð Þ C ⊗ SU ð Þ L ⊗ U ð Þ Y ⊗ U ð Þ B − L , the superpotential of the B-LSSM iswritten as W ¼ W MSSM þ W ð B − L Þ : ð Þ Where W MSSM denotes the superpotential of the MSSM[36], W ð B − L Þ denotes the additional terms in this model,and can be written as W ð B − L Þ ¼ Y ν ;ij ˆ L i ˆ H ˆ ν cj þ μ ˆ η ˆ η þ Y x;ij ˆ ν ci ˆ η ˆ ν cj : ð Þ The quantum numbers of the superfields of the quarks andleptons are assigned as ˆ Q i ¼ (cid:1) ˆ u i ˆ d i (cid:3) ∼ ð ; ; = ; = Þ ; ˆ L i ¼ (cid:1) ˆ ν i ˆ e i (cid:3) ∼ ð ; ; − = ; − = Þ ; ˆ U ci ∼ ð ; ; − = ; − = Þ ; ˆ D ci ∼ ð ; ; = ; − = Þ ; ˆ E ci ∼ ð ; ; ; = Þ ; ð Þ with i ¼ , 2, 3 denoting the generation indices. Inaddition, the quantum numbers of two Higgs doublets are ˆ H ¼ (cid:1) ˆ H ˆ H (cid:3) ∼ ð ; ; − = ; Þ ; ˆ H ¼ (cid:1) ˆ H ˆ H (cid:3) ∼ ð ; ; = ; Þ : ð Þ The quantum numbers of chiral singlet superfields are ˆ η ∼ ð ; ; ; − Þ , ˆ η ∼ ð ; ; ; Þ , and that of three gen-erations of right-handed neutrinos is ˆ ν ci ∼ ð ; ; ; = Þ .Correspondingly, the soft breaking terms in the B-LSSMare written as L soft ¼ L MSSMsoft þ L B − L soft ; ð Þ where the L MSSMsoft denotes the soft breaking terms in theMSSM [36], and L B − L soft ¼ (cid:4) − M BB ˜ λ B ˜ λ B − M B ˜ λ B ˜ λ B − B μ ˜ η ˜ η þ T ij ν H ˜ ν ci ˜ L j þ T ijx ˜ η ˜ ν ci ˜ ν cj þ H : c : (cid:5) − m ˜ ν ;ij ð ˜ ν ci Þ (cid:2) ˜ ν cj − m ˜ η j ˜ η j − m ˜ η j ˜ η j ; ð Þ where λ B , λ B represent the gauginos of U ð Þ Y and U ð Þ B − L ,respectively. The local gauge symmetry SU ð Þ C ⊗ SU ð Þ L ⊗ U ð Þ Y ⊗ U ð Þ B − L is broken down to the electromagneticsymmetry U ð Þ em when the Higgs fields acquires the nonzerovacuum expectation values (VEVs): H ¼ ffiffiffi p ð v þ ReH þ i Im H Þ ;H ¼ ffiffiffi p ð v þ ReH þ i Im H Þ ; ˜ η ¼ ffiffiffi p ð u þ Re ˜ η þ i Im ˜ η Þ ; ˜ η ¼ ffiffiffi p ð u þ Re ˜ η þ i Im ˜ η Þ : ð Þ Similar to the ratio of nonzero VEVs of H and H , we taketan β ¼ u u denoting the ratio of nonzero VEVs of two chiralsinglet superfields ˜ η and ˜ η here.There is the gauge kinetic mixing − κ Y;BL A Y μ A μ ;BL fromtwo local U ð Þ gauge groups, and the mixing term satisfiesthe gauge invariance, where A Y μ , A μ ;BL represent the gaugefields of two gauge groups U ð Þ Y and U ð Þ B − L , respec-tively, and the antisymmetric tensor − κ Y;BL representsthe mixing between two U ð Þ gauge fields. The choice κ Y;BL ¼ is unnatural because the mixing at low energyscale still can acquire a nonzero value through the evolutionof renormalization group equations (RGEs) [37 – κ Y;BL ¼ at the great uniform theory scale.The soft breaking parameters T ð T ij ν ; T ijx ; T d; ; T u; Þ areproportional to the corresponding Yukawa couplings, i.e., T ij ν ¼ Y ν ;ij A ν , T ijx ¼ Y x;ij A x , T d; ¼ Y b A b and T u; ¼ Y t A t (the trilinear scalar terms in the soft supersymmetrybreaking potential).Because of the reasons above, the covariant derivative isusually written as D μ ¼ ∂ μ − i ð Y; B − L Þ (cid:1) g Y g YB g BY g B − L (cid:3) RR T (cid:1) A Y μ A BL μ (cid:3) ; ð Þ where Y , B − L correspond to the hypercharge and B − L charge, respectively. Furthermore, R denotes a × CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D rthogonal matrix. We can choose a proper R and thenrewrite the coupling matrix as (cid:1) g Y g YB g BY g B − L (cid:3) R ¼ (cid:1) g g YB g B (cid:3) ; ð Þ where g is the hypercharge coupling constant of the SM,which can be modified in the B-LSSM. Meanwhile, twoU(1) gauge fields are redefined as (cid:1) A Y μ A BL μ (cid:3) ¼ R T (cid:1) A Y μ A BL μ (cid:3) : ð Þ The gauge kinetic mixing induces some interestingphenomenology. First, A BL μ boson can mix with the A Y μ and V μ bosons at the tree level. In the interaction basis ( A Y μ , V μ , A BL μ ), the mass squared matrix of neutral gauge bosonsis written as g v − g g v g g YB v − g g v g v − g g YB v g g YB v − g g YB v g YB v þ g B u : ð Þ This mass squared matrix can be diagonalized by a unitarymatrix, and the mass eigenstates can be written as linearcombinations of ( A Y μ , V μ , A BL μ ): γ μ Z μ Z μ ¼ cos θ W sin θ W − sin θ W cos θ W cos θ W cos θ W sin θ W sin θ W sin θ W − cos θ W sin θ W cos θ W × A Y μ V μ A BL μ : ð Þ Here, θ W , θ W represent two mixing angles [44]:sin θ W ¼ g g þ g ; ð Þ sin θ W ¼ − g B ½ð g YB − g − g Þ x þ g B (cid:3)½ g YB ð g þ g Þ (cid:3) x þ g B ð g YB − g − g Þ x þ g B ; ð Þ with x ¼ vu , v ¼ v þ v and u ¼ u þ u . When x ≪ ,the eigenvalues of Eq. (11) can be written as [45] m γ ¼ ;m Z ≃ ð g þ g Þ v − g B ð g þ g þ g YB Þ x v ;m Z ≃ (cid:4) g YB v þ g B u þ g B ð g þ g þ g YB Þ x v (cid:5) : ð Þ The effective potential can be written as [46]: V ¼ V þ Δ V ;t þ Δ V ; ˜ t þ Δ V ; ν þ Δ V ; ˜ ν þ Δ V ;b þ Δ V ; ˜ b : ð Þ Here, V denotes the scalar potential at tree level, Δ V ;t represents the correction from top quark, and Δ V ; ˜ t represents the corrections from scalar top quarks, Δ V ; ν denotes the corrections from neutrinos, and Δ V ; ˜ ν denotesthe corrections from sneutrinos Δ V ;b represents thecorrection from bottom quark, and Δ V ; ˜ b represents thecorrections from scalar bottom quarks, respectively.The concrete expressions of those pieces are V ¼ (cid:4) g ðj H j − j H j Þ þ g B ðj ˜ η j − j ˜ η j Þþ g B g YB ðj H j − j H j Þðj ˜ η j − j ˜ η j Þ (cid:5) þ ½j μ j ðj H j þ j H j Þ þ j μ j ðj ˜ η j þ j ˜ η j Þþ m H j H j þ m H j H j þ m ˜ η j ˜ η j þ m ˜ η j ˜ η j þ ð − B μ ˜ η ˜ η − B μ H H þ H : c : Þ(cid:3) ; ð Þ Δ V ;t ¼ − π m t (cid:4) ln (cid:1) m t Q (cid:3) − (cid:5) ; Δ V ; ˜ t ¼ π X i ¼ m ˜ t i (cid:4) ln (cid:1) m ˜ t i Q (cid:3) − (cid:5) ; Δ V ; ν ¼ − π X i ¼ m ν iR (cid:4) ln (cid:1) m ν iR Q (cid:3) − (cid:5) ; Δ V ; ˜ ν ¼ π X i ¼ m ˜ ν iR (cid:4) ln (cid:1) m ˜ ν iR Q (cid:3) − (cid:5) ; Δ V ;b ¼ − π m b (cid:4) ln (cid:1) m b Q (cid:3) − (cid:5) ; Δ V ; ˜ b ¼ π X i ¼ m ˜ b i (cid:4) ln (cid:1) m ˜ b i Q (cid:3) − (cid:5) ; ð Þ where Q denotes the renormalization scale; m t , m b re-present the masses of top and bottom quark, respectively,;and m ˜ t ; , m ˜ b ; denote the masses of scalar top and scalarbottom quarks, respectively. In addition, m ν iR represents the e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020)
In this section, we briefly introduce the basic character-istics of the B-LSSM. About the B-LSSM, there are severaldifferent versions. Here, we apply the version describedin Refs. [27 – – SU ð Þ C ⊗ SU ð Þ L ⊗ U ð Þ Y ⊗ U ð Þ B − L , where the U ð Þ B − L is an additional gaugesymmetry.With the local gauge group SU ð Þ C ⊗ SU ð Þ L ⊗ U ð Þ Y ⊗ U ð Þ B − L , the superpotential of the B-LSSM iswritten as W ¼ W MSSM þ W ð B − L Þ : ð Þ Where W MSSM denotes the superpotential of the MSSM[36], W ð B − L Þ denotes the additional terms in this model,and can be written as W ð B − L Þ ¼ Y ν ;ij ˆ L i ˆ H ˆ ν cj þ μ ˆ η ˆ η þ Y x;ij ˆ ν ci ˆ η ˆ ν cj : ð Þ The quantum numbers of the superfields of the quarks andleptons are assigned as ˆ Q i ¼ (cid:1) ˆ u i ˆ d i (cid:3) ∼ ð ; ; = ; = Þ ; ˆ L i ¼ (cid:1) ˆ ν i ˆ e i (cid:3) ∼ ð ; ; − = ; − = Þ ; ˆ U ci ∼ ð ; ; − = ; − = Þ ; ˆ D ci ∼ ð ; ; = ; − = Þ ; ˆ E ci ∼ ð ; ; ; = Þ ; ð Þ with i ¼ , 2, 3 denoting the generation indices. Inaddition, the quantum numbers of two Higgs doublets are ˆ H ¼ (cid:1) ˆ H ˆ H (cid:3) ∼ ð ; ; − = ; Þ ; ˆ H ¼ (cid:1) ˆ H ˆ H (cid:3) ∼ ð ; ; = ; Þ : ð Þ The quantum numbers of chiral singlet superfields are ˆ η ∼ ð ; ; ; − Þ , ˆ η ∼ ð ; ; ; Þ , and that of three gen-erations of right-handed neutrinos is ˆ ν ci ∼ ð ; ; ; = Þ .Correspondingly, the soft breaking terms in the B-LSSMare written as L soft ¼ L MSSMsoft þ L B − L soft ; ð Þ where the L MSSMsoft denotes the soft breaking terms in theMSSM [36], and L B − L soft ¼ (cid:4) − M BB ˜ λ B ˜ λ B − M B ˜ λ B ˜ λ B − B μ ˜ η ˜ η þ T ij ν H ˜ ν ci ˜ L j þ T ijx ˜ η ˜ ν ci ˜ ν cj þ H : c : (cid:5) − m ˜ ν ;ij ð ˜ ν ci Þ (cid:2) ˜ ν cj − m ˜ η j ˜ η j − m ˜ η j ˜ η j ; ð Þ where λ B , λ B represent the gauginos of U ð Þ Y and U ð Þ B − L ,respectively. The local gauge symmetry SU ð Þ C ⊗ SU ð Þ L ⊗ U ð Þ Y ⊗ U ð Þ B − L is broken down to the electromagneticsymmetry U ð Þ em when the Higgs fields acquires the nonzerovacuum expectation values (VEVs): H ¼ ffiffiffi p ð v þ ReH þ i Im H Þ ;H ¼ ffiffiffi p ð v þ ReH þ i Im H Þ ; ˜ η ¼ ffiffiffi p ð u þ Re ˜ η þ i Im ˜ η Þ ; ˜ η ¼ ffiffiffi p ð u þ Re ˜ η þ i Im ˜ η Þ : ð Þ Similar to the ratio of nonzero VEVs of H and H , we taketan β ¼ u u denoting the ratio of nonzero VEVs of two chiralsinglet superfields ˜ η and ˜ η here.There is the gauge kinetic mixing − κ Y;BL A Y μ A μ ;BL fromtwo local U ð Þ gauge groups, and the mixing term satisfiesthe gauge invariance, where A Y μ , A μ ;BL represent the gaugefields of two gauge groups U ð Þ Y and U ð Þ B − L , respec-tively, and the antisymmetric tensor − κ Y;BL representsthe mixing between two U ð Þ gauge fields. The choice κ Y;BL ¼ is unnatural because the mixing at low energyscale still can acquire a nonzero value through the evolutionof renormalization group equations (RGEs) [37 – κ Y;BL ¼ at the great uniform theory scale.The soft breaking parameters T ð T ij ν ; T ijx ; T d; ; T u; Þ areproportional to the corresponding Yukawa couplings, i.e., T ij ν ¼ Y ν ;ij A ν , T ijx ¼ Y x;ij A x , T d; ¼ Y b A b and T u; ¼ Y t A t (the trilinear scalar terms in the soft supersymmetrybreaking potential).Because of the reasons above, the covariant derivative isusually written as D μ ¼ ∂ μ − i ð Y; B − L Þ (cid:1) g Y g YB g BY g B − L (cid:3) RR T (cid:1) A Y μ A BL μ (cid:3) ; ð Þ where Y , B − L correspond to the hypercharge and B − L charge, respectively. Furthermore, R denotes a × CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D rthogonal matrix. We can choose a proper R and thenrewrite the coupling matrix as (cid:1) g Y g YB g BY g B − L (cid:3) R ¼ (cid:1) g g YB g B (cid:3) ; ð Þ where g is the hypercharge coupling constant of the SM,which can be modified in the B-LSSM. Meanwhile, twoU(1) gauge fields are redefined as (cid:1) A Y μ A BL μ (cid:3) ¼ R T (cid:1) A Y μ A BL μ (cid:3) : ð Þ The gauge kinetic mixing induces some interestingphenomenology. First, A BL μ boson can mix with the A Y μ and V μ bosons at the tree level. In the interaction basis ( A Y μ , V μ , A BL μ ), the mass squared matrix of neutral gauge bosonsis written as g v − g g v g g YB v − g g v g v − g g YB v g g YB v − g g YB v g YB v þ g B u : ð Þ This mass squared matrix can be diagonalized by a unitarymatrix, and the mass eigenstates can be written as linearcombinations of ( A Y μ , V μ , A BL μ ): γ μ Z μ Z μ ¼ cos θ W sin θ W − sin θ W cos θ W cos θ W cos θ W sin θ W sin θ W sin θ W − cos θ W sin θ W cos θ W × A Y μ V μ A BL μ : ð Þ Here, θ W , θ W represent two mixing angles [44]:sin θ W ¼ g g þ g ; ð Þ sin θ W ¼ − g B ½ð g YB − g − g Þ x þ g B (cid:3)½ g YB ð g þ g Þ (cid:3) x þ g B ð g YB − g − g Þ x þ g B ; ð Þ with x ¼ vu , v ¼ v þ v and u ¼ u þ u . When x ≪ ,the eigenvalues of Eq. (11) can be written as [45] m γ ¼ ;m Z ≃ ð g þ g Þ v − g B ð g þ g þ g YB Þ x v ;m Z ≃ (cid:4) g YB v þ g B u þ g B ð g þ g þ g YB Þ x v (cid:5) : ð Þ The effective potential can be written as [46]: V ¼ V þ Δ V ;t þ Δ V ; ˜ t þ Δ V ; ν þ Δ V ; ˜ ν þ Δ V ;b þ Δ V ; ˜ b : ð Þ Here, V denotes the scalar potential at tree level, Δ V ;t represents the correction from top quark, and Δ V ; ˜ t represents the corrections from scalar top quarks, Δ V ; ν denotes the corrections from neutrinos, and Δ V ; ˜ ν denotesthe corrections from sneutrinos Δ V ;b represents thecorrection from bottom quark, and Δ V ; ˜ b represents thecorrections from scalar bottom quarks, respectively.The concrete expressions of those pieces are V ¼ (cid:4) g ðj H j − j H j Þ þ g B ðj ˜ η j − j ˜ η j Þþ g B g YB ðj H j − j H j Þðj ˜ η j − j ˜ η j Þ (cid:5) þ ½j μ j ðj H j þ j H j Þ þ j μ j ðj ˜ η j þ j ˜ η j Þþ m H j H j þ m H j H j þ m ˜ η j ˜ η j þ m ˜ η j ˜ η j þ ð − B μ ˜ η ˜ η − B μ H H þ H : c : Þ(cid:3) ; ð Þ Δ V ;t ¼ − π m t (cid:4) ln (cid:1) m t Q (cid:3) − (cid:5) ; Δ V ; ˜ t ¼ π X i ¼ m ˜ t i (cid:4) ln (cid:1) m ˜ t i Q (cid:3) − (cid:5) ; Δ V ; ν ¼ − π X i ¼ m ν iR (cid:4) ln (cid:1) m ν iR Q (cid:3) − (cid:5) ; Δ V ; ˜ ν ¼ π X i ¼ m ˜ ν iR (cid:4) ln (cid:1) m ˜ ν iR Q (cid:3) − (cid:5) ; Δ V ;b ¼ − π m b (cid:4) ln (cid:1) m b Q (cid:3) − (cid:5) ; Δ V ; ˜ b ¼ π X i ¼ m ˜ b i (cid:4) ln (cid:1) m ˜ b i Q (cid:3) − (cid:5) ; ð Þ where Q denotes the renormalization scale; m t , m b re-present the masses of top and bottom quark, respectively,;and m ˜ t ; , m ˜ b ; denote the masses of scalar top and scalarbottom quarks, respectively. In addition, m ν iR represents the e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020) asses of right-handed neutrinos, and m ˜ ν iR represents themasses of right-handed sneutrinos.The stability conditions are ∂ V ∂ ReH ¼ ; ∂ V ∂ ReH ¼ ; ∂ V ∂ Re ˜ η ¼ ; ∂ V ∂ Re ˜ η ¼ ð Þ the detailed expression about Eq. (19) is given inAppendix A. Furthermore, the gauge kinetic mixinginduces the mixing among the H , H , ˜ η , ˜ η at treelevel. In the interaction basis (Re H , Re H , Re ˜ η , Re ˜ η ),the mass squared matrix for CP -even Higgs bosons iswritten as M h ¼ Δ m h þ g v c β þ ReB μ t β − g v s β c β − ReB μ g B g YB vuc β c β − g B g YB vuc β s β − g v s β c β − ReB μ g v s β þ ReB μ ctg β g B g YB vuc β s β g B g YB vus β s β g B g YB vuc β c β g B g YB vuc β s β g B u c β þ ReB μ t β − g B u s β c β − ReB μ − g B g YB vuc β s β g B g YB vus β s β − g B u s β c β − ReB μ g B u s β þ ReB μ ctg β ; ð Þ where the abbreviations are c β ¼ cos β , s β ¼ sin β , t β ¼ tan β , c β ¼ cos β , s β ¼ sin β , and Δ m h ¼ Δ m Φ d Φ d Δ m Φ u Φ d Δ m Φ η Φ d Δ m Φ ¯ η Φ d Δ m Φ d Φ u Δ m Φ u Φ u Δ m Φ η Φ u Δ m Φ ¯ η Φ u Δ m Φ d Φ η Δ m Φ u Φ η Δ m Φ η Φ η Δ m Φ ¯ η Φ η Δ m Φ d Φ ¯ η Δ m Φ u Φ ¯ η Δ m Φ η Φ ¯ η Δ m Φ ¯ η Φ ¯ η : ð Þ Here Δ m h represents the one-loop correction to massmatrix squared; the detailed expression about this is givenin Appendix A, and it can be obtained by the secondderivative of the effective potential. In addition, g ¼ g þ g þ g YB . The mass matrix M h can be diagonalized by the × unitary matrix Z H .In the interaction basis (ImH ; ImH ; Im ˜ η ; Im ˜ η ), themass squared matrix of CP -odd Higgs can be written as: m A ¼ t β B μ B μ B μ t β B μ t β B μ B μ B μ t β B μ þ Δ m A ; ð Þ here, Δ m A represents the one-loop correction to masssquared matrix of CP -odd Higgs, and Δ m A ¼ − Δ m h . Themass matrix m A can be diagonalized by the × unitarymatrix Z A . The eigenvalues of Eq. (22) can be written as m η G ; ¼ ;m A ; ¼ ð m ð Þ A ; Þ þ θ ð Δ m Þ ; ð Þ where m ð Þ A ; is the contribution under tree level approxi-mation, and θ ð Δ m Þ ¼ ½ t (cid:4) ð t − ð Δ m ϕ d ϕ d Δ m ϕ u ϕ u þ Δ m ϕ d ϕ d Δ m ϕ η ϕ η þ Δ m ϕ u ϕ u Δ m ϕ η ϕ η þ ð Δ m ϕ d ϕ d þ Δ m ϕ u ϕ u þ Δ m ϕ η ϕ η Þ Δ m ϕ ¯ η ϕ ¯ η þ Δ m ϕ η ϕ d þ Δ m ϕ ¯ η ϕ u ÞÞ (cid:3) ; ð Þ with t ¼ Δ m ϕ d ϕ d þ Δ m ϕ u ϕ u þ Δ m ϕ η ϕ η þ Δ m ϕ ¯ η ϕ ¯ η ; the detailed expression about θ ð Δ m Þ is given in Appendix A. Theeigenstates corresponding to Goldstone are A uj ¼ c β ImH − s β ImH ;A η j ¼ c β Im ˜ η − s β Im ˜ η : ð Þ The Higgs self-coupling can be defined asCUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D h i h j h k ¼ ∂ V ∂ ð ReH Þ Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ ∂ ReH Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ ∂ ReH Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ ∂ ReH Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ ∂ ReH Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ ∂ ReH Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ ∂ ReH Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ReH ∂ ReH ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ReH ∂ ReH ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ReH ∂ Re ˜ η ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ReH ∂ Re ˜ η ∂ Re ˜ η Z H i Z H j Z H k ; ð Þ where Y t ¼ ffiffi p m t v , Y b ¼ ffiffi p m b v , A t is the trilinear couplingsbetween Higgs and scalar top quarks, and μ denotes the massparameter of Higgsino. In addition, the detailed expressionabout tree level correction λ ð Þ h i h j h k is given in Appendix B.The issues we discuss also involve the coupling of two CP -odd Higgs and one CP -even Higgs. The correspondingexpression can be found in Appendix C. III. CROSS SECTION OF THE HIGGS BOSON PAIRPRODUCTION THROUGH e + e − → hhZ In this section, we will introduce the production ofthe Higgs boson pair through e þ e − → hhZ . The channel ofthe production of the Higgs boson pair is open when thecollision energy E cm of the initial state particle is more thanabout 340 GeV. The decay and production of Higgs bosonhave been discussed extensively [47 – e þ e − → hhZ here. We will carefullyanalyze the influence of relevant parameters on the totalreaction cross section and angular distribution of thedifferential cross section in this model.The Feynman diagrams contributing to this process aregiven in Figs. 1 and 2, where N denotes Z and Z bosons, A represents CP -odd Higgs fields. The diagrams in Fig. 1originate from the SM sector and the new physics sector,while those of Fig. 2 originate from the new physics sector,respectively.In our calculation, we choose collision energy E cm ¼ GeV, so we ignore the masses of positron and electron.In addition, we neglect the Feynman diagrams generated byYukawa couplings of electron.In the B-LSSM, additional Feynman diagrams thatcontribute to this process are already given in Fig. 2.Taking Fig. 2( d ) as an example, one derives the effectiveoperator from the diagram ( d ) as (a) (b) (c) FIG. 1. Feynman diagrams for the Higgs boson pairs production through the process of e þ e − → hhZ in the SM and in the B-LSSM. (a ´ ´ ´ ´ ) (b ) (c ) (d ) FIG. 2. Additional Feynman diagrams for the Higgs boson pairs production through the process of e þ e − → hhZ in the B-LSSM. e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020)
In this section, we briefly introduce the basic character-istics of the B-LSSM. About the B-LSSM, there are severaldifferent versions. Here, we apply the version describedin Refs. [27 – – SU ð Þ C ⊗ SU ð Þ L ⊗ U ð Þ Y ⊗ U ð Þ B − L , where the U ð Þ B − L is an additional gaugesymmetry.With the local gauge group SU ð Þ C ⊗ SU ð Þ L ⊗ U ð Þ Y ⊗ U ð Þ B − L , the superpotential of the B-LSSM iswritten as W ¼ W MSSM þ W ð B − L Þ : ð Þ Where W MSSM denotes the superpotential of the MSSM[36], W ð B − L Þ denotes the additional terms in this model,and can be written as W ð B − L Þ ¼ Y ν ;ij ˆ L i ˆ H ˆ ν cj þ μ ˆ η ˆ η þ Y x;ij ˆ ν ci ˆ η ˆ ν cj : ð Þ The quantum numbers of the superfields of the quarks andleptons are assigned as ˆ Q i ¼ (cid:1) ˆ u i ˆ d i (cid:3) ∼ ð ; ; = ; = Þ ; ˆ L i ¼ (cid:1) ˆ ν i ˆ e i (cid:3) ∼ ð ; ; − = ; − = Þ ; ˆ U ci ∼ ð ; ; − = ; − = Þ ; ˆ D ci ∼ ð ; ; = ; − = Þ ; ˆ E ci ∼ ð ; ; ; = Þ ; ð Þ with i ¼ , 2, 3 denoting the generation indices. Inaddition, the quantum numbers of two Higgs doublets are ˆ H ¼ (cid:1) ˆ H ˆ H (cid:3) ∼ ð ; ; − = ; Þ ; ˆ H ¼ (cid:1) ˆ H ˆ H (cid:3) ∼ ð ; ; = ; Þ : ð Þ The quantum numbers of chiral singlet superfields are ˆ η ∼ ð ; ; ; − Þ , ˆ η ∼ ð ; ; ; Þ , and that of three gen-erations of right-handed neutrinos is ˆ ν ci ∼ ð ; ; ; = Þ .Correspondingly, the soft breaking terms in the B-LSSMare written as L soft ¼ L MSSMsoft þ L B − L soft ; ð Þ where the L MSSMsoft denotes the soft breaking terms in theMSSM [36], and L B − L soft ¼ (cid:4) − M BB ˜ λ B ˜ λ B − M B ˜ λ B ˜ λ B − B μ ˜ η ˜ η þ T ij ν H ˜ ν ci ˜ L j þ T ijx ˜ η ˜ ν ci ˜ ν cj þ H : c : (cid:5) − m ˜ ν ;ij ð ˜ ν ci Þ (cid:2) ˜ ν cj − m ˜ η j ˜ η j − m ˜ η j ˜ η j ; ð Þ where λ B , λ B represent the gauginos of U ð Þ Y and U ð Þ B − L ,respectively. The local gauge symmetry SU ð Þ C ⊗ SU ð Þ L ⊗ U ð Þ Y ⊗ U ð Þ B − L is broken down to the electromagneticsymmetry U ð Þ em when the Higgs fields acquires the nonzerovacuum expectation values (VEVs): H ¼ ffiffiffi p ð v þ ReH þ i Im H Þ ;H ¼ ffiffiffi p ð v þ ReH þ i Im H Þ ; ˜ η ¼ ffiffiffi p ð u þ Re ˜ η þ i Im ˜ η Þ ; ˜ η ¼ ffiffiffi p ð u þ Re ˜ η þ i Im ˜ η Þ : ð Þ Similar to the ratio of nonzero VEVs of H and H , we taketan β ¼ u u denoting the ratio of nonzero VEVs of two chiralsinglet superfields ˜ η and ˜ η here.There is the gauge kinetic mixing − κ Y;BL A Y μ A μ ;BL fromtwo local U ð Þ gauge groups, and the mixing term satisfiesthe gauge invariance, where A Y μ , A μ ;BL represent the gaugefields of two gauge groups U ð Þ Y and U ð Þ B − L , respec-tively, and the antisymmetric tensor − κ Y;BL representsthe mixing between two U ð Þ gauge fields. The choice κ Y;BL ¼ is unnatural because the mixing at low energyscale still can acquire a nonzero value through the evolutionof renormalization group equations (RGEs) [37 – κ Y;BL ¼ at the great uniform theory scale.The soft breaking parameters T ð T ij ν ; T ijx ; T d; ; T u; Þ areproportional to the corresponding Yukawa couplings, i.e., T ij ν ¼ Y ν ;ij A ν , T ijx ¼ Y x;ij A x , T d; ¼ Y b A b and T u; ¼ Y t A t (the trilinear scalar terms in the soft supersymmetrybreaking potential).Because of the reasons above, the covariant derivative isusually written as D μ ¼ ∂ μ − i ð Y; B − L Þ (cid:1) g Y g YB g BY g B − L (cid:3) RR T (cid:1) A Y μ A BL μ (cid:3) ; ð Þ where Y , B − L correspond to the hypercharge and B − L charge, respectively. Furthermore, R denotes a × CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D rthogonal matrix. We can choose a proper R and thenrewrite the coupling matrix as (cid:1) g Y g YB g BY g B − L (cid:3) R ¼ (cid:1) g g YB g B (cid:3) ; ð Þ where g is the hypercharge coupling constant of the SM,which can be modified in the B-LSSM. Meanwhile, twoU(1) gauge fields are redefined as (cid:1) A Y μ A BL μ (cid:3) ¼ R T (cid:1) A Y μ A BL μ (cid:3) : ð Þ The gauge kinetic mixing induces some interestingphenomenology. First, A BL μ boson can mix with the A Y μ and V μ bosons at the tree level. In the interaction basis ( A Y μ , V μ , A BL μ ), the mass squared matrix of neutral gauge bosonsis written as g v − g g v g g YB v − g g v g v − g g YB v g g YB v − g g YB v g YB v þ g B u : ð Þ This mass squared matrix can be diagonalized by a unitarymatrix, and the mass eigenstates can be written as linearcombinations of ( A Y μ , V μ , A BL μ ): γ μ Z μ Z μ ¼ cos θ W sin θ W − sin θ W cos θ W cos θ W cos θ W sin θ W sin θ W sin θ W − cos θ W sin θ W cos θ W × A Y μ V μ A BL μ : ð Þ Here, θ W , θ W represent two mixing angles [44]:sin θ W ¼ g g þ g ; ð Þ sin θ W ¼ − g B ½ð g YB − g − g Þ x þ g B (cid:3)½ g YB ð g þ g Þ (cid:3) x þ g B ð g YB − g − g Þ x þ g B ; ð Þ with x ¼ vu , v ¼ v þ v and u ¼ u þ u . When x ≪ ,the eigenvalues of Eq. (11) can be written as [45] m γ ¼ ;m Z ≃ ð g þ g Þ v − g B ð g þ g þ g YB Þ x v ;m Z ≃ (cid:4) g YB v þ g B u þ g B ð g þ g þ g YB Þ x v (cid:5) : ð Þ The effective potential can be written as [46]: V ¼ V þ Δ V ;t þ Δ V ; ˜ t þ Δ V ; ν þ Δ V ; ˜ ν þ Δ V ;b þ Δ V ; ˜ b : ð Þ Here, V denotes the scalar potential at tree level, Δ V ;t represents the correction from top quark, and Δ V ; ˜ t represents the corrections from scalar top quarks, Δ V ; ν denotes the corrections from neutrinos, and Δ V ; ˜ ν denotesthe corrections from sneutrinos Δ V ;b represents thecorrection from bottom quark, and Δ V ; ˜ b represents thecorrections from scalar bottom quarks, respectively.The concrete expressions of those pieces are V ¼ (cid:4) g ðj H j − j H j Þ þ g B ðj ˜ η j − j ˜ η j Þþ g B g YB ðj H j − j H j Þðj ˜ η j − j ˜ η j Þ (cid:5) þ ½j μ j ðj H j þ j H j Þ þ j μ j ðj ˜ η j þ j ˜ η j Þþ m H j H j þ m H j H j þ m ˜ η j ˜ η j þ m ˜ η j ˜ η j þ ð − B μ ˜ η ˜ η − B μ H H þ H : c : Þ(cid:3) ; ð Þ Δ V ;t ¼ − π m t (cid:4) ln (cid:1) m t Q (cid:3) − (cid:5) ; Δ V ; ˜ t ¼ π X i ¼ m ˜ t i (cid:4) ln (cid:1) m ˜ t i Q (cid:3) − (cid:5) ; Δ V ; ν ¼ − π X i ¼ m ν iR (cid:4) ln (cid:1) m ν iR Q (cid:3) − (cid:5) ; Δ V ; ˜ ν ¼ π X i ¼ m ˜ ν iR (cid:4) ln (cid:1) m ˜ ν iR Q (cid:3) − (cid:5) ; Δ V ;b ¼ − π m b (cid:4) ln (cid:1) m b Q (cid:3) − (cid:5) ; Δ V ; ˜ b ¼ π X i ¼ m ˜ b i (cid:4) ln (cid:1) m ˜ b i Q (cid:3) − (cid:5) ; ð Þ where Q denotes the renormalization scale; m t , m b re-present the masses of top and bottom quark, respectively,;and m ˜ t ; , m ˜ b ; denote the masses of scalar top and scalarbottom quarks, respectively. In addition, m ν iR represents the e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020) asses of right-handed neutrinos, and m ˜ ν iR represents themasses of right-handed sneutrinos.The stability conditions are ∂ V ∂ ReH ¼ ; ∂ V ∂ ReH ¼ ; ∂ V ∂ Re ˜ η ¼ ; ∂ V ∂ Re ˜ η ¼ ð Þ the detailed expression about Eq. (19) is given inAppendix A. Furthermore, the gauge kinetic mixinginduces the mixing among the H , H , ˜ η , ˜ η at treelevel. In the interaction basis (Re H , Re H , Re ˜ η , Re ˜ η ),the mass squared matrix for CP -even Higgs bosons iswritten as M h ¼ Δ m h þ g v c β þ ReB μ t β − g v s β c β − ReB μ g B g YB vuc β c β − g B g YB vuc β s β − g v s β c β − ReB μ g v s β þ ReB μ ctg β g B g YB vuc β s β g B g YB vus β s β g B g YB vuc β c β g B g YB vuc β s β g B u c β þ ReB μ t β − g B u s β c β − ReB μ − g B g YB vuc β s β g B g YB vus β s β − g B u s β c β − ReB μ g B u s β þ ReB μ ctg β ; ð Þ where the abbreviations are c β ¼ cos β , s β ¼ sin β , t β ¼ tan β , c β ¼ cos β , s β ¼ sin β , and Δ m h ¼ Δ m Φ d Φ d Δ m Φ u Φ d Δ m Φ η Φ d Δ m Φ ¯ η Φ d Δ m Φ d Φ u Δ m Φ u Φ u Δ m Φ η Φ u Δ m Φ ¯ η Φ u Δ m Φ d Φ η Δ m Φ u Φ η Δ m Φ η Φ η Δ m Φ ¯ η Φ η Δ m Φ d Φ ¯ η Δ m Φ u Φ ¯ η Δ m Φ η Φ ¯ η Δ m Φ ¯ η Φ ¯ η : ð Þ Here Δ m h represents the one-loop correction to massmatrix squared; the detailed expression about this is givenin Appendix A, and it can be obtained by the secondderivative of the effective potential. In addition, g ¼ g þ g þ g YB . The mass matrix M h can be diagonalized by the × unitary matrix Z H .In the interaction basis (ImH ; ImH ; Im ˜ η ; Im ˜ η ), themass squared matrix of CP -odd Higgs can be written as: m A ¼ t β B μ B μ B μ t β B μ t β B μ B μ B μ t β B μ þ Δ m A ; ð Þ here, Δ m A represents the one-loop correction to masssquared matrix of CP -odd Higgs, and Δ m A ¼ − Δ m h . Themass matrix m A can be diagonalized by the × unitarymatrix Z A . The eigenvalues of Eq. (22) can be written as m η G ; ¼ ;m A ; ¼ ð m ð Þ A ; Þ þ θ ð Δ m Þ ; ð Þ where m ð Þ A ; is the contribution under tree level approxi-mation, and θ ð Δ m Þ ¼ ½ t (cid:4) ð t − ð Δ m ϕ d ϕ d Δ m ϕ u ϕ u þ Δ m ϕ d ϕ d Δ m ϕ η ϕ η þ Δ m ϕ u ϕ u Δ m ϕ η ϕ η þ ð Δ m ϕ d ϕ d þ Δ m ϕ u ϕ u þ Δ m ϕ η ϕ η Þ Δ m ϕ ¯ η ϕ ¯ η þ Δ m ϕ η ϕ d þ Δ m ϕ ¯ η ϕ u ÞÞ (cid:3) ; ð Þ with t ¼ Δ m ϕ d ϕ d þ Δ m ϕ u ϕ u þ Δ m ϕ η ϕ η þ Δ m ϕ ¯ η ϕ ¯ η ; the detailed expression about θ ð Δ m Þ is given in Appendix A. Theeigenstates corresponding to Goldstone are A uj ¼ c β ImH − s β ImH ;A η j ¼ c β Im ˜ η − s β Im ˜ η : ð Þ The Higgs self-coupling can be defined asCUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D h i h j h k ¼ ∂ V ∂ ð ReH Þ Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ ∂ ReH Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ ∂ ReH Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ð ReH Þ ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ ∂ ReH Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ ∂ ReH Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ ∂ ReH Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ ∂ ReH Z H i Z H j Z H k þ ∂ V ∂ ð Re ˜ η Þ ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ReH ∂ ReH ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ReH ∂ ReH ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ReH ∂ Re ˜ η ∂ Re ˜ η Z H i Z H j Z H k þ ∂ V ∂ ReH ∂ Re ˜ η ∂ Re ˜ η Z H i Z H j Z H k ; ð Þ where Y t ¼ ffiffi p m t v , Y b ¼ ffiffi p m b v , A t is the trilinear couplingsbetween Higgs and scalar top quarks, and μ denotes the massparameter of Higgsino. In addition, the detailed expressionabout tree level correction λ ð Þ h i h j h k is given in Appendix B.The issues we discuss also involve the coupling of two CP -odd Higgs and one CP -even Higgs. The correspondingexpression can be found in Appendix C. III. CROSS SECTION OF THE HIGGS BOSON PAIRPRODUCTION THROUGH e + e − → hhZ In this section, we will introduce the production ofthe Higgs boson pair through e þ e − → hhZ . The channel ofthe production of the Higgs boson pair is open when thecollision energy E cm of the initial state particle is more thanabout 340 GeV. The decay and production of Higgs bosonhave been discussed extensively [47 – e þ e − → hhZ here. We will carefullyanalyze the influence of relevant parameters on the totalreaction cross section and angular distribution of thedifferential cross section in this model.The Feynman diagrams contributing to this process aregiven in Figs. 1 and 2, where N denotes Z and Z bosons, A represents CP -odd Higgs fields. The diagrams in Fig. 1originate from the SM sector and the new physics sector,while those of Fig. 2 originate from the new physics sector,respectively.In our calculation, we choose collision energy E cm ¼ GeV, so we ignore the masses of positron and electron.In addition, we neglect the Feynman diagrams generated byYukawa couplings of electron.In the B-LSSM, additional Feynman diagrams thatcontribute to this process are already given in Fig. 2.Taking Fig. 2( d ) as an example, one derives the effectiveoperator from the diagram ( d ) as (a) (b) (c) FIG. 1. Feynman diagrams for the Higgs boson pairs production through the process of e þ e − → hhZ in the SM and in the B-LSSM. (a ´ ´ ´ ´ ) (b ) (c ) (d ) FIG. 2. Additional Feynman diagrams for the Higgs boson pairs production through the process of e þ e − → hhZ in the B-LSSM. e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020) L;R ¼ ¯ v ð p Þ q α p L;R u ð p Þ ε (cid:2) α ð k Þ ; ð Þ with p L;R ¼ ∓ γ , where the p , p , q represent themomenta of the initial state particles and final stateHiggs boson, k denotes the momentum of the final statevector boson, respectively. The corresponding effectiveamplitude can be written as: M ð d Þ ¼ C d L O L þ C d R O R : ð Þ The differential cross section of this process can bewritten as d σ d Ω ¼ π E cm Z π d φ Z E max E min dE Z E max E min dE j M j : ð Þ Here, M denotes the amplitude of all of these diagramsdrawn in Figs. 1 and 2, and it is written as: M ¼ C ð Þ L;R O L;R þ C ð Þ L;R O L;R ; ð Þ the Wilson coefficients of those operators are given inAppendix D. In addition, φ denotes the angle between theprojection of the momentum direction of the final state Higgson the x-y plane and the x axis, Ω is the spatial solid anglebetween the initial state electron and the final state Higgs, and d Ω ¼ sin θ d θ d φ , respectively. We take the momentumdirection of the final state Higgs as z axis, and the momentumdirection of the initial state electron on the x-z plane. Here θ stands for the angle between initial state electron and finalstate Higgs, φ is the angle between the projection of themomentum direction of the initial state electron on the x-yplane and the x axis, as shown in Fig. 3. Furthermore, E , E both are the energy of the final state particles Higgs, where E min ¼ m h ;E max ¼ E cm − m h m z − m z E cm ;E min ¼ − ð E − E cm Þð E E cm − E cm − m h þ m z Þ E E cm − ð E cm þ m h Þþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E − m h p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð E E cm − E cm þ m z Þ − ð m h m z Þ p E E cm − ð E cm þ m h Þ ;E max ¼ − ð E − E cm Þð E E cm − E cm − m h þ m z Þ E E cm − ð E cm þ m h Þ − ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E − m h p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð E E cm − E cm þ m z Þ − ð m h m z Þ p E E cm − ð E cm þ m h Þ : ð Þ In addition, the cross section about this process is σ ¼ π E cm Z π d φ Z E max E min dE × Z E max E min dE Z j M j d ð sin θ Þ : ð Þ FIG. 3. The picture of kinematics. (a) (b)
FIG. 4. The diagrams of cross section σ versus tan β , where (a) g YB ¼ − . (solid line), g YB ¼ − . (dashed line), g YB ¼ − . (dottedline), and (b) g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dotted line). CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D
V. NUMERICAL RESULTS
In this section, we will present the numerical results of theprocess e þ e − → hhZ . The input parameters related to theSM are chosen as m W ¼ . GeV, m Z ¼ . GeV, α em ð m Z Þ ¼ = . , α s ð m Z Þ ¼ . , m t ¼ . GeV, m b ¼ . GeV. The SM-like Higgs mass is [50] m h ¼ . (cid:4) . GeV ; ð Þ which constrains the parameter space of our model con-cretely [51]. We choose these parameters so that thecorresponding theoretical prediction of the mass of thelightest CP -even Higgs fits the experimental data with 3standard deviations: . GeV ≤ m h ≤ . GeV.The updated experimental data [52] on searching Z indicates M Z ≥ . TeV at 95% Confidence Level (CL),and we choose M Z ¼ . TeV in our following numericalanalysis. In addition, Refs. [53,54] give us a lower boundon the ratio between the Z mass and its gauge coupling at99% CL as M Z =g B ≥ TeV ; ð Þ then the scope of g B is limited to < g B ≤ . . The LHCexperimental data also constrains the parameter tan β astan β < . [29]. In order to coincide with the constraintsfrom the direct searches of the squarks at the LHC [55,56]and the observed Higgs signal in Ref. [57], for thoseparameters in the soft breaking terms, we take A ν ¼ A x ¼ TeV, A b ¼ A t ¼ TeV, μ ¼ GeV, μ ¼ GeV, B μ ¼ × GeV , B μ ¼ × GeV , m ˜ q ¼ m ˜ u ¼ diag ð ; ; . Þ TeV, and u ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u þ u p ; it can be obtainedfrom Eq. (15). Now, we present our numerical results.We adopt the latest predicted data of the SM to proceedin our analysis; it is about 0.12 – ¯ c T ð m Z Þ ∈ ½ − . ; . (cid:3) × − ; ð ¯ c W ð m Z Þ þ ¯ c B ð m Z ÞÞ ∈ ½ − . ; . (cid:3) × − ; ¯ c W ∈ ½ − . ; . (cid:3) ; ¯ c HW ∈ ½ − . ; . (cid:3) ; ¯ c HB ∈ ½ − . ; . (cid:3) : ð Þ Combining the tree diagrams and the correction of thesehigh-dimensional operators, the SM theoretical predictionon σ ð e þ e − → hhZ Þ ≃ . (cid:4) . fb can be obtained.Taking E cm ¼ GeV, tan β ¼ . , m ˜ ν ; ¼ GeV, m ˜ L; ¼ GeV, Y ν ; ¼ × − , and Y x; ¼ . , weplot the total cross section σ of e þ e − → hhZ versus theparameter tan β in the Fig. 4, where the gray band (a) (b) FIG. 6. The diagrams of cross section σ versus g B and g YB , where (a) g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dottedline), and (b) tan β ¼ (solid line), tan β ¼ (dashed line), tan β ¼ (dotted line).FIG. 5. The diagrams of cross section σ versus tan β ,where g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dotted line). e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020)
In this section, we will present the numerical results of theprocess e þ e − → hhZ . The input parameters related to theSM are chosen as m W ¼ . GeV, m Z ¼ . GeV, α em ð m Z Þ ¼ = . , α s ð m Z Þ ¼ . , m t ¼ . GeV, m b ¼ . GeV. The SM-like Higgs mass is [50] m h ¼ . (cid:4) . GeV ; ð Þ which constrains the parameter space of our model con-cretely [51]. We choose these parameters so that thecorresponding theoretical prediction of the mass of thelightest CP -even Higgs fits the experimental data with 3standard deviations: . GeV ≤ m h ≤ . GeV.The updated experimental data [52] on searching Z indicates M Z ≥ . TeV at 95% Confidence Level (CL),and we choose M Z ¼ . TeV in our following numericalanalysis. In addition, Refs. [53,54] give us a lower boundon the ratio between the Z mass and its gauge coupling at99% CL as M Z =g B ≥ TeV ; ð Þ then the scope of g B is limited to < g B ≤ . . The LHCexperimental data also constrains the parameter tan β astan β < . [29]. In order to coincide with the constraintsfrom the direct searches of the squarks at the LHC [55,56]and the observed Higgs signal in Ref. [57], for thoseparameters in the soft breaking terms, we take A ν ¼ A x ¼ TeV, A b ¼ A t ¼ TeV, μ ¼ GeV, μ ¼ GeV, B μ ¼ × GeV , B μ ¼ × GeV , m ˜ q ¼ m ˜ u ¼ diag ð ; ; . Þ TeV, and u ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u þ u p ; it can be obtainedfrom Eq. (15). Now, we present our numerical results.We adopt the latest predicted data of the SM to proceedin our analysis; it is about 0.12 – ¯ c T ð m Z Þ ∈ ½ − . ; . (cid:3) × − ; ð ¯ c W ð m Z Þ þ ¯ c B ð m Z ÞÞ ∈ ½ − . ; . (cid:3) × − ; ¯ c W ∈ ½ − . ; . (cid:3) ; ¯ c HW ∈ ½ − . ; . (cid:3) ; ¯ c HB ∈ ½ − . ; . (cid:3) : ð Þ Combining the tree diagrams and the correction of thesehigh-dimensional operators, the SM theoretical predictionon σ ð e þ e − → hhZ Þ ≃ . (cid:4) . fb can be obtained.Taking E cm ¼ GeV, tan β ¼ . , m ˜ ν ; ¼ GeV, m ˜ L; ¼ GeV, Y ν ; ¼ × − , and Y x; ¼ . , weplot the total cross section σ of e þ e − → hhZ versus theparameter tan β in the Fig. 4, where the gray band (a) (b) FIG. 6. The diagrams of cross section σ versus g B and g YB , where (a) g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dottedline), and (b) tan β ¼ (solid line), tan β ¼ (dashed line), tan β ¼ (dotted line).FIG. 5. The diagrams of cross section σ versus tan β ,where g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dotted line). e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020) epresents the SM prediction with 3 standard deviations [58].In the Fig. 4(a), we take g B ¼ . , and g YB ¼ − . (solidline), g YB ¼ − . (dashed line), g YB ¼ − . (dotted line),respectively. In the Fig. 4(b), we take g YB ¼ − . , and g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dottedline), respectively. Obviously the theoretical prediction onthe total cross section σ depends on the parameter tan β strongly. Along with the increasing of tan β , the cross sectiondecreases steeply as tan β ≤ . As tan β > , the depend-ence of total cross section on tan β is mild. Furthermore, thedifference between the prediction from the SM and that fromthe B-LSSM exceeds 3 standard deviations.Taking E cm ¼ GeV, tan β ¼ , g YB ¼ − . , m ˜ ν ; ¼ GeV, m ˜ L; ¼ GeV, Y ν ; ¼ × − , and Y x; ¼ . , we plot the total cross section σ of e þ e − → hhZ versus the parameter tan β in the Fig. 5, where the gray bandrepresents the SM prediction with 3 standard deviations[58]. In the Fig. 5, g B ¼ . (solid line), g B ¼ . (dashedline) and g B ¼ . (dotted line), respectively. With theincreasing of tan β , the cross section increases steeply. In order to further analyze how the new parameters g YB and g B in the B-LSSM affect the cross section σ , we plot theFig. 6, and the gray band also represents the SM predictionwith 3 standard deviations [58]. Taking E cm ¼ GeV,tan β ¼ . , m ˜ ν ; ¼ GeV, m ˜ L; ¼ GeV, Y ν ; ¼ × − , and Y x; ¼ . , we plot the total cross section of e þ e − → hhZ versus the new parameters g YB and g B in theFig. 6. In the Fig. 6(a), we take tan β ¼ , and g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dotted line),respectively. In the Fig. 6(b), we take g YB ¼ − . andtan β ¼ (solid line), tan β ¼ (dashed line), tan β ¼ (dotted line), respectively. The total cross section σ in theB-LSSM can exceed that in the SM easily when g B and j g YB j is small. In addition, the theoretical prediction on thetotal cross section σ depends on the new parameters g B and g YB strongly. In the Fig. 6(a), with the decreasing of j g YB j ,the total cross section increases sharply. In the Fig. 6(b), thetotal cross section σ decreases steeply when g B increases.Taking E cm ¼ GeV, tan β ¼ . , m ˜ ν ; ¼ GeV, m ˜ L; ¼ GeV, g B ¼ . , tan β ¼ , Y ν ; ¼ × − ,we plot the Fig. 7, where the gray band represents the SMprediction with 3 standard deviations [58]. In the Fig. 7, wetake g YB ¼ − . (solid line), g YB ¼ − . (dashed line), g YB ¼ − . (dotted line), respectively. Obviously, thedependence of total cross section on Yukawa coupling Y x; is mild, and, with the decreasing of j g YB j , the totalcross section increases. The mass of sneutrino in the one-loop effective potential is much smaller than the mass of thestop quark and can be almost ignored, so Y x; has a smalleffect on the total cross section σ . In addition, Y ν ; is in theorder of − , so it also has little effect on the result.We also plot the figure with the total cross section σ of e þ e − → hhZ versus the parameter tan β in the MSSM.In order to compare with the MSSM in Ref. [59], weadopt μ ¼ − TeV, E cm ¼ GeV, m ˜ L; ¼ GeV, g YB ¼ , g B ¼ , tan β ¼ , Y x; ¼ , Y ν ; ¼ to draw Y x ,33 f b FIG. 7. The diagrams of cross section σ versus Y x; , where g YB ¼ − . (solid line), g YB ¼ − . (dashed line), g YB ¼ − . (dotted line).FIG. 8. The diagrams of cross section σ versus tan β in theMSSM, where μ ¼ − TeV, E cm ¼ GeV, m ˜ L; ¼ GeV, g YB ¼ , g B ¼ , tan β ¼ , Y x; ¼ , Y ν ; ¼ . FIG. 9. The picture of differential cross section d σ d Ω versus angledistribution θ , where the solid line denotes tan β ¼ , thedashed line denotes tan β ¼ , and the dotted line denotestan β ¼ . CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D ig. 8. Along with the increasing of tan β , the total crosssection increases as tan β ≤ . When tan β > , thedependence of total cross section on tan β is mild. Inaddition, the total cross section is always within theprediction range of SM.Furthermore, we analyze the influence of the parameterson the angular distribution of differential cross section.Taking E cm ¼ GeV, tan β ¼ . , m ˜ ν ; ¼ GeV, m ˜ L; ¼ GeV, Y ν ; ¼ × − , and Y x; ¼ . , weplot the Figs. 9 and 10. In the Fig. 9, where g B ¼ . , g YB ¼ − . , and tan β ¼ (solid line), tan β ¼ (dashed line), tan β ¼ (dotted line), respectively. Inthe Fig. 10(a), we take tan β ¼ , g B ¼ . and g YB ¼ − . (solid line), g YB ¼ − . (dashed line), g YB ¼ − . (dotted line), respectively. In the Fig. 10(b), we taketan β ¼ , g YB ¼ − . and g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dotted line), respectively.Obviously, when θ ¼ π , the differential cross sectionreaches maximum, and when θ ¼ and π , the differentialcross section reaches minimum. The theoretical prediction on the differential cross section depends on the parameters tan β , g YB , and g B strongly, along with when the parameters tan β , j g YB j , and g B decrease, the differential cross section increases. (a) tan h h h (b) tan ^ FIG. 11. The picture of λ ð Þ h h h and α versus tan β , where (a) tan β ¼ . , g B ¼ . , and g YB ¼ − . , and (b) tan β ¼ . , g B ¼ . , g YB ¼ − . , Y ν ; ¼ × − , and Y x; ¼ . . (a) (b) FIG. 10. The picture of differential cross section d σ d Ω versus angle distribution θ , where (a) g YB ¼ − . (solid line), g YB ¼ − . (dashed line), g YB ¼ − . (dotted line), and (b) g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dotted line).FIG. 12. The picture of δ versus tan β , where E cm ¼ GeV,tan β ¼ . , g B ¼ . , g YB ¼ − . , Y ν ; ¼ × − , and Y x; ¼ . . e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020)
In this section, we will present the numerical results of theprocess e þ e − → hhZ . The input parameters related to theSM are chosen as m W ¼ . GeV, m Z ¼ . GeV, α em ð m Z Þ ¼ = . , α s ð m Z Þ ¼ . , m t ¼ . GeV, m b ¼ . GeV. The SM-like Higgs mass is [50] m h ¼ . (cid:4) . GeV ; ð Þ which constrains the parameter space of our model con-cretely [51]. We choose these parameters so that thecorresponding theoretical prediction of the mass of thelightest CP -even Higgs fits the experimental data with 3standard deviations: . GeV ≤ m h ≤ . GeV.The updated experimental data [52] on searching Z indicates M Z ≥ . TeV at 95% Confidence Level (CL),and we choose M Z ¼ . TeV in our following numericalanalysis. In addition, Refs. [53,54] give us a lower boundon the ratio between the Z mass and its gauge coupling at99% CL as M Z =g B ≥ TeV ; ð Þ then the scope of g B is limited to < g B ≤ . . The LHCexperimental data also constrains the parameter tan β astan β < . [29]. In order to coincide with the constraintsfrom the direct searches of the squarks at the LHC [55,56]and the observed Higgs signal in Ref. [57], for thoseparameters in the soft breaking terms, we take A ν ¼ A x ¼ TeV, A b ¼ A t ¼ TeV, μ ¼ GeV, μ ¼ GeV, B μ ¼ × GeV , B μ ¼ × GeV , m ˜ q ¼ m ˜ u ¼ diag ð ; ; . Þ TeV, and u ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u þ u p ; it can be obtainedfrom Eq. (15). Now, we present our numerical results.We adopt the latest predicted data of the SM to proceedin our analysis; it is about 0.12 – ¯ c T ð m Z Þ ∈ ½ − . ; . (cid:3) × − ; ð ¯ c W ð m Z Þ þ ¯ c B ð m Z ÞÞ ∈ ½ − . ; . (cid:3) × − ; ¯ c W ∈ ½ − . ; . (cid:3) ; ¯ c HW ∈ ½ − . ; . (cid:3) ; ¯ c HB ∈ ½ − . ; . (cid:3) : ð Þ Combining the tree diagrams and the correction of thesehigh-dimensional operators, the SM theoretical predictionon σ ð e þ e − → hhZ Þ ≃ . (cid:4) . fb can be obtained.Taking E cm ¼ GeV, tan β ¼ . , m ˜ ν ; ¼ GeV, m ˜ L; ¼ GeV, Y ν ; ¼ × − , and Y x; ¼ . , weplot the total cross section σ of e þ e − → hhZ versus theparameter tan β in the Fig. 4, where the gray band (a) (b) FIG. 6. The diagrams of cross section σ versus g B and g YB , where (a) g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dottedline), and (b) tan β ¼ (solid line), tan β ¼ (dashed line), tan β ¼ (dotted line).FIG. 5. The diagrams of cross section σ versus tan β ,where g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dotted line). e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020) epresents the SM prediction with 3 standard deviations [58].In the Fig. 4(a), we take g B ¼ . , and g YB ¼ − . (solidline), g YB ¼ − . (dashed line), g YB ¼ − . (dotted line),respectively. In the Fig. 4(b), we take g YB ¼ − . , and g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dottedline), respectively. Obviously the theoretical prediction onthe total cross section σ depends on the parameter tan β strongly. Along with the increasing of tan β , the cross sectiondecreases steeply as tan β ≤ . As tan β > , the depend-ence of total cross section on tan β is mild. Furthermore, thedifference between the prediction from the SM and that fromthe B-LSSM exceeds 3 standard deviations.Taking E cm ¼ GeV, tan β ¼ , g YB ¼ − . , m ˜ ν ; ¼ GeV, m ˜ L; ¼ GeV, Y ν ; ¼ × − , and Y x; ¼ . , we plot the total cross section σ of e þ e − → hhZ versus the parameter tan β in the Fig. 5, where the gray bandrepresents the SM prediction with 3 standard deviations[58]. In the Fig. 5, g B ¼ . (solid line), g B ¼ . (dashedline) and g B ¼ . (dotted line), respectively. With theincreasing of tan β , the cross section increases steeply. In order to further analyze how the new parameters g YB and g B in the B-LSSM affect the cross section σ , we plot theFig. 6, and the gray band also represents the SM predictionwith 3 standard deviations [58]. Taking E cm ¼ GeV,tan β ¼ . , m ˜ ν ; ¼ GeV, m ˜ L; ¼ GeV, Y ν ; ¼ × − , and Y x; ¼ . , we plot the total cross section of e þ e − → hhZ versus the new parameters g YB and g B in theFig. 6. In the Fig. 6(a), we take tan β ¼ , and g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dotted line),respectively. In the Fig. 6(b), we take g YB ¼ − . andtan β ¼ (solid line), tan β ¼ (dashed line), tan β ¼ (dotted line), respectively. The total cross section σ in theB-LSSM can exceed that in the SM easily when g B and j g YB j is small. In addition, the theoretical prediction on thetotal cross section σ depends on the new parameters g B and g YB strongly. In the Fig. 6(a), with the decreasing of j g YB j ,the total cross section increases sharply. In the Fig. 6(b), thetotal cross section σ decreases steeply when g B increases.Taking E cm ¼ GeV, tan β ¼ . , m ˜ ν ; ¼ GeV, m ˜ L; ¼ GeV, g B ¼ . , tan β ¼ , Y ν ; ¼ × − ,we plot the Fig. 7, where the gray band represents the SMprediction with 3 standard deviations [58]. In the Fig. 7, wetake g YB ¼ − . (solid line), g YB ¼ − . (dashed line), g YB ¼ − . (dotted line), respectively. Obviously, thedependence of total cross section on Yukawa coupling Y x; is mild, and, with the decreasing of j g YB j , the totalcross section increases. The mass of sneutrino in the one-loop effective potential is much smaller than the mass of thestop quark and can be almost ignored, so Y x; has a smalleffect on the total cross section σ . In addition, Y ν ; is in theorder of − , so it also has little effect on the result.We also plot the figure with the total cross section σ of e þ e − → hhZ versus the parameter tan β in the MSSM.In order to compare with the MSSM in Ref. [59], weadopt μ ¼ − TeV, E cm ¼ GeV, m ˜ L; ¼ GeV, g YB ¼ , g B ¼ , tan β ¼ , Y x; ¼ , Y ν ; ¼ to draw Y x ,33 f b FIG. 7. The diagrams of cross section σ versus Y x; , where g YB ¼ − . (solid line), g YB ¼ − . (dashed line), g YB ¼ − . (dotted line).FIG. 8. The diagrams of cross section σ versus tan β in theMSSM, where μ ¼ − TeV, E cm ¼ GeV, m ˜ L; ¼ GeV, g YB ¼ , g B ¼ , tan β ¼ , Y x; ¼ , Y ν ; ¼ . FIG. 9. The picture of differential cross section d σ d Ω versus angledistribution θ , where the solid line denotes tan β ¼ , thedashed line denotes tan β ¼ , and the dotted line denotestan β ¼ . CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D ig. 8. Along with the increasing of tan β , the total crosssection increases as tan β ≤ . When tan β > , thedependence of total cross section on tan β is mild. Inaddition, the total cross section is always within theprediction range of SM.Furthermore, we analyze the influence of the parameterson the angular distribution of differential cross section.Taking E cm ¼ GeV, tan β ¼ . , m ˜ ν ; ¼ GeV, m ˜ L; ¼ GeV, Y ν ; ¼ × − , and Y x; ¼ . , weplot the Figs. 9 and 10. In the Fig. 9, where g B ¼ . , g YB ¼ − . , and tan β ¼ (solid line), tan β ¼ (dashed line), tan β ¼ (dotted line), respectively. Inthe Fig. 10(a), we take tan β ¼ , g B ¼ . and g YB ¼ − . (solid line), g YB ¼ − . (dashed line), g YB ¼ − . (dotted line), respectively. In the Fig. 10(b), we taketan β ¼ , g YB ¼ − . and g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dotted line), respectively.Obviously, when θ ¼ π , the differential cross sectionreaches maximum, and when θ ¼ and π , the differentialcross section reaches minimum. The theoretical prediction on the differential cross section depends on the parameters tan β , g YB , and g B strongly, along with when the parameters tan β , j g YB j , and g B decrease, the differential cross section increases. (a) tan h h h (b) tan ^ FIG. 11. The picture of λ ð Þ h h h and α versus tan β , where (a) tan β ¼ . , g B ¼ . , and g YB ¼ − . , and (b) tan β ¼ . , g B ¼ . , g YB ¼ − . , Y ν ; ¼ × − , and Y x; ¼ . . (a) (b) FIG. 10. The picture of differential cross section d σ d Ω versus angle distribution θ , where (a) g YB ¼ − . (solid line), g YB ¼ − . (dashed line), g YB ¼ − . (dotted line), and (b) g B ¼ . (solid line), g B ¼ . (dashed line), g B ¼ . (dotted line).FIG. 12. The picture of δ versus tan β , where E cm ¼ GeV,tan β ¼ . , g B ¼ . , g YB ¼ − . , Y ν ; ¼ × − , and Y x; ¼ . . e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020) ow we analyze the self-coupling of Higgs, where λ ð Þ h h h denotes the tree level contribution of triple Higgsself-coupling, and α ¼ λ h h h − λ ð Þ h h h λ ð Þ h h h . Taking tan β ¼ . , g B ¼ . , and g YB ¼ − . , we plot the dependence ofself-coupling on the parameter tan β in the Fig. 11(a).Obviously, λ ð Þ h h h increases steeply as < tan β < .Meanwhile, the dependence of self-coupling on the param-eter tan β is mild when tan β > . Taking tan β ¼ . , g B ¼ . , g YB ¼ − . , m ˜ ν ; ¼ GeV, m ˜ L; ¼ GeV, Y ν ; ¼ × − , and Y x; ¼ . , we plot dependenceof the relative corrections of self-coupling from the one-loop effective potential varying with tan β accordingly.Obviously the relative correction increases steeply whentan β increases.Finally we present the relative correction on the pro-duction cross section δ ¼ σ − σ σ versus the parameter tan β ,where σ denotes the theoretical prediction of the total crosssection in the SM. Taking E cm ¼ GeV, tan β ¼ . , g B ¼ . , g YB ¼ − . , m ˜ ν ; ¼ GeV, m ˜ L; ¼ GeV, Y ν ; ¼ × − , and Y x; ¼ . , we plot the relativecorrection versus tan β in the Fig. 12. The theoreticalprediction on the production cross section of theB-LSSM deviates that of the SM obviously as tan β < .The dependence of the relative correction on tan β changesmildly as tan β > . V. CONCLUSION
In this work we analyze the production cross sectionof e þ e − → hhZ and the self-coupling of Higgs in theB-LSSM. Some parameters affect the theoretical prediction on the production cross section of e þ e − → hhZ strongly,for example the new gauge coupling g YB . Actually thetheoretical prediction on the cross section deviates fromthat of the SM obviously under some assumptions on theparameters of the model. Nevertheless, the correction fromone-loop effective potential to the self-coupling of thelightest Higgs can be neglected safely.Although the cross section has not been measured whenthe center-of-mass energy is 500 GeV, the trilinear Higgsboson coupling can be measured at international linearcollider through this process. In the future, if the exper-imental value greatly exceeds the SM, the model can beused to explain the deviation. If the experimental value isconsistent with the SM, it will constrain our param-eter space. ACKNOWLEDGMENTS
We are very grateful to Shu-min Zhao the professor ofHebei University, for giving us some useful discussions. Thiswork is supported by National Natural Science Foundationof China (NNSFC) (Grants No. 11535002, No. 11605037,and No. 11705045), Hebei Key Lab of Optic-ElectronicInformation and Materials, the midwest universities compre-hensive strength promotion project and the youth top-notchtalent support program of the Hebei Province, AdvancedTalents Incubation Program of the Hebei University,521000981396.
APPENDIX A: THE ONE LOOP CORRECTIONTO MASS SQUARED MATRIX FOR CP -EVENHIGGS BOSONS The detailed expression about stability condition: ∂ V ∂ ReH ¼ − v B μ þ ð g YB g B u cos β þ g v cos β Þ v þ ð m H þ μ Þ v þ π f ð Q ; m ˜ t Þ (cid:4) g v þ m ˜ t − m ˜ t (cid:1) − M tm C v þ ffiffiffi p F (cid:3)(cid:5) þ π f ð Q ; m ˜ t Þ (cid:4) g v − m ˜ t − m ˜ t (cid:1) − M tm C v þ ffiffiffi p F (cid:3)(cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) g v þ m ˜ b − m ˜ b (cid:1) − M bm C v þ ffiffiffi p F (cid:3)(cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) g v − m ˜ b − m ˜ b (cid:1) − M bm C v þ ffiffiffi p F (cid:3)(cid:5) − π f ð Q ; m b Þ Y b m b þ π f ð Q ; m ˜ ν R Þ (cid:7) g v − Y − (cid:4) ð m ˜ ν ILL þ m ˜ ν IRR Þ g v − ð g þ g YB g B Þ v m ˜ ν IRR − g YB g B v m ˜ ν ILL − ffiffiffi p μ Y ν ;ij m ˜ ν ILR (cid:5)(cid:8) ; ð A1 Þ CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D V ∂ ReH ¼ ð g YB g B u cos β þ g v cos β Þ v − v B μ þ v ð m H þ μ Þþ π f ð Q ; m ˜ t Þ (cid:4) − g v þ Y t v þ m ˜ t − m ˜ t (cid:1) M tm C v þ ffiffiffi p F (cid:3)(cid:5) þ π (cid:7) f ð Q ; m ˜ t Þ (cid:4) − g v þ Y t v − m ˜ t − m ˜ t (cid:1) M tm C v þ ffiffiffi p F (cid:3)(cid:5) − f ð Q ; m t Þ Y t m t (cid:8) þ π f ð Q ; m ˜ b Þ (cid:4) − g v þ Y b v þ m ˜ b − m ˜ b (cid:1) M bm C v þ ffiffiffi p F (cid:3)(cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) − g v þ Y b v − m ˜ b − m ˜ b (cid:1) M bm C v þ ffiffiffi p F (cid:3)(cid:5) þ π f ð Q ; m ˜ ν R Þ (cid:7)(cid:1) − g þ Y ν ;ij (cid:3) v − Y − (cid:4)(cid:1) − g þ Y ν ;ij (cid:3) ð m ˜ ν ILL þ m ˜ ν IRR Þ v þ ð g þ g YB g B − Y ν ;ij Þ v m ˜ ν IRR − ð g YB g B þ Y ν ;ij Þ v m ˜ ν ILL þ ð ffiffiffi p T ij ν þ u Y ν ;ij Y x;ij Þ m ˜ ν ILR (cid:5)(cid:8) ; ð A2 Þ ∂ V ∂ Re ˜ η ¼ ð g B u cos β þ g YB g B v cos β Þ u þ ð m ˜ η þ μ Þ u − u B μ þ π f ð Q ; m ˜ t Þ (cid:4) g YB g B u þ M tm C u ð m ˜ t − m ˜ t Þ (cid:5) þ π f ð Q ; m ˜ t Þ (cid:4) g YB g B u − M tm C u ð m ˜ t − m ˜ t Þ (cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) g YB g B u þ M bm C u ð m ˜ b − m ˜ b Þ (cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) g YB g B u − M bm C u ð m ˜ b − m ˜ b Þ (cid:5) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) g YB g B u þ ffiffiffi p T ijx þ u Y x;ij − Y − ½ð g YB g B u þ ffiffiffi p T ijx þ u Y x;ij Þð m ˜ ν ILL þ m ˜ ν IRR Þ − ð g YB g B þ g B Þ u m ˜ ν IRR − m ˜ ν ILL ð − g B u þ ffiffiffi p T ijx þ u Y x;ij Þ(cid:3) (cid:8) − f ð Q; m ν R Þ u Y x;ij (cid:8) ; ð A3 Þ ∂ V ∂ Re ˜ η ¼ ð g YB g B v cos β − g B u cos β Þ u þ u ð m ˜ η þ μ Þ − u B μ þ π f ð Q ; m ˜ t Þ (cid:4) − g YB g B u − M tm C u ð m ˜ t − m ˜ t Þ (cid:5) þ π f ð Q ; m ˜ t Þ (cid:4) − g YB g B u þ M tm C u ð m ˜ t − m ˜ t Þ (cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) − g YB g B u − M bm C u ð m ˜ b − m ˜ b Þ (cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) − g YB g B u þ M bm C u ð m ˜ b − m ˜ b Þ (cid:5) þ π f ð Q ; m ˜ ν R Þf − g YB g B u − ffiffiffi p μ Y x;ij − Y − ½ð − g B u − g YB g B u þ g B u − ffiffiffi p μ Y x;ij Þð m ˜ ν ILL þ m ˜ ν IRR Þ − ð g B þ g YB g B Þ u m ˜ ν IRR − m ˜ ν ILL ð g B u − ffiffiffi p μ Y x;ij Þ(cid:3)g : ð A4 Þ The detailed expression about Δ m h : e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020)
APPENDIX A: THE ONE LOOP CORRECTIONTO MASS SQUARED MATRIX FOR CP -EVENHIGGS BOSONS The detailed expression about stability condition: ∂ V ∂ ReH ¼ − v B μ þ ð g YB g B u cos β þ g v cos β Þ v þ ð m H þ μ Þ v þ π f ð Q ; m ˜ t Þ (cid:4) g v þ m ˜ t − m ˜ t (cid:1) − M tm C v þ ffiffiffi p F (cid:3)(cid:5) þ π f ð Q ; m ˜ t Þ (cid:4) g v − m ˜ t − m ˜ t (cid:1) − M tm C v þ ffiffiffi p F (cid:3)(cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) g v þ m ˜ b − m ˜ b (cid:1) − M bm C v þ ffiffiffi p F (cid:3)(cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) g v − m ˜ b − m ˜ b (cid:1) − M bm C v þ ffiffiffi p F (cid:3)(cid:5) − π f ð Q ; m b Þ Y b m b þ π f ð Q ; m ˜ ν R Þ (cid:7) g v − Y − (cid:4) ð m ˜ ν ILL þ m ˜ ν IRR Þ g v − ð g þ g YB g B Þ v m ˜ ν IRR − g YB g B v m ˜ ν ILL − ffiffiffi p μ Y ν ;ij m ˜ ν ILR (cid:5)(cid:8) ; ð A1 Þ CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D V ∂ ReH ¼ ð g YB g B u cos β þ g v cos β Þ v − v B μ þ v ð m H þ μ Þþ π f ð Q ; m ˜ t Þ (cid:4) − g v þ Y t v þ m ˜ t − m ˜ t (cid:1) M tm C v þ ffiffiffi p F (cid:3)(cid:5) þ π (cid:7) f ð Q ; m ˜ t Þ (cid:4) − g v þ Y t v − m ˜ t − m ˜ t (cid:1) M tm C v þ ffiffiffi p F (cid:3)(cid:5) − f ð Q ; m t Þ Y t m t (cid:8) þ π f ð Q ; m ˜ b Þ (cid:4) − g v þ Y b v þ m ˜ b − m ˜ b (cid:1) M bm C v þ ffiffiffi p F (cid:3)(cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) − g v þ Y b v − m ˜ b − m ˜ b (cid:1) M bm C v þ ffiffiffi p F (cid:3)(cid:5) þ π f ð Q ; m ˜ ν R Þ (cid:7)(cid:1) − g þ Y ν ;ij (cid:3) v − Y − (cid:4)(cid:1) − g þ Y ν ;ij (cid:3) ð m ˜ ν ILL þ m ˜ ν IRR Þ v þ ð g þ g YB g B − Y ν ;ij Þ v m ˜ ν IRR − ð g YB g B þ Y ν ;ij Þ v m ˜ ν ILL þ ð ffiffiffi p T ij ν þ u Y ν ;ij Y x;ij Þ m ˜ ν ILR (cid:5)(cid:8) ; ð A2 Þ ∂ V ∂ Re ˜ η ¼ ð g B u cos β þ g YB g B v cos β Þ u þ ð m ˜ η þ μ Þ u − u B μ þ π f ð Q ; m ˜ t Þ (cid:4) g YB g B u þ M tm C u ð m ˜ t − m ˜ t Þ (cid:5) þ π f ð Q ; m ˜ t Þ (cid:4) g YB g B u − M tm C u ð m ˜ t − m ˜ t Þ (cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) g YB g B u þ M bm C u ð m ˜ b − m ˜ b Þ (cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) g YB g B u − M bm C u ð m ˜ b − m ˜ b Þ (cid:5) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) g YB g B u þ ffiffiffi p T ijx þ u Y x;ij − Y − ½ð g YB g B u þ ffiffiffi p T ijx þ u Y x;ij Þð m ˜ ν ILL þ m ˜ ν IRR Þ − ð g YB g B þ g B Þ u m ˜ ν IRR − m ˜ ν ILL ð − g B u þ ffiffiffi p T ijx þ u Y x;ij Þ(cid:3) (cid:8) − f ð Q; m ν R Þ u Y x;ij (cid:8) ; ð A3 Þ ∂ V ∂ Re ˜ η ¼ ð g YB g B v cos β − g B u cos β Þ u þ u ð m ˜ η þ μ Þ − u B μ þ π f ð Q ; m ˜ t Þ (cid:4) − g YB g B u − M tm C u ð m ˜ t − m ˜ t Þ (cid:5) þ π f ð Q ; m ˜ t Þ (cid:4) − g YB g B u þ M tm C u ð m ˜ t − m ˜ t Þ (cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) − g YB g B u − M bm C u ð m ˜ b − m ˜ b Þ (cid:5) þ π f ð Q ; m ˜ b Þ (cid:4) − g YB g B u þ M bm C u ð m ˜ b − m ˜ b Þ (cid:5) þ π f ð Q ; m ˜ ν R Þf − g YB g B u − ffiffiffi p μ Y x;ij − Y − ½ð − g B u − g YB g B u þ g B u − ffiffiffi p μ Y x;ij Þð m ˜ ν ILL þ m ˜ ν IRR Þ − ð g B þ g YB g B Þ u m ˜ ν IRR − m ˜ ν ILL ð g B u − ffiffiffi p μ Y x;ij Þ(cid:3)g : ð A4 Þ The detailed expression about Δ m h : e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020) m Φ d Φ d ¼ π (cid:7) ½ f ð Q ;m ˜ t Þ − f ð Q ;m ˜ t Þ(cid:3) (cid:4) − C v ð m ˜ t − m ˜ t Þ − v ð F − C M tm Þ ð m ˜ t − m ˜ t Þ (cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v þ F − C M tm v m ˜ t − m ˜ t (cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v − F − C M tm v m ˜ t − m ˜ t (cid:5) (cid:8) þ π (cid:7) ½ f ð Q ;m ˜ b Þ − f ð Q ;m ˜ b Þ(cid:3) (cid:4) − C v ð m ˜ b − m ˜ b Þ − v ð F − C M bm Þ ð m ˜ b − m ˜ b Þ (cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v þ F − C M bm v m ˜ b − m ˜ b (cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v − F − C M bm v m ˜ b − m ˜ b (cid:5) − f ð Q ;m b Þ Y b (cid:8) þ π (cid:7) g f ð Q ;m ˜ ν R Þ þ f ð Q ;m ˜ ν R Þ (cid:7) Y − y d (cid:4)(cid:1) g − g YB g B (cid:3) v m ˜ ν ILL − (cid:1) g þ g YB g B (cid:3) v m ˜ ν IRR − ffiffiffi p μ Y ν ;ij m ˜ ν ILR (cid:5) þ Y − (cid:4)(cid:1) g − g YB g B (cid:3) ð m ˜ ν ILL þ m ˜ ν IRR
Þ þ g v − g m ˜ ν IRR − ð g þ g YB g B Þ g YB g B v þ μ Y ν ;ij (cid:5)(cid:8) þ ln m ˜ ν R Q y d R (cid:8) ; ð A5 Þ Δ m Φ u Φ d ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − C v v ð m ˜ t − m ˜ t Þ þ M tm C v − F ð m ˜ t − m ˜ t Þ ð M tm C v þ F Þ (cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v þ F − C M tm v ð m ˜ t − m ˜ t Þ (cid:1) C v þ F þ C M tm v m ˜ t − m ˜ t (cid:3)(cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v − F − C M tm v ð m ˜ t − m ˜ t Þ (cid:1) C v − F þ C M tm v m ˜ t − m ˜ t (cid:3)(cid:5)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − C v v ð m ˜ b − m ˜ b Þ þ M bm C v − F ð m ˜ b − m ˜ b Þ ð M bm C v þ F (cid:3)(cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v þ F − C M bm v ð m ˜ b − m ˜ b Þ (cid:1) C v þ F þ C M bm v m ˜ b − m ˜ b (cid:3)(cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v − F − C M bm v ð m ˜ b − m ˜ b Þ (cid:1) C v − F þ C M bm v m ˜ b − m ˜ b (cid:3)(cid:5)(cid:8) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) Y − y d y u − Y − (cid:4) − ð T ij ν þ ffiffiffi p u Y x;ij Y ν ;ij Þ μ Y ν ;ij − (cid:1) g þ g YB g B − Y ν ;ij (cid:3) g YB g B v v − ð g þ g YB g B Þ (cid:1) g YB g B þ Y ν ;ij (cid:3) v v þ (cid:1) Y ν ;ij − g v (cid:3) g v v (cid:5)(cid:8) þ ln m ˜ ν R Q y d R (cid:1) − g v þ v Y ν ;ij − Y − y u (cid:3)(cid:8) ; ð A6 Þ CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D m Φ η Φ d ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − C C v u m ˜ t − m ˜ t − M tm C u ð F − C M tm v Þð m ˜ t − m ˜ t Þ (cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v þ F − C M tm v m ˜ t − m ˜ t (cid:5) × (cid:1) C þ M tm C m ˜ t − m ˜ t (cid:3) u þ ln m ˜ t Q × (cid:4) ð Y t − C Þ v − F − C M tm v m ˜ t − m ˜ t (cid:5) × (cid:1) C − M tm C m ˜ t − m ˜ t (cid:3) u (cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − C C v u m ˜ b − m ˜ b − M bm C u ð F − C M bm v Þð m ˜ b − m ˜ b Þ (cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v þ F − C M bm v m ˜ b − m ˜ b (cid:5) × (cid:1) C þ M bm C m ˜ b − m ˜ b (cid:3) u þ ln m ˜ b Q × (cid:4) ð Y b − C Þ v − F − C M bm v m ˜ b − m ˜ b (cid:5) × (cid:1) C − M bm C m ˜ b − m ˜ b (cid:3) u (cid:8) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) Y − y d y η − Y − (cid:4)(cid:1) g YB g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) g v − ð g þ g YB g B Þ (cid:1) − g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) v − ð g B þ g YB g B Þ g YB g B v u − ffiffiffi p μ Y ν ;ii Y x;ij v (cid:3) (cid:8) þ ln m ˜ ν R Q y d R × (cid:1) g YB g B u þ ffiffiffi p T ijx þ u Y x;ij − Y − y η (cid:3)(cid:8) ; ð A7 Þ Δ m Φ ¯ η Φ d ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) C C v u ð m ˜ t − m ˜ t Þ þ M tm v C ð m ˜ t − m ˜ t Þ (cid:1) − C M tm u þ F (cid:3)(cid:5) − ln m ˜ t Q × (cid:1) C þ M tm C m ˜ t − m ˜ t (cid:3) u × (cid:4) ð Y t − C Þ v þ m ˜ t − m ˜ t (cid:1) − C M tm v þ F (cid:3)(cid:5) þ ln m ˜ t Q × (cid:1) − C þ M tm C m ˜ t − m ˜ t (cid:3) u (cid:4) ð Y t − C Þ v − m ˜ t − m ˜ t (cid:1) − C M tm v þ F (cid:3)(cid:5)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) C C v u ð m ˜ b − m ˜ b Þ þ M bm v C ð m ˜ b − m ˜ b Þ (cid:1) − C M bm u þ F (cid:3)(cid:5) − ln m ˜ b Q × (cid:1) C þ M bm C m ˜ b − m ˜ b (cid:3) u × (cid:4) ð Y b − C Þ v þ m ˜ b − m ˜ b (cid:1) − C M bm v þ F (cid:3)(cid:5) þ ln m ˜ b Q × (cid:1) − C þ M bm C m ˜ b − m ˜ b (cid:3) u (cid:4) ð Y b − C Þ v − m ˜ b − m ˜ b (cid:1) − C M bm v þ F (cid:3)(cid:5)(cid:8) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) Y − y d y ¯ η − Y − (cid:4)(cid:1) − g YB g B u − ffiffiffi p μ Y x;ij (cid:3) g v − ð g þ g YB g B Þ (cid:1) g B u − ffiffiffi p μ Y x;ij (cid:3) v − ð g B þ g YB g B Þ g YB g B v u (cid:5)(cid:8) þ ln m ˜ ν R Q y d R (cid:1) − g YB g B u − ffiffiffi p μ Y x;ij − Y − y ¯ η (cid:3)(cid:8) ; ð A8 Þ e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020)
Þ þ g v − g m ˜ ν IRR − ð g þ g YB g B Þ g YB g B v þ μ Y ν ;ij (cid:5)(cid:8) þ ln m ˜ ν R Q y d R (cid:8) ; ð A5 Þ Δ m Φ u Φ d ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − C v v ð m ˜ t − m ˜ t Þ þ M tm C v − F ð m ˜ t − m ˜ t Þ ð M tm C v þ F Þ (cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v þ F − C M tm v ð m ˜ t − m ˜ t Þ (cid:1) C v þ F þ C M tm v m ˜ t − m ˜ t (cid:3)(cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v − F − C M tm v ð m ˜ t − m ˜ t Þ (cid:1) C v − F þ C M tm v m ˜ t − m ˜ t (cid:3)(cid:5)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − C v v ð m ˜ b − m ˜ b Þ þ M bm C v − F ð m ˜ b − m ˜ b Þ ð M bm C v þ F (cid:3)(cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v þ F − C M bm v ð m ˜ b − m ˜ b Þ (cid:1) C v þ F þ C M bm v m ˜ b − m ˜ b (cid:3)(cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v − F − C M bm v ð m ˜ b − m ˜ b Þ (cid:1) C v − F þ C M bm v m ˜ b − m ˜ b (cid:3)(cid:5)(cid:8) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) Y − y d y u − Y − (cid:4) − ð T ij ν þ ffiffiffi p u Y x;ij Y ν ;ij Þ μ Y ν ;ij − (cid:1) g þ g YB g B − Y ν ;ij (cid:3) g YB g B v v − ð g þ g YB g B Þ (cid:1) g YB g B þ Y ν ;ij (cid:3) v v þ (cid:1) Y ν ;ij − g v (cid:3) g v v (cid:5)(cid:8) þ ln m ˜ ν R Q y d R (cid:1) − g v þ v Y ν ;ij − Y − y u (cid:3)(cid:8) ; ð A6 Þ CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D m Φ η Φ d ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − C C v u m ˜ t − m ˜ t − M tm C u ð F − C M tm v Þð m ˜ t − m ˜ t Þ (cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v þ F − C M tm v m ˜ t − m ˜ t (cid:5) × (cid:1) C þ M tm C m ˜ t − m ˜ t (cid:3) u þ ln m ˜ t Q × (cid:4) ð Y t − C Þ v − F − C M tm v m ˜ t − m ˜ t (cid:5) × (cid:1) C − M tm C m ˜ t − m ˜ t (cid:3) u (cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − C C v u m ˜ b − m ˜ b − M bm C u ð F − C M bm v Þð m ˜ b − m ˜ b Þ (cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v þ F − C M bm v m ˜ b − m ˜ b (cid:5) × (cid:1) C þ M bm C m ˜ b − m ˜ b (cid:3) u þ ln m ˜ b Q × (cid:4) ð Y b − C Þ v − F − C M bm v m ˜ b − m ˜ b (cid:5) × (cid:1) C − M bm C m ˜ b − m ˜ b (cid:3) u (cid:8) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) Y − y d y η − Y − (cid:4)(cid:1) g YB g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) g v − ð g þ g YB g B Þ (cid:1) − g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) v − ð g B þ g YB g B Þ g YB g B v u − ffiffiffi p μ Y ν ;ii Y x;ij v (cid:3) (cid:8) þ ln m ˜ ν R Q y d R × (cid:1) g YB g B u þ ffiffiffi p T ijx þ u Y x;ij − Y − y η (cid:3)(cid:8) ; ð A7 Þ Δ m Φ ¯ η Φ d ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) C C v u ð m ˜ t − m ˜ t Þ þ M tm v C ð m ˜ t − m ˜ t Þ (cid:1) − C M tm u þ F (cid:3)(cid:5) − ln m ˜ t Q × (cid:1) C þ M tm C m ˜ t − m ˜ t (cid:3) u × (cid:4) ð Y t − C Þ v þ m ˜ t − m ˜ t (cid:1) − C M tm v þ F (cid:3)(cid:5) þ ln m ˜ t Q × (cid:1) − C þ M tm C m ˜ t − m ˜ t (cid:3) u (cid:4) ð Y t − C Þ v − m ˜ t − m ˜ t (cid:1) − C M tm v þ F (cid:3)(cid:5)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) C C v u ð m ˜ b − m ˜ b Þ þ M bm v C ð m ˜ b − m ˜ b Þ (cid:1) − C M bm u þ F (cid:3)(cid:5) − ln m ˜ b Q × (cid:1) C þ M bm C m ˜ b − m ˜ b (cid:3) u × (cid:4) ð Y b − C Þ v þ m ˜ b − m ˜ b (cid:1) − C M bm v þ F (cid:3)(cid:5) þ ln m ˜ b Q × (cid:1) − C þ M bm C m ˜ b − m ˜ b (cid:3) u (cid:4) ð Y b − C Þ v − m ˜ b − m ˜ b (cid:1) − C M bm v þ F (cid:3)(cid:5)(cid:8) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) Y − y d y ¯ η − Y − (cid:4)(cid:1) − g YB g B u − ffiffiffi p μ Y x;ij (cid:3) g v − ð g þ g YB g B Þ (cid:1) g B u − ffiffiffi p μ Y x;ij (cid:3) v − ð g B þ g YB g B Þ g YB g B v u (cid:5)(cid:8) þ ln m ˜ ν R Q y d R (cid:1) − g YB g B u − ffiffiffi p μ Y x;ij − Y − y ¯ η (cid:3)(cid:8) ; ð A8 Þ e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020) m Φ u Φ u ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) C v ð m ˜ t − m ˜ t Þ − v ð m ˜ t − m ˜ t Þ × (cid:1) C M tm þ A t Y t (cid:3) (cid:5) þ ln m ˜ t Q (cid:1) C v þ C M tm v þ F m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) C v − C M tm v þ F m ˜ t − m ˜ t (cid:3) (cid:8) − f ð Q ; m t Þ Y t þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) C v ð m ˜ b − m ˜ b Þ − v ð m ˜ b − m ˜ b Þ × (cid:1) C M bm þ A b Y b (cid:3) (cid:5) þ ln m ˜ b Q (cid:1) C v þ C M bm v þ F m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) C v − C M bm v þ F m ˜ b − m ˜ b (cid:3) (cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) − g þ Y ν ;ij (cid:3) þ f ð Q ; m ˜ ν R Þ (cid:1) Y − y u − Y − y u (cid:3) þ ln m ˜ ν R Q y u R (cid:5) ; ð A9 Þ Δ m Φ η Φ u ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) C C u v ð m ˜ t − m ˜ t Þ − M tm C u v ð m ˜ t − m ˜ t Þ (cid:1) C M tm þ A t Y t (cid:3)(cid:5) þ ln m ˜ t Q × (cid:1) C þ M tm C m ˜ t − m ˜ t (cid:3) u × (cid:1) C v þ C M tm v þ F m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q × (cid:1) C − M tm C m ˜ t − m ˜ t (cid:3) u × (cid:1) C v − C M tm v þ F m ˜ t − m ˜ t (cid:3)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) C C u v ð m ˜ b − m ˜ b Þ − M bm C u v ð m ˜ b − m ˜ b Þ (cid:1) C M bm þ A b Y b (cid:3)(cid:5) þ ln m ˜ b Q × (cid:1) C þ M bm C m ˜ b − m ˜ b (cid:3) u × (cid:1) C v þ C M bm v þ F m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q × (cid:1) C − M bm C m ˜ b − m ˜ b (cid:3) u × (cid:1) C v − C M bm v þ F m ˜ b − m ˜ b (cid:3)(cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) Y − y η y u − Y − y u η (cid:3) þ ln m ˜ ν R Q y u R y η R (cid:5) ; ð A10 Þ Δ m Φ ¯ η Φ u ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − C C v u ð m ˜ t − m ˜ t Þ þ M tm C u v ð m ˜ t − m ˜ t Þ (cid:1) C M tm þ F (cid:3)(cid:5) þ ln m ˜ t Q (cid:1) − C − M tm C m ˜ t − m ˜ t (cid:3) u × (cid:1) C v þ C M tm v þ F m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) − C þ M tm C m ˜ t − m ˜ t (cid:3) u × (cid:1) C v − C M tm v þ F m ˜ t − m ˜ t (cid:3)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − C C v u ð m ˜ b − m ˜ b Þ þ M bm C u v ð m ˜ b − m ˜ b Þ (cid:1) C M bm þ F (cid:3)(cid:5) þ ln m ˜ b Q (cid:1) − C − M bm C m ˜ b − m ˜ b (cid:3) u × (cid:1) C v þ C M bm v þ F m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) − C þ M bm C m ˜ b − m ˜ b (cid:3) u × (cid:1) C v − C M bm v þ F m ˜ b − m ˜ b (cid:3)(cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) Y − y ¯ η y u − Y − y u η (cid:3) þ ln m ˜ ν R Q y u R y ¯ η R (cid:5) ; ð A11 Þ CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D m Φ η Φ η ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) m ˜ t − m ˜ t − M tm ð m ˜ t − m ˜ t Þ (cid:5) C u þ ln m ˜ t Q (cid:1) C u þ M tm C u m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) C u − M tm C u m ˜ t − m ˜ t (cid:3) (cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) m ˜ b − m ˜ b − M bm ð m ˜ b − m ˜ b Þ (cid:5) C u þ ln m ˜ b Q (cid:1) C u þ M bm C u m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) C u − M bm C u m ˜ b − m ˜ b (cid:3) (cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) g YB g B þ Y x;ij (cid:3) þ f ð Q; m ˜ ν R Þ (cid:1) Y − y η − Y − y η (cid:3) þ ln m ˜ ν R Q y η R − f ð Q ; m ν R Þ Y x;ij þ ln m ν R Q u Y x;ij (cid:5) ; ð A12 Þ Δ m Φ ¯ η Φ η ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − m ˜ t − m ˜ t þ M tm ð m ˜ t − m ˜ t Þ (cid:5) C u u þ ln m ˜ t Q (cid:1) − C u − M tm C u m ˜ t − m ˜ t (cid:3) × (cid:1) C u þ M tm C u m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) − C u þ M tm C u m ˜ t − m ˜ t (cid:3) × (cid:1) C u − M tm C u m ˜ t − m ˜ t (cid:3)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − m ˜ b − m ˜ b þ M bm ð m ˜ b − m ˜ b Þ (cid:5) C u u þ ln m ˜ b Q (cid:1) − C u − M bm C u m ˜ b − m ˜ b (cid:3) × (cid:1) C u þ M bm C u m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) − C u þ M bm C u m ˜ b − m ˜ b (cid:3) × (cid:1) C u − M bm C u m ˜ b − m ˜ b (cid:3)(cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) Y − y η y ¯ η − Y − y η η (cid:3) þ ln m ˜ ν R Q y η R y ¯ η R (cid:5) ; ð A13 Þ Δ m Φ ¯ η Φ ¯ η ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) m ˜ t − m ˜ t − M tm ð m ˜ t − m ˜ t Þ (cid:5) C u þ ln m ˜ t Q (cid:1) − C u − M tm C u m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) − C u þ M tm C u m ˜ t − m ˜ t (cid:3) (cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) m ˜ b − m ˜ b − M bm ð m ˜ b − m ˜ b Þ (cid:5) C u þ ln m ˜ b Q (cid:1) − C u − M bm C u m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) − C u þ M bm C u m ˜ b − m ˜ b (cid:3) (cid:8) þ π (cid:4) − f ð Q ; m ˜ ν R Þ g YB g B þ f ð Q; m ˜ ν R Þ (cid:1) Y − y η − Y − y ¯ η (cid:3) þ ln m ˜ ν R Q y η R (cid:5) : ð A14 Þ Here, f ð Q ;m t; ˜ t ; Þ¼ m t; ˜ t ; ð ln m t; ˜ t ; Q − Þ , f ð Q ; m b; ˜ b ; Þ ¼ m b; ˜ b ; ð ln m b; ˜ b ; Q − Þ , f ð Q ; m ν iR ; ˜ ν iR Þ ¼ m ν iR ; ˜ ν iR ð ln m ν iR; ˜ ν iR Q − Þ ,and e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020)
Þ þ g v − g m ˜ ν IRR − ð g þ g YB g B Þ g YB g B v þ μ Y ν ;ij (cid:5)(cid:8) þ ln m ˜ ν R Q y d R (cid:8) ; ð A5 Þ Δ m Φ u Φ d ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − C v v ð m ˜ t − m ˜ t Þ þ M tm C v − F ð m ˜ t − m ˜ t Þ ð M tm C v þ F Þ (cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v þ F − C M tm v ð m ˜ t − m ˜ t Þ (cid:1) C v þ F þ C M tm v m ˜ t − m ˜ t (cid:3)(cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v − F − C M tm v ð m ˜ t − m ˜ t Þ (cid:1) C v − F þ C M tm v m ˜ t − m ˜ t (cid:3)(cid:5)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − C v v ð m ˜ b − m ˜ b Þ þ M bm C v − F ð m ˜ b − m ˜ b Þ ð M bm C v þ F (cid:3)(cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v þ F − C M bm v ð m ˜ b − m ˜ b Þ (cid:1) C v þ F þ C M bm v m ˜ b − m ˜ b (cid:3)(cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v − F − C M bm v ð m ˜ b − m ˜ b Þ (cid:1) C v − F þ C M bm v m ˜ b − m ˜ b (cid:3)(cid:5)(cid:8) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) Y − y d y u − Y − (cid:4) − ð T ij ν þ ffiffiffi p u Y x;ij Y ν ;ij Þ μ Y ν ;ij − (cid:1) g þ g YB g B − Y ν ;ij (cid:3) g YB g B v v − ð g þ g YB g B Þ (cid:1) g YB g B þ Y ν ;ij (cid:3) v v þ (cid:1) Y ν ;ij − g v (cid:3) g v v (cid:5)(cid:8) þ ln m ˜ ν R Q y d R (cid:1) − g v þ v Y ν ;ij − Y − y u (cid:3)(cid:8) ; ð A6 Þ CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D m Φ η Φ d ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − C C v u m ˜ t − m ˜ t − M tm C u ð F − C M tm v Þð m ˜ t − m ˜ t Þ (cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v þ F − C M tm v m ˜ t − m ˜ t (cid:5) × (cid:1) C þ M tm C m ˜ t − m ˜ t (cid:3) u þ ln m ˜ t Q × (cid:4) ð Y t − C Þ v − F − C M tm v m ˜ t − m ˜ t (cid:5) × (cid:1) C − M tm C m ˜ t − m ˜ t (cid:3) u (cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − C C v u m ˜ b − m ˜ b − M bm C u ð F − C M bm v Þð m ˜ b − m ˜ b Þ (cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v þ F − C M bm v m ˜ b − m ˜ b (cid:5) × (cid:1) C þ M bm C m ˜ b − m ˜ b (cid:3) u þ ln m ˜ b Q × (cid:4) ð Y b − C Þ v − F − C M bm v m ˜ b − m ˜ b (cid:5) × (cid:1) C − M bm C m ˜ b − m ˜ b (cid:3) u (cid:8) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) Y − y d y η − Y − (cid:4)(cid:1) g YB g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) g v − ð g þ g YB g B Þ (cid:1) − g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) v − ð g B þ g YB g B Þ g YB g B v u − ffiffiffi p μ Y ν ;ii Y x;ij v (cid:3) (cid:8) þ ln m ˜ ν R Q y d R × (cid:1) g YB g B u þ ffiffiffi p T ijx þ u Y x;ij − Y − y η (cid:3)(cid:8) ; ð A7 Þ Δ m Φ ¯ η Φ d ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) C C v u ð m ˜ t − m ˜ t Þ þ M tm v C ð m ˜ t − m ˜ t Þ (cid:1) − C M tm u þ F (cid:3)(cid:5) − ln m ˜ t Q × (cid:1) C þ M tm C m ˜ t − m ˜ t (cid:3) u × (cid:4) ð Y t − C Þ v þ m ˜ t − m ˜ t (cid:1) − C M tm v þ F (cid:3)(cid:5) þ ln m ˜ t Q × (cid:1) − C þ M tm C m ˜ t − m ˜ t (cid:3) u (cid:4) ð Y t − C Þ v − m ˜ t − m ˜ t (cid:1) − C M tm v þ F (cid:3)(cid:5)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) C C v u ð m ˜ b − m ˜ b Þ þ M bm v C ð m ˜ b − m ˜ b Þ (cid:1) − C M bm u þ F (cid:3)(cid:5) − ln m ˜ b Q × (cid:1) C þ M bm C m ˜ b − m ˜ b (cid:3) u × (cid:4) ð Y b − C Þ v þ m ˜ b − m ˜ b (cid:1) − C M bm v þ F (cid:3)(cid:5) þ ln m ˜ b Q × (cid:1) − C þ M bm C m ˜ b − m ˜ b (cid:3) u (cid:4) ð Y b − C Þ v − m ˜ b − m ˜ b (cid:1) − C M bm v þ F (cid:3)(cid:5)(cid:8) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) Y − y d y ¯ η − Y − (cid:4)(cid:1) − g YB g B u − ffiffiffi p μ Y x;ij (cid:3) g v − ð g þ g YB g B Þ (cid:1) g B u − ffiffiffi p μ Y x;ij (cid:3) v − ð g B þ g YB g B Þ g YB g B v u (cid:5)(cid:8) þ ln m ˜ ν R Q y d R (cid:1) − g YB g B u − ffiffiffi p μ Y x;ij − Y − y ¯ η (cid:3)(cid:8) ; ð A8 Þ e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020) m Φ u Φ u ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) C v ð m ˜ t − m ˜ t Þ − v ð m ˜ t − m ˜ t Þ × (cid:1) C M tm þ A t Y t (cid:3) (cid:5) þ ln m ˜ t Q (cid:1) C v þ C M tm v þ F m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) C v − C M tm v þ F m ˜ t − m ˜ t (cid:3) (cid:8) − f ð Q ; m t Þ Y t þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) C v ð m ˜ b − m ˜ b Þ − v ð m ˜ b − m ˜ b Þ × (cid:1) C M bm þ A b Y b (cid:3) (cid:5) þ ln m ˜ b Q (cid:1) C v þ C M bm v þ F m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) C v − C M bm v þ F m ˜ b − m ˜ b (cid:3) (cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) − g þ Y ν ;ij (cid:3) þ f ð Q ; m ˜ ν R Þ (cid:1) Y − y u − Y − y u (cid:3) þ ln m ˜ ν R Q y u R (cid:5) ; ð A9 Þ Δ m Φ η Φ u ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) C C u v ð m ˜ t − m ˜ t Þ − M tm C u v ð m ˜ t − m ˜ t Þ (cid:1) C M tm þ A t Y t (cid:3)(cid:5) þ ln m ˜ t Q × (cid:1) C þ M tm C m ˜ t − m ˜ t (cid:3) u × (cid:1) C v þ C M tm v þ F m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q × (cid:1) C − M tm C m ˜ t − m ˜ t (cid:3) u × (cid:1) C v − C M tm v þ F m ˜ t − m ˜ t (cid:3)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) C C u v ð m ˜ b − m ˜ b Þ − M bm C u v ð m ˜ b − m ˜ b Þ (cid:1) C M bm þ A b Y b (cid:3)(cid:5) þ ln m ˜ b Q × (cid:1) C þ M bm C m ˜ b − m ˜ b (cid:3) u × (cid:1) C v þ C M bm v þ F m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q × (cid:1) C − M bm C m ˜ b − m ˜ b (cid:3) u × (cid:1) C v − C M bm v þ F m ˜ b − m ˜ b (cid:3)(cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) Y − y η y u − Y − y u η (cid:3) þ ln m ˜ ν R Q y u R y η R (cid:5) ; ð A10 Þ Δ m Φ ¯ η Φ u ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − C C v u ð m ˜ t − m ˜ t Þ þ M tm C u v ð m ˜ t − m ˜ t Þ (cid:1) C M tm þ F (cid:3)(cid:5) þ ln m ˜ t Q (cid:1) − C − M tm C m ˜ t − m ˜ t (cid:3) u × (cid:1) C v þ C M tm v þ F m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) − C þ M tm C m ˜ t − m ˜ t (cid:3) u × (cid:1) C v − C M tm v þ F m ˜ t − m ˜ t (cid:3)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − C C v u ð m ˜ b − m ˜ b Þ þ M bm C u v ð m ˜ b − m ˜ b Þ (cid:1) C M bm þ F (cid:3)(cid:5) þ ln m ˜ b Q (cid:1) − C − M bm C m ˜ b − m ˜ b (cid:3) u × (cid:1) C v þ C M bm v þ F m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) − C þ M bm C m ˜ b − m ˜ b (cid:3) u × (cid:1) C v − C M bm v þ F m ˜ b − m ˜ b (cid:3)(cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) Y − y ¯ η y u − Y − y u η (cid:3) þ ln m ˜ ν R Q y u R y ¯ η R (cid:5) ; ð A11 Þ CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D m Φ η Φ η ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) m ˜ t − m ˜ t − M tm ð m ˜ t − m ˜ t Þ (cid:5) C u þ ln m ˜ t Q (cid:1) C u þ M tm C u m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) C u − M tm C u m ˜ t − m ˜ t (cid:3) (cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) m ˜ b − m ˜ b − M bm ð m ˜ b − m ˜ b Þ (cid:5) C u þ ln m ˜ b Q (cid:1) C u þ M bm C u m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) C u − M bm C u m ˜ b − m ˜ b (cid:3) (cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) g YB g B þ Y x;ij (cid:3) þ f ð Q; m ˜ ν R Þ (cid:1) Y − y η − Y − y η (cid:3) þ ln m ˜ ν R Q y η R − f ð Q ; m ν R Þ Y x;ij þ ln m ν R Q u Y x;ij (cid:5) ; ð A12 Þ Δ m Φ ¯ η Φ η ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − m ˜ t − m ˜ t þ M tm ð m ˜ t − m ˜ t Þ (cid:5) C u u þ ln m ˜ t Q (cid:1) − C u − M tm C u m ˜ t − m ˜ t (cid:3) × (cid:1) C u þ M tm C u m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) − C u þ M tm C u m ˜ t − m ˜ t (cid:3) × (cid:1) C u − M tm C u m ˜ t − m ˜ t (cid:3)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − m ˜ b − m ˜ b þ M bm ð m ˜ b − m ˜ b Þ (cid:5) C u u þ ln m ˜ b Q (cid:1) − C u − M bm C u m ˜ b − m ˜ b (cid:3) × (cid:1) C u þ M bm C u m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) − C u þ M bm C u m ˜ b − m ˜ b (cid:3) × (cid:1) C u − M bm C u m ˜ b − m ˜ b (cid:3)(cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) Y − y η y ¯ η − Y − y η η (cid:3) þ ln m ˜ ν R Q y η R y ¯ η R (cid:5) ; ð A13 Þ Δ m Φ ¯ η Φ ¯ η ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) m ˜ t − m ˜ t − M tm ð m ˜ t − m ˜ t Þ (cid:5) C u þ ln m ˜ t Q (cid:1) − C u − M tm C u m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) − C u þ M tm C u m ˜ t − m ˜ t (cid:3) (cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) m ˜ b − m ˜ b − M bm ð m ˜ b − m ˜ b Þ (cid:5) C u þ ln m ˜ b Q (cid:1) − C u − M bm C u m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) − C u þ M bm C u m ˜ b − m ˜ b (cid:3) (cid:8) þ π (cid:4) − f ð Q ; m ˜ ν R Þ g YB g B þ f ð Q; m ˜ ν R Þ (cid:1) Y − y η − Y − y ¯ η (cid:3) þ ln m ˜ ν R Q y η R (cid:5) : ð A14 Þ Here, f ð Q ;m t; ˜ t ; Þ¼ m t; ˜ t ; ð ln m t; ˜ t ; Q − Þ , f ð Q ; m b; ˜ b ; Þ ¼ m b; ˜ b ; ð ln m b; ˜ b ; Q − Þ , f ð Q ; m ν iR ; ˜ ν iR Þ ¼ m ν iR ; ˜ ν iR ð ln m ν iR; ˜ ν iR Q − Þ ,and e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020) ¼ − g þ ð g þ g YB Þþ g YB g B ; C ¼ − g − ð g þ g YB Þþ Y t ; C ¼ − g − ð g þ g YB Þþ Y b ;C ¼ − g B − g YB g B ; C ¼ g YB g B ; F ¼ ð v A t − v μ Þ A t Y t ; F ¼ ð v A b − v μ Þ A b Y b ;F ¼ − A t Y t μ ; F ¼ ð v μ − v A t Þ μ Y t ; F ¼ ð v μ − v A b Þ μ Y b ; F ¼ μ Y t ; F ¼ μ Y b ;Y ¼ ð m ˜ ν ILL þ m ˜ ν IRR Þ − ð m ˜ ν ILL m ˜ ν IRR − m ˜ ν ILR Þ ; y d ¼ ð m ˜ ν ILL þ m ˜ ν IRR Þ g v − ð g þ g YB g B Þ v m ˜ ν IRR þ m ˜ ν ILL g YB g B ð v − ffiffiffi p μ Y ν ;ij Þ ;y u ¼ (cid:1) − g v þ v Y ν ;ij (cid:3) ð m ˜ ν ILL þ m ˜ ν IRR Þ − (cid:4) − ð g þ g YB g B Þ v m ˜ ν IRR þ v Y ν ;ij m ˜ ν IRR þ (cid:1) g YB g B þ Y ν ;ij (cid:3) m ˜ ν ILL v (cid:5) þð ffiffiffi p T ij ν þ u Y x;ij Y ν ;ij Þ m ˜ ν ILR ;y η ¼ ð g YB g B u þ ffiffiffi p T ijx þ u Y x;ij Þð m ˜ ν IRR þ m ˜ ν ILL Þ − ð g B þ g YB g B Þ u m ˜ ν IRR − m ˜ ν ILL ð − g B u þ ffiffiffi p T ijx þ u Y x;ij Þþ v Y x;ij Y ν ;ij m ˜ ν ILR ;y ¯ η ¼ ½ g B u − ffiffiffi p μ Y x;ij − ð g B þ g YB g B Þ u (cid:3)ð m ˜ ν IRR þ m ˜ ν ILL Þ − ð g B þ g YB g B Þ u m ˜ ν IRR þ m ˜ ν ILL ð g B u − ffiffiffi p μ Y x;ij Þ ;y d ¼ g ð m ˜ ν ILL þ m ˜ ν IRR Þþ g v − ð g þ g YB g B Þ m ˜ ν IRR þ g YB g B ð g þ g YB g B Þ v þ g YB g B m ˜ ν ILL þ μ Y ν ;ij ;y u ¼ (cid:1) Y ν ;ij − g (cid:3) ð m ˜ ν ILL þ m ˜ ν IRR Þþ (cid:1) Y ν ;ij − g (cid:3)(cid:1) Y ν ;ij − g (cid:3) v − (cid:4) − ð g þ g YB g B þ Y ν ;ij Þ m ˜ ν IRR þ m ˜ ν ILL (cid:1) g YB g B þ Y ν ;ij (cid:3) − ð g þ g YB g B þ Y ν ;ij Þ (cid:1) g YB g B þ Y ν ;ij (cid:3) v (cid:5) þ (cid:1) ffiffiffi p T ij ν þ u Y x;ij Y ν ;ij (cid:3) ;y η ¼ ð g YB g B þ Y x;ij Þð m ˜ ν ILL þ m ˜ ν IRR Þ − ð g B þ g YB g B Þ m ˜ ν IRR − u ð g B þ g YB g B Þ (cid:1) − g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) − ð g B þ g YB g B Þ (cid:1) − g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) u − m ˜ ν ILL ð − g B þ Y x;ij Þþð v Y x;ij Y ν ;ij Þ ;y ¯ η ¼ − g YB g B ð m ˜ ν ILL þ m ˜ ν IRR Þþ ð g B þ g YB g B Þ m ˜ ν IRR þ g B m ˜ ν ILL þ u ð g B þ g YB g B Þð g B u − ffiffiffi p μ Y x;ij Þþð − g B u þ ffiffiffi p μ Y x;ij Þð g B þ g YB g B Þ u ;y u η ¼ (cid:1) − g v þ v Y ν ;ij (cid:3)(cid:1) g YB g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) − ð g þ g YB g B þ Y ν ;ij Þ (cid:1) g B u − ffiffiffi p T ijx − u Y x;ij (cid:3) v − ð g B þ g YB g B Þ (cid:1) g YB g B þ Y ν ;ij (cid:3) v u þ (cid:1) u Y x;ij Y ν ;ij − ffiffiffi p μ Y ν ;ij (cid:3) v ;y η η ¼ − ð g B þ g YB g B Þ (cid:1) g B u − ffiffiffi p μ Y x;ij (cid:3) u − ð g B þ g YB g B Þð − g B u þ ffiffiffi p T ijx þ u Y x;ij Þ u ;y d R ¼ g v − Y − y d ; y u R ¼ − g v − Y − y u ; y η R ¼ g YB g B u þ ffiffiffi p T ijx þ u Y x;ij − Y − y η ;y ¯ η R ¼ − g YB g B u − ffiffiffi p μ Y x;ij − Y − y ¯ η : ð A15 Þ In addition, m ˜ t ; ¼ M ts (cid:4) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M tm þ M tLR q ; ð A16 Þ M ts ¼ ð g v cos β þ g YB g B u cos β Þ þ m ˜ q ; þ v Y t ; ð A17 Þ CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D tm ¼ f½ g − ð g þ g YB Þ − g YB g B (cid:3) v cos β þ ð g B − g YB g B Þ u cos β þ m ˜ u ; þ v Y t g ; ð A18 Þ M tLR ¼ ð − v μ Y t þ v A t Y t Þ ; ð A19 Þ m ˜ b ; ¼ M bs (cid:4) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M bm þ M bLR q ; ð A20 Þ M bs ¼ ð g v cos β þ g YB g B u cos β Þ þ m ˜ q ; þ v Y b ; ð A21 Þ M bm ¼ f½ g − ð g þ g YB Þ − g YB g B (cid:3) v cos β þ ð g B − g YB g B Þ u cos β þ m ˜ d ; þ v Y b g ; ð A22 Þ M bLR ¼ ð − v μ Y b þ v A b Y b Þ ; ð A23 Þ and m ν R ¼ u Y x;ij : ð A24 Þ m ˜ ν R ¼ ð m ˜ ν ILL þ m ˜ ν IRR Þ − ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð m ˜ ν ILL þ m ˜ ν IRR Þ − ð m ˜ ν ILL m ˜ ν IRR − m ˜ ν ILR Þ q ; ð A25 Þ m ˜ ν ILL ¼ ð g B þ g YB g B Þ u cos β þ ð g þ g YB g B Þ v cos β þ m ˜ L;ij þ v Y ν ;ij ; ð A26 Þ m ˜ ν IRR ¼ − g B u cos β − g YB g B v cos β þ m ˜ ν ;ij − ffiffiffi p u μ Y x;ij þ v Y ν ;ij þ ffiffiffi p u T ijx ; ð A27 Þ m ˜ ν ILR ¼ ffiffiffi p v T ij ν − ffiffiffi p μ v Y ν ;ij þ u v Y ν ;ij Y x;ij : ð A28 Þ APPENDIX B: THE TREE LEVEL CORRECTION OF HIGGS SELF-COUPLING λ ð Þ h i h j h k ¼ ð Z Hi ðð g þ g YB þ g Þ Z Hj ð v Z Hk þ v Z Hk Þ − g YB g B ð Z Hj ð v Z Hk þ u Z Hk Þ − Z Hj ð u Z Hk þ v Z Hk ÞÞþ Z Hj ð − g YB g B ð − u Z Hk þ u Z Hk Þ − ð g þ g YB þ g Þ v Z Hk þ ð g þ g YB þ g Þ v Z Hk ÞÞþ Z Hi ðð g þ g YB þ g Þ Z Hj ð v Z Hk þ v Z Hk Þ þ g YB g B ð Z Hj ð u Z Hk þ v Z Hk Þ − Z Hj ð u Z Hk þ v Z Hk ÞÞþ Z Hj ð g YB g B Þð − u Z Hk þ u Z Hk Þ − ð g þ g YB þ g Þ v Z Hk þ ð g þ g YB þ g Þ v Z Hk ÞÞ − ð − Z Hi ð − g YB g B u Z Hj Z Hk þ g B u Z Hj Z Hk − g YB g B v Z Hj Z Hk þ g B u Z Hj Z Hk þ g YB g B Z Hj ð u Z Hk þ v Z Hk Þþ Z Hj g B ð − u Z Hk þ u Z Hk Þ þ g YB g B v Z Hk − g YB g B v Z Hk ÞÞþ Z Hi ð − g YB g B u Z Hj Z Hk − g YB g B v Z Hj Z Hk − g B u Z Hj Z Hk þ g YB g B Z Hj ð v Z Hk þ u Z Hk Þ − g B u Z Hj Z Hk þ Z Hj ð g B ð u Z Hk − u Z Hk Þ þ g YB g B v Z Hk − g YB g B v Z Hk ÞÞÞÞ : ð B1 Þ e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020)
Þ þ g v − g m ˜ ν IRR − ð g þ g YB g B Þ g YB g B v þ μ Y ν ;ij (cid:5)(cid:8) þ ln m ˜ ν R Q y d R (cid:8) ; ð A5 Þ Δ m Φ u Φ d ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − C v v ð m ˜ t − m ˜ t Þ þ M tm C v − F ð m ˜ t − m ˜ t Þ ð M tm C v þ F Þ (cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v þ F − C M tm v ð m ˜ t − m ˜ t Þ (cid:1) C v þ F þ C M tm v m ˜ t − m ˜ t (cid:3)(cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v − F − C M tm v ð m ˜ t − m ˜ t Þ (cid:1) C v − F þ C M tm v m ˜ t − m ˜ t (cid:3)(cid:5)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − C v v ð m ˜ b − m ˜ b Þ þ M bm C v − F ð m ˜ b − m ˜ b Þ ð M bm C v þ F (cid:3)(cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v þ F − C M bm v ð m ˜ b − m ˜ b Þ (cid:1) C v þ F þ C M bm v m ˜ b − m ˜ b (cid:3)(cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v − F − C M bm v ð m ˜ b − m ˜ b Þ (cid:1) C v − F þ C M bm v m ˜ b − m ˜ b (cid:3)(cid:5)(cid:8) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) Y − y d y u − Y − (cid:4) − ð T ij ν þ ffiffiffi p u Y x;ij Y ν ;ij Þ μ Y ν ;ij − (cid:1) g þ g YB g B − Y ν ;ij (cid:3) g YB g B v v − ð g þ g YB g B Þ (cid:1) g YB g B þ Y ν ;ij (cid:3) v v þ (cid:1) Y ν ;ij − g v (cid:3) g v v (cid:5)(cid:8) þ ln m ˜ ν R Q y d R (cid:1) − g v þ v Y ν ;ij − Y − y u (cid:3)(cid:8) ; ð A6 Þ CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D m Φ η Φ d ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − C C v u m ˜ t − m ˜ t − M tm C u ð F − C M tm v Þð m ˜ t − m ˜ t Þ (cid:5) þ ln m ˜ t Q (cid:4) ð Y t − C Þ v þ F − C M tm v m ˜ t − m ˜ t (cid:5) × (cid:1) C þ M tm C m ˜ t − m ˜ t (cid:3) u þ ln m ˜ t Q × (cid:4) ð Y t − C Þ v − F − C M tm v m ˜ t − m ˜ t (cid:5) × (cid:1) C − M tm C m ˜ t − m ˜ t (cid:3) u (cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − C C v u m ˜ b − m ˜ b − M bm C u ð F − C M bm v Þð m ˜ b − m ˜ b Þ (cid:5) þ ln m ˜ b Q (cid:4) ð Y b − C Þ v þ F − C M bm v m ˜ b − m ˜ b (cid:5) × (cid:1) C þ M bm C m ˜ b − m ˜ b (cid:3) u þ ln m ˜ b Q × (cid:4) ð Y b − C Þ v − F − C M bm v m ˜ b − m ˜ b (cid:5) × (cid:1) C − M bm C m ˜ b − m ˜ b (cid:3) u (cid:8) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) Y − y d y η − Y − (cid:4)(cid:1) g YB g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) g v − ð g þ g YB g B Þ (cid:1) − g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) v − ð g B þ g YB g B Þ g YB g B v u − ffiffiffi p μ Y ν ;ii Y x;ij v (cid:3) (cid:8) þ ln m ˜ ν R Q y d R × (cid:1) g YB g B u þ ffiffiffi p T ijx þ u Y x;ij − Y − y η (cid:3)(cid:8) ; ð A7 Þ Δ m Φ ¯ η Φ d ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) C C v u ð m ˜ t − m ˜ t Þ þ M tm v C ð m ˜ t − m ˜ t Þ (cid:1) − C M tm u þ F (cid:3)(cid:5) − ln m ˜ t Q × (cid:1) C þ M tm C m ˜ t − m ˜ t (cid:3) u × (cid:4) ð Y t − C Þ v þ m ˜ t − m ˜ t (cid:1) − C M tm v þ F (cid:3)(cid:5) þ ln m ˜ t Q × (cid:1) − C þ M tm C m ˜ t − m ˜ t (cid:3) u (cid:4) ð Y t − C Þ v − m ˜ t − m ˜ t (cid:1) − C M tm v þ F (cid:3)(cid:5)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) C C v u ð m ˜ b − m ˜ b Þ þ M bm v C ð m ˜ b − m ˜ b Þ (cid:1) − C M bm u þ F (cid:3)(cid:5) − ln m ˜ b Q × (cid:1) C þ M bm C m ˜ b − m ˜ b (cid:3) u × (cid:4) ð Y b − C Þ v þ m ˜ b − m ˜ b (cid:1) − C M bm v þ F (cid:3)(cid:5) þ ln m ˜ b Q × (cid:1) − C þ M bm C m ˜ b − m ˜ b (cid:3) u (cid:4) ð Y b − C Þ v − m ˜ b − m ˜ b (cid:1) − C M bm v þ F (cid:3)(cid:5)(cid:8) þ π (cid:7) f ð Q ; m ˜ ν R Þ (cid:7) Y − y d y ¯ η − Y − (cid:4)(cid:1) − g YB g B u − ffiffiffi p μ Y x;ij (cid:3) g v − ð g þ g YB g B Þ (cid:1) g B u − ffiffiffi p μ Y x;ij (cid:3) v − ð g B þ g YB g B Þ g YB g B v u (cid:5)(cid:8) þ ln m ˜ ν R Q y d R (cid:1) − g YB g B u − ffiffiffi p μ Y x;ij − Y − y ¯ η (cid:3)(cid:8) ; ð A8 Þ e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020) m Φ u Φ u ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) C v ð m ˜ t − m ˜ t Þ − v ð m ˜ t − m ˜ t Þ × (cid:1) C M tm þ A t Y t (cid:3) (cid:5) þ ln m ˜ t Q (cid:1) C v þ C M tm v þ F m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) C v − C M tm v þ F m ˜ t − m ˜ t (cid:3) (cid:8) − f ð Q ; m t Þ Y t þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) C v ð m ˜ b − m ˜ b Þ − v ð m ˜ b − m ˜ b Þ × (cid:1) C M bm þ A b Y b (cid:3) (cid:5) þ ln m ˜ b Q (cid:1) C v þ C M bm v þ F m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) C v − C M bm v þ F m ˜ b − m ˜ b (cid:3) (cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) − g þ Y ν ;ij (cid:3) þ f ð Q ; m ˜ ν R Þ (cid:1) Y − y u − Y − y u (cid:3) þ ln m ˜ ν R Q y u R (cid:5) ; ð A9 Þ Δ m Φ η Φ u ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) C C u v ð m ˜ t − m ˜ t Þ − M tm C u v ð m ˜ t − m ˜ t Þ (cid:1) C M tm þ A t Y t (cid:3)(cid:5) þ ln m ˜ t Q × (cid:1) C þ M tm C m ˜ t − m ˜ t (cid:3) u × (cid:1) C v þ C M tm v þ F m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q × (cid:1) C − M tm C m ˜ t − m ˜ t (cid:3) u × (cid:1) C v − C M tm v þ F m ˜ t − m ˜ t (cid:3)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) C C u v ð m ˜ b − m ˜ b Þ − M bm C u v ð m ˜ b − m ˜ b Þ (cid:1) C M bm þ A b Y b (cid:3)(cid:5) þ ln m ˜ b Q × (cid:1) C þ M bm C m ˜ b − m ˜ b (cid:3) u × (cid:1) C v þ C M bm v þ F m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q × (cid:1) C − M bm C m ˜ b − m ˜ b (cid:3) u × (cid:1) C v − C M bm v þ F m ˜ b − m ˜ b (cid:3)(cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) Y − y η y u − Y − y u η (cid:3) þ ln m ˜ ν R Q y u R y η R (cid:5) ; ð A10 Þ Δ m Φ ¯ η Φ u ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − C C v u ð m ˜ t − m ˜ t Þ þ M tm C u v ð m ˜ t − m ˜ t Þ (cid:1) C M tm þ F (cid:3)(cid:5) þ ln m ˜ t Q (cid:1) − C − M tm C m ˜ t − m ˜ t (cid:3) u × (cid:1) C v þ C M tm v þ F m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) − C þ M tm C m ˜ t − m ˜ t (cid:3) u × (cid:1) C v − C M tm v þ F m ˜ t − m ˜ t (cid:3)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − C C v u ð m ˜ b − m ˜ b Þ þ M bm C u v ð m ˜ b − m ˜ b Þ (cid:1) C M bm þ F (cid:3)(cid:5) þ ln m ˜ b Q (cid:1) − C − M bm C m ˜ b − m ˜ b (cid:3) u × (cid:1) C v þ C M bm v þ F m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) − C þ M bm C m ˜ b − m ˜ b (cid:3) u × (cid:1) C v − C M bm v þ F m ˜ b − m ˜ b (cid:3)(cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) Y − y ¯ η y u − Y − y u η (cid:3) þ ln m ˜ ν R Q y u R y ¯ η R (cid:5) ; ð A11 Þ CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D m Φ η Φ η ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) m ˜ t − m ˜ t − M tm ð m ˜ t − m ˜ t Þ (cid:5) C u þ ln m ˜ t Q (cid:1) C u þ M tm C u m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) C u − M tm C u m ˜ t − m ˜ t (cid:3) (cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) m ˜ b − m ˜ b − M bm ð m ˜ b − m ˜ b Þ (cid:5) C u þ ln m ˜ b Q (cid:1) C u þ M bm C u m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) C u − M bm C u m ˜ b − m ˜ b (cid:3) (cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) g YB g B þ Y x;ij (cid:3) þ f ð Q; m ˜ ν R Þ (cid:1) Y − y η − Y − y η (cid:3) þ ln m ˜ ν R Q y η R − f ð Q ; m ν R Þ Y x;ij þ ln m ν R Q u Y x;ij (cid:5) ; ð A12 Þ Δ m Φ ¯ η Φ η ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) − m ˜ t − m ˜ t þ M tm ð m ˜ t − m ˜ t Þ (cid:5) C u u þ ln m ˜ t Q (cid:1) − C u − M tm C u m ˜ t − m ˜ t (cid:3) × (cid:1) C u þ M tm C u m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) − C u þ M tm C u m ˜ t − m ˜ t (cid:3) × (cid:1) C u − M tm C u m ˜ t − m ˜ t (cid:3)(cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) − m ˜ b − m ˜ b þ M bm ð m ˜ b − m ˜ b Þ (cid:5) C u u þ ln m ˜ b Q (cid:1) − C u − M bm C u m ˜ b − m ˜ b (cid:3) × (cid:1) C u þ M bm C u m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) − C u þ M bm C u m ˜ b − m ˜ b (cid:3) × (cid:1) C u − M bm C u m ˜ b − m ˜ b (cid:3)(cid:8) þ π (cid:4) f ð Q ; m ˜ ν R Þ (cid:1) Y − y η y ¯ η − Y − y η η (cid:3) þ ln m ˜ ν R Q y η R y ¯ η R (cid:5) ; ð A13 Þ Δ m Φ ¯ η Φ ¯ η ¼ π (cid:7) ½ f ð Q ; m ˜ t Þ − f ð Q ; m ˜ t Þ(cid:3) (cid:4) m ˜ t − m ˜ t − M tm ð m ˜ t − m ˜ t Þ (cid:5) C u þ ln m ˜ t Q (cid:1) − C u − M tm C u m ˜ t − m ˜ t (cid:3) þ ln m ˜ t Q (cid:1) − C u þ M tm C u m ˜ t − m ˜ t (cid:3) (cid:8) þ π (cid:7) ½ f ð Q ; m ˜ b Þ − f ð Q ; m ˜ b Þ(cid:3) (cid:4) m ˜ b − m ˜ b − M bm ð m ˜ b − m ˜ b Þ (cid:5) C u þ ln m ˜ b Q (cid:1) − C u − M bm C u m ˜ b − m ˜ b (cid:3) þ ln m ˜ b Q (cid:1) − C u þ M bm C u m ˜ b − m ˜ b (cid:3) (cid:8) þ π (cid:4) − f ð Q ; m ˜ ν R Þ g YB g B þ f ð Q; m ˜ ν R Þ (cid:1) Y − y η − Y − y ¯ η (cid:3) þ ln m ˜ ν R Q y η R (cid:5) : ð A14 Þ Here, f ð Q ;m t; ˜ t ; Þ¼ m t; ˜ t ; ð ln m t; ˜ t ; Q − Þ , f ð Q ; m b; ˜ b ; Þ ¼ m b; ˜ b ; ð ln m b; ˜ b ; Q − Þ , f ð Q ; m ν iR ; ˜ ν iR Þ ¼ m ν iR ; ˜ ν iR ð ln m ν iR; ˜ ν iR Q − Þ ,and e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020) ¼ − g þ ð g þ g YB Þþ g YB g B ; C ¼ − g − ð g þ g YB Þþ Y t ; C ¼ − g − ð g þ g YB Þþ Y b ;C ¼ − g B − g YB g B ; C ¼ g YB g B ; F ¼ ð v A t − v μ Þ A t Y t ; F ¼ ð v A b − v μ Þ A b Y b ;F ¼ − A t Y t μ ; F ¼ ð v μ − v A t Þ μ Y t ; F ¼ ð v μ − v A b Þ μ Y b ; F ¼ μ Y t ; F ¼ μ Y b ;Y ¼ ð m ˜ ν ILL þ m ˜ ν IRR Þ − ð m ˜ ν ILL m ˜ ν IRR − m ˜ ν ILR Þ ; y d ¼ ð m ˜ ν ILL þ m ˜ ν IRR Þ g v − ð g þ g YB g B Þ v m ˜ ν IRR þ m ˜ ν ILL g YB g B ð v − ffiffiffi p μ Y ν ;ij Þ ;y u ¼ (cid:1) − g v þ v Y ν ;ij (cid:3) ð m ˜ ν ILL þ m ˜ ν IRR Þ − (cid:4) − ð g þ g YB g B Þ v m ˜ ν IRR þ v Y ν ;ij m ˜ ν IRR þ (cid:1) g YB g B þ Y ν ;ij (cid:3) m ˜ ν ILL v (cid:5) þð ffiffiffi p T ij ν þ u Y x;ij Y ν ;ij Þ m ˜ ν ILR ;y η ¼ ð g YB g B u þ ffiffiffi p T ijx þ u Y x;ij Þð m ˜ ν IRR þ m ˜ ν ILL Þ − ð g B þ g YB g B Þ u m ˜ ν IRR − m ˜ ν ILL ð − g B u þ ffiffiffi p T ijx þ u Y x;ij Þþ v Y x;ij Y ν ;ij m ˜ ν ILR ;y ¯ η ¼ ½ g B u − ffiffiffi p μ Y x;ij − ð g B þ g YB g B Þ u (cid:3)ð m ˜ ν IRR þ m ˜ ν ILL Þ − ð g B þ g YB g B Þ u m ˜ ν IRR þ m ˜ ν ILL ð g B u − ffiffiffi p μ Y x;ij Þ ;y d ¼ g ð m ˜ ν ILL þ m ˜ ν IRR Þþ g v − ð g þ g YB g B Þ m ˜ ν IRR þ g YB g B ð g þ g YB g B Þ v þ g YB g B m ˜ ν ILL þ μ Y ν ;ij ;y u ¼ (cid:1) Y ν ;ij − g (cid:3) ð m ˜ ν ILL þ m ˜ ν IRR Þþ (cid:1) Y ν ;ij − g (cid:3)(cid:1) Y ν ;ij − g (cid:3) v − (cid:4) − ð g þ g YB g B þ Y ν ;ij Þ m ˜ ν IRR þ m ˜ ν ILL (cid:1) g YB g B þ Y ν ;ij (cid:3) − ð g þ g YB g B þ Y ν ;ij Þ (cid:1) g YB g B þ Y ν ;ij (cid:3) v (cid:5) þ (cid:1) ffiffiffi p T ij ν þ u Y x;ij Y ν ;ij (cid:3) ;y η ¼ ð g YB g B þ Y x;ij Þð m ˜ ν ILL þ m ˜ ν IRR Þ − ð g B þ g YB g B Þ m ˜ ν IRR − u ð g B þ g YB g B Þ (cid:1) − g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) − ð g B þ g YB g B Þ (cid:1) − g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) u − m ˜ ν ILL ð − g B þ Y x;ij Þþð v Y x;ij Y ν ;ij Þ ;y ¯ η ¼ − g YB g B ð m ˜ ν ILL þ m ˜ ν IRR Þþ ð g B þ g YB g B Þ m ˜ ν IRR þ g B m ˜ ν ILL þ u ð g B þ g YB g B Þð g B u − ffiffiffi p μ Y x;ij Þþð − g B u þ ffiffiffi p μ Y x;ij Þð g B þ g YB g B Þ u ;y u η ¼ (cid:1) − g v þ v Y ν ;ij (cid:3)(cid:1) g YB g B u þ ffiffiffi p T ijx þ u Y x;ij (cid:3) − ð g þ g YB g B þ Y ν ;ij Þ (cid:1) g B u − ffiffiffi p T ijx − u Y x;ij (cid:3) v − ð g B þ g YB g B Þ (cid:1) g YB g B þ Y ν ;ij (cid:3) v u þ (cid:1) u Y x;ij Y ν ;ij − ffiffiffi p μ Y ν ;ij (cid:3) v ;y η η ¼ − ð g B þ g YB g B Þ (cid:1) g B u − ffiffiffi p μ Y x;ij (cid:3) u − ð g B þ g YB g B Þð − g B u þ ffiffiffi p T ijx þ u Y x;ij Þ u ;y d R ¼ g v − Y − y d ; y u R ¼ − g v − Y − y u ; y η R ¼ g YB g B u þ ffiffiffi p T ijx þ u Y x;ij − Y − y η ;y ¯ η R ¼ − g YB g B u − ffiffiffi p μ Y x;ij − Y − y ¯ η : ð A15 Þ In addition, m ˜ t ; ¼ M ts (cid:4) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M tm þ M tLR q ; ð A16 Þ M ts ¼ ð g v cos β þ g YB g B u cos β Þ þ m ˜ q ; þ v Y t ; ð A17 Þ CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D tm ¼ f½ g − ð g þ g YB Þ − g YB g B (cid:3) v cos β þ ð g B − g YB g B Þ u cos β þ m ˜ u ; þ v Y t g ; ð A18 Þ M tLR ¼ ð − v μ Y t þ v A t Y t Þ ; ð A19 Þ m ˜ b ; ¼ M bs (cid:4) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M bm þ M bLR q ; ð A20 Þ M bs ¼ ð g v cos β þ g YB g B u cos β Þ þ m ˜ q ; þ v Y b ; ð A21 Þ M bm ¼ f½ g − ð g þ g YB Þ − g YB g B (cid:3) v cos β þ ð g B − g YB g B Þ u cos β þ m ˜ d ; þ v Y b g ; ð A22 Þ M bLR ¼ ð − v μ Y b þ v A b Y b Þ ; ð A23 Þ and m ν R ¼ u Y x;ij : ð A24 Þ m ˜ ν R ¼ ð m ˜ ν ILL þ m ˜ ν IRR Þ − ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð m ˜ ν ILL þ m ˜ ν IRR Þ − ð m ˜ ν ILL m ˜ ν IRR − m ˜ ν ILR Þ q ; ð A25 Þ m ˜ ν ILL ¼ ð g B þ g YB g B Þ u cos β þ ð g þ g YB g B Þ v cos β þ m ˜ L;ij þ v Y ν ;ij ; ð A26 Þ m ˜ ν IRR ¼ − g B u cos β − g YB g B v cos β þ m ˜ ν ;ij − ffiffiffi p u μ Y x;ij þ v Y ν ;ij þ ffiffiffi p u T ijx ; ð A27 Þ m ˜ ν ILR ¼ ffiffiffi p v T ij ν − ffiffiffi p μ v Y ν ;ij þ u v Y ν ;ij Y x;ij : ð A28 Þ APPENDIX B: THE TREE LEVEL CORRECTION OF HIGGS SELF-COUPLING λ ð Þ h i h j h k ¼ ð Z Hi ðð g þ g YB þ g Þ Z Hj ð v Z Hk þ v Z Hk Þ − g YB g B ð Z Hj ð v Z Hk þ u Z Hk Þ − Z Hj ð u Z Hk þ v Z Hk ÞÞþ Z Hj ð − g YB g B ð − u Z Hk þ u Z Hk Þ − ð g þ g YB þ g Þ v Z Hk þ ð g þ g YB þ g Þ v Z Hk ÞÞþ Z Hi ðð g þ g YB þ g Þ Z Hj ð v Z Hk þ v Z Hk Þ þ g YB g B ð Z Hj ð u Z Hk þ v Z Hk Þ − Z Hj ð u Z Hk þ v Z Hk ÞÞþ Z Hj ð g YB g B Þð − u Z Hk þ u Z Hk Þ − ð g þ g YB þ g Þ v Z Hk þ ð g þ g YB þ g Þ v Z Hk ÞÞ − ð − Z Hi ð − g YB g B u Z Hj Z Hk þ g B u Z Hj Z Hk − g YB g B v Z Hj Z Hk þ g B u Z Hj Z Hk þ g YB g B Z Hj ð u Z Hk þ v Z Hk Þþ Z Hj g B ð − u Z Hk þ u Z Hk Þ þ g YB g B v Z Hk − g YB g B v Z Hk ÞÞþ Z Hi ð − g YB g B u Z Hj Z Hk − g YB g B v Z Hj Z Hk − g B u Z Hj Z Hk þ g YB g B Z Hj ð v Z Hk þ u Z Hk Þ − g B u Z Hj Z Hk þ Z Hj ð g B ð u Z Hk − u Z Hk Þ þ g YB g B v Z Hk − g YB g B v Z Hk ÞÞÞÞ : ð B1 Þ e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020)
PPENDIX C: THE COUPLING OF TWO CP -ODD HIGGS AND ONE CP -EVEN HIGGS λ A i A j h k ¼ ∂ V ∂ ð ImH Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂ ImH ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ Im ˜ η ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂ ImH ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ Im ˜ η ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂ ImH ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ Im ˜ η ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂ ImH ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ Im ˜ η ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k : ð C1 Þ Here, the detailed expression about tree level correction λ ð Þ A i A j h k is written as: λ ð Þ A i A j h k ¼ ð Z Ai Z Aj ð − g YB g B ð − u Z Hk þ u Z Hk Þ − ð g þ g YB þ g Þ v Z Hk þ ð g þ g YB þ g Þ v Z Hk Þþ Z Ai Z Aj ð g YB g B ð − u Z Hk þ u Z Hk Þ þ ð g þ g YB þ g Þ v Z Hk − ð g þ g YB þ g Þ v Z Hk Þ − ð Z Ai Z Aj − Z Ai Z Aj Þð g B ð − u Z Hk þ u Z Hk Þ þ g YB g B v Z Hk − g YB g B v Z Hk ÞÞ : ð C2 Þ APPENDIX D: THE WILSON COEFFICIENTS OFTHE PROCESS e + e − → hhZ C ð Þ L;R denotes the Wilson coefficient corresponding toFig. 1: C ð Þ L;R ¼ C a L;R þ C b L;R þ C c L;R : ð D1 Þ C a L;R ¼ i λ h i h j h k g NZh g eeN ; ½ð p þ p Þ − m N (cid:3)½ð q þ q Þ − m h i (cid:3) ;C b L;R ¼ ig NZh g NNh g eeN ; ½ð p þ p Þ − m N (cid:3)½ð p þ p − q Þ − m N (cid:3) ;C c L;R ¼ − ig ZZhh g eeZ ; ð p þ p Þ − m Z : ð D2 Þ CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D ffective operator: O L;R ¼ ¯ v ð p Þð γ α Þ p L;R u ð p Þ ε (cid:2) α ð k Þ : ð D3 Þ C ð Þ L;R represents the Wilson coefficient corresponding toFig. 2: C ð Þ L;R ¼ C a L;R þ C b L;R þ C c L;R þ C d L;R : ð D4 Þ C a L;R ¼ − ig ZhA g NhA g eeN ; ½ð p þ p Þ − m N (cid:3)½ð p þ p − q Þ − m A k (cid:3) ;C b L;R ¼ i λ h i h j h k g ZhA g eeA ; ½ð p þ p Þ − m A k (cid:3)½ð p þ p − k Þ − m h i (cid:3) ;C c L;R ¼ − i λ A i A j h k g eeA ; g ZhA ½ð p þ p Þ − m A k (cid:3)½ð p þ p − q Þ − m A i (cid:3) ;C d L;R ¼ − ig NZh g eeA ; g NhA ½ð p þ p Þ − m A k (cid:3)½ð p þ p − q Þ − m N (cid:3) : ð D5 Þ Effective operator: O L;R ¼ ¯ v ð p Þð p α þ p α Þ p L;R u ð p Þ ε (cid:2) α ð k Þ ; O L;R ¼ ¯ v ð p Þð q α − p α − p α − q α Þ p L;R u ð p Þ ε (cid:2) α ð k Þ ; ð D6 Þ O L;R , O L;R have been given in Eq. (D3) and Eq. (27),respectively. Furthermore, λ h i h j h k has been given in Eq. (26)and Appendix B, and λ A i A j h k denotes the triple Higgs self-coupling of CP -even and CP -odd Higgs; the detailedexpression about this has been given in Appendix C.The superscripts (a, b, c, a ; b ; c ; d ) respectively representthe corresponding Feynman diagram labels in Figs. 1 and 2,and m h , m A denote the masses for Higgs and pseudoscalarHiggs, with i , k ¼ , 2, 3, 4 denoting the index ofgeneration. g ZZhh ¼ g cos θ W cos θ W Z Hi Z Hj þ g g cos θ W cos θ W sin θ W Z Hi Z Hj þ g cos θ W sin θ W Z Hi Z Hj − g YB g cos θ W cos θ W sin θ W Z Hi Z Hj − g g YB cos θ W sin θ W sin θ W Z Hi Z Hj þ g YB sin θ W Z Hi Z Hj þ g cos θ W cos θ W Z Hi Z Hj þ g g cos θ W cos θ W sin θ W Z Hi Z Hj þ g cos θ W sin θ W Z Hi Z Hj − g YB g cos θ W cos θ W sin θ W Z Hi Z Hj − g g YB cos θ W sin θ W sin θ W Z Hi Z Hj þ g YB sin θ W Z Hi Z Hj þ þ g B sin θ W Z Hi Z Hj þ g B sin θ W Z Hi Z Hj : ð D7 Þ g eeZ ¼ − g cos θ W sin θ W þ g cos θ W cos θ W þ ð g YB þ g B Þ sin θ W ;g eeZ ¼ − g cos θ W sin θ W þ ð g YB þ g B Þ sin θ W ;g eeZ ¼ ðð g sin θ W − g cos θ W Þ sin θ W þ ð g YB þ g B Þ cos θ W Þ ;g eeZ ¼ g sin θ W sin θ W − ð g YB þ g B Þ cos θ W : ð D8 Þ g ZZh ¼ ð v ð g cos θ W sin θ W þ g cos θ W cos θ W − g YB sin θ W Þ Z Hi þ v ð g cos θ W sin θ W þ g cos θ W cos θ W − g YB sin θ W Þ Z Hi þ g B sin θ W ð u Z Hi þ u Z Hi ÞÞ ;g Z Z h ¼ ð v ðð g sin θ W þ g cos θ W Þ sin θ W þ g YB cos θ W Þ Z Hi þ v ðð g sin θ W þ g cos θ W Þ sin θ W þ g YB cos θ W Þ Z Hi þ g B cos θ W ð u Z Hi þ u Z Hi ÞÞ ;g ZZ h ¼ ð − v ð g g YB cos θ W sin θ W þ g cos θ W cos θ W sin θ W þ cos θ W ð g sin θ W − g YB Þ sin θ W − g g YB sin θ W sin θ W þ g cos θ W ð g sin θ W sin θ W þ g YB cos θ W − g YB sin θ W ÞÞ Z Hi − v ð g g YB cos θ W sin θ W þ g cos θ W cos θ W sin θ W þ cos θ W ð g sin θ W − g YB Þ sin θ W − g g YB sin θ W sin θ W þ g cos θ W ð g sin θ W sin θ W þ g YB cos θ W − g YB sin θ W ÞÞ Z Hi þ g B sin θ W ð u Z Hi þ u Z Hi ÞÞ : ð D9 Þ e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020)
PPENDIX C: THE COUPLING OF TWO CP -ODD HIGGS AND ONE CP -EVEN HIGGS λ A i A j h k ¼ ∂ V ∂ ð ImH Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂ ImH ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ Im ˜ η ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂ ImH ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ Im ˜ η ∂∂ Im ˜ η ∂ ReH Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂ ImH ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ Im ˜ η ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ImH Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ð ∂ Im ˜ η Þ ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂ ImH ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ ImH ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k þ ∂ V ∂ Im ˜ η ∂∂ Im ˜ η ∂ Re ˜ η Z A i Z A j Z H k : ð C1 Þ Here, the detailed expression about tree level correction λ ð Þ A i A j h k is written as: λ ð Þ A i A j h k ¼ ð Z Ai Z Aj ð − g YB g B ð − u Z Hk þ u Z Hk Þ − ð g þ g YB þ g Þ v Z Hk þ ð g þ g YB þ g Þ v Z Hk Þþ Z Ai Z Aj ð g YB g B ð − u Z Hk þ u Z Hk Þ þ ð g þ g YB þ g Þ v Z Hk − ð g þ g YB þ g Þ v Z Hk Þ − ð Z Ai Z Aj − Z Ai Z Aj Þð g B ð − u Z Hk þ u Z Hk Þ þ g YB g B v Z Hk − g YB g B v Z Hk ÞÞ : ð C2 Þ APPENDIX D: THE WILSON COEFFICIENTS OFTHE PROCESS e + e − → hhZ C ð Þ L;R denotes the Wilson coefficient corresponding toFig. 1: C ð Þ L;R ¼ C a L;R þ C b L;R þ C c L;R : ð D1 Þ C a L;R ¼ i λ h i h j h k g NZh g eeN ; ½ð p þ p Þ − m N (cid:3)½ð q þ q Þ − m h i (cid:3) ;C b L;R ¼ ig NZh g NNh g eeN ; ½ð p þ p Þ − m N (cid:3)½ð p þ p − q Þ − m N (cid:3) ;C c L;R ¼ − ig ZZhh g eeZ ; ð p þ p Þ − m Z : ð D2 Þ CUI, FENG, YAN, ZHANG, NING, and YANG PHYS. REV. D ffective operator: O L;R ¼ ¯ v ð p Þð γ α Þ p L;R u ð p Þ ε (cid:2) α ð k Þ : ð D3 Þ C ð Þ L;R represents the Wilson coefficient corresponding toFig. 2: C ð Þ L;R ¼ C a L;R þ C b L;R þ C c L;R þ C d L;R : ð D4 Þ C a L;R ¼ − ig ZhA g NhA g eeN ; ½ð p þ p Þ − m N (cid:3)½ð p þ p − q Þ − m A k (cid:3) ;C b L;R ¼ i λ h i h j h k g ZhA g eeA ; ½ð p þ p Þ − m A k (cid:3)½ð p þ p − k Þ − m h i (cid:3) ;C c L;R ¼ − i λ A i A j h k g eeA ; g ZhA ½ð p þ p Þ − m A k (cid:3)½ð p þ p − q Þ − m A i (cid:3) ;C d L;R ¼ − ig NZh g eeA ; g NhA ½ð p þ p Þ − m A k (cid:3)½ð p þ p − q Þ − m N (cid:3) : ð D5 Þ Effective operator: O L;R ¼ ¯ v ð p Þð p α þ p α Þ p L;R u ð p Þ ε (cid:2) α ð k Þ ; O L;R ¼ ¯ v ð p Þð q α − p α − p α − q α Þ p L;R u ð p Þ ε (cid:2) α ð k Þ ; ð D6 Þ O L;R , O L;R have been given in Eq. (D3) and Eq. (27),respectively. Furthermore, λ h i h j h k has been given in Eq. (26)and Appendix B, and λ A i A j h k denotes the triple Higgs self-coupling of CP -even and CP -odd Higgs; the detailedexpression about this has been given in Appendix C.The superscripts (a, b, c, a ; b ; c ; d ) respectively representthe corresponding Feynman diagram labels in Figs. 1 and 2,and m h , m A denote the masses for Higgs and pseudoscalarHiggs, with i , k ¼ , 2, 3, 4 denoting the index ofgeneration. g ZZhh ¼ g cos θ W cos θ W Z Hi Z Hj þ g g cos θ W cos θ W sin θ W Z Hi Z Hj þ g cos θ W sin θ W Z Hi Z Hj − g YB g cos θ W cos θ W sin θ W Z Hi Z Hj − g g YB cos θ W sin θ W sin θ W Z Hi Z Hj þ g YB sin θ W Z Hi Z Hj þ g cos θ W cos θ W Z Hi Z Hj þ g g cos θ W cos θ W sin θ W Z Hi Z Hj þ g cos θ W sin θ W Z Hi Z Hj − g YB g cos θ W cos θ W sin θ W Z Hi Z Hj − g g YB cos θ W sin θ W sin θ W Z Hi Z Hj þ g YB sin θ W Z Hi Z Hj þ þ g B sin θ W Z Hi Z Hj þ g B sin θ W Z Hi Z Hj : ð D7 Þ g eeZ ¼ − g cos θ W sin θ W þ g cos θ W cos θ W þ ð g YB þ g B Þ sin θ W ;g eeZ ¼ − g cos θ W sin θ W þ ð g YB þ g B Þ sin θ W ;g eeZ ¼ ðð g sin θ W − g cos θ W Þ sin θ W þ ð g YB þ g B Þ cos θ W Þ ;g eeZ ¼ g sin θ W sin θ W − ð g YB þ g B Þ cos θ W : ð D8 Þ g ZZh ¼ ð v ð g cos θ W sin θ W þ g cos θ W cos θ W − g YB sin θ W Þ Z Hi þ v ð g cos θ W sin θ W þ g cos θ W cos θ W − g YB sin θ W Þ Z Hi þ g B sin θ W ð u Z Hi þ u Z Hi ÞÞ ;g Z Z h ¼ ð v ðð g sin θ W þ g cos θ W Þ sin θ W þ g YB cos θ W Þ Z Hi þ v ðð g sin θ W þ g cos θ W Þ sin θ W þ g YB cos θ W Þ Z Hi þ g B cos θ W ð u Z Hi þ u Z Hi ÞÞ ;g ZZ h ¼ ð − v ð g g YB cos θ W sin θ W þ g cos θ W cos θ W sin θ W þ cos θ W ð g sin θ W − g YB Þ sin θ W − g g YB sin θ W sin θ W þ g cos θ W ð g sin θ W sin θ W þ g YB cos θ W − g YB sin θ W ÞÞ Z Hi − v ð g g YB cos θ W sin θ W þ g cos θ W cos θ W sin θ W þ cos θ W ð g sin θ W − g YB Þ sin θ W − g g YB sin θ W sin θ W þ g cos θ W ð g sin θ W sin θ W þ g YB cos θ W − g YB sin θ W ÞÞ Z Hi þ g B sin θ W ð u Z Hi þ u Z Hi ÞÞ : ð D9 Þ e þ e − → hhZ IN THE B − L SYMMETRIC … PHYS. REV. D075002 (2020)
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