Higgs boson mass corrections in the μνSSM with effective potential methods
Hai-Bin Zhang, Tai-Fu Feng, Xiu-Yi Yang, Shu-Min Zhao, Guo-Zhu Ning
aa r X i v : . [ h e p - ph ] A p r Higgs boson mass corrections in the µν SSM with effectivepotential methods
Hai-Bin Zhang a ∗ , Tai-Fu Feng a † , Xiu-Yi Yang b , Shu-Min Zhao a , Guo-Zhu Ning aa Department of Physics, Hebei University, Baoding, 071002, China b College of Science, University of Science and Technology Liaoning, Anshan, 114051, China
Abstract
To solve the µ problem of the MSSM, the µ from ν Supersymmetric Standard Model ( µν SSM)introduces three singlet right-handed neutrino superfields ˆ ν ci , which lead to the mixing of theneutral components of the Higgs doublets with the sneutrinos, producing a relatively large CP-even neutral scalar mass matrix. In this work, we analytically diagonalize the CP-even neutralscalar mass matrix and analyze in detail how the mixing impacts the lightest Higgs boson mass.We also give an approximate expression for the lightest Higgs boson mass. Simultaneously, weconsider the radiative corrections to the Higgs boson masses with effective potential methods. PACS numbers: 12.60.Jv, 14.80.DaKeywords: Supersymmetry, Higgs bosons ∗ email:[email protected] † email:[email protected] . INTRODUCTION Since the ATLAS and CMS Collaborations reported the significant discovery of a newneutral Higgs boson [1, 2], the Higgs boson mass is now precisely measured by [3] m h = 125 . ± .
24 GeV . (1)Therefore, the accurate Higgs boson mass will give most stringent constraints on parameterspace for the standard model and its various extensions.As a supersymmetric model, the “ µ from ν supersymmetric standard model” ( µν SSM)has the superpotential: [4–10] W = ǫ ab (cid:16) Y u ij ˆ H bu ˆ Q ai ˆ u cj + Y d ij ˆ H ad ˆ Q bi ˆ d cj + Y e ij ˆ H ad ˆ L bi ˆ e cj + Y ν ij ˆ H bu ˆ L ai ˆ ν cj (cid:17) − ǫ ab λ i ˆ ν ci ˆ H ad ˆ H bu + 13 κ ijk ˆ ν ci ˆ ν cj ˆ ν ck , (2)where ˆ H Tu = (cid:16) ˆ H + u , ˆ H u (cid:17) , ˆ H Td = (cid:16) ˆ H d , ˆ H − d (cid:17) , ˆ Q Ti = (cid:16) ˆ u i , ˆ d i (cid:17) , ˆ L Ti = (cid:16) ˆ ν i , ˆ e i (cid:17) are SU (2) doubletsuperfields, and Y u,d,e,ν , λ , and κ are dimensionless matrices, a vector, and a totally sym-metric tensor, respectively. a, b = 1 , ǫ = 1,and i, j, k = 1 , , µν SSM introducesthree singlet right-handed neutrino superfields ˆ ν ci to solve the µ problem [16] of the MSSM.Once the electroweak symmetry is broken (EWSB), the effective µ term − ǫ ab µ ˆ H ad ˆ H bu is gen-erated spontaneously through right-handed sneutrino vacuum expectation values (VEVs), µ = λ i h ˜ ν ci i . Additionally, three tiny neutrino masses can be generated at the tree levelthrough a TeV scale seesaw mechanism [4–9, 17–23].In the µν SSM, the left- and right-handed sneutrino VEVs lead to the mixing of theneutral components of the Higgs doublets with the sneutrinos producing an 8 × µν SSM, including the Higgs boson mass corrections. We present the diagonalizationof the neutral scalar mass matrix analytically in Sec. III. The numerical analyses are givenin Sec. IV, and Sec. V provides a summary. The tedious formulas are collected in theAppendixes.
II. THE HIGGS SECTOR
The Higgs sector of the µν SSM contains the usual two Higgs doublets with the left- andright-handed sneutrinos: ˆ H Td = (cid:16) ˆ H d , ˆ H − d (cid:17) , ˆ H Tu = (cid:16) ˆ H + u , ˆ H u (cid:17) , ˆ ν i and ˆ ν ci . Once EWSB, theneutral scalars have the VEVs: h H d i = υ d , h H u i = υ u , h ˜ ν i i = υ ν i , h ˜ ν ci i = υ ν ci . (3)One can define the neutral scalars as H d = 1 √ (cid:16) h d + iP d (cid:17) + υ d , ˜ ν i = 1 √ (cid:16) (˜ ν i ) ℜ + i (˜ ν i ) ℑ (cid:17) + υ ν i ,H u = 1 √ (cid:16) h u + iP u (cid:17) + υ u , ˜ ν ci = 1 √ (cid:16) (˜ ν ci ) ℜ + i (˜ ν ci ) ℑ (cid:17) + υ ν ci , (4)Considering that the neutrino oscillation data constrain neutrino Yukawa couplings Y ν i ∼O (10 − ) and left-handed sneutrino VEVs υ ν i ∼ O (10 − GeV) [4–7, 17–22], in the followingwe could reasonably neglect the small terms including Y ν or υ ν i in the Higgs sector. Then,the superpotential in Eq. (2) approximately leads to the tree-level neutral scalar (Higgs)potential: V = V F + V D + V soft , (5)with V F = λ i λ ∗ i H d H ∗ d H u H ∗ u + λ i λ ∗ j ˜ ν ci ˜ ν c ∗ j ( H d H ∗ d + H u H ∗ u )+ κ ijk κ ∗ ljm ˜ ν ci ˜ ν ck ˜ ν c ∗ l ˜ ν c ∗ m − ( κ ijk λ ∗ j ˜ ν ci ˜ ν ck H ∗ d H ∗ u + H . c . ) , (6)3 D = G ν i ˜ ν ∗ i + H d H ∗ d − H u H ∗ u ) , (7) V soft = m H d H d H ∗ d + m H u H u H ∗ u + m L ij ˜ ν i ˜ ν ∗ j + m ν cij ˜ ν ci ˜ ν c ∗ j − (cid:16) ( A λ λ ) i ν ci H d H u −
13 ( A κ κ ) ijk ˜ ν ci ˜ ν cj ˜ ν ck + H . c . (cid:17) , (8)where G = g + g and g c W = g s W = e , V F and V D are the usual F and D terms derivedfrom the superpotential, and V soft denotes the soft supersymmetry breaking terms. Forsimplicity, we will assume that all parameters in the potential are real in the following.With effective potential methods [24–39], the one-loop effective potential can be given by V = 132 π n X ˜ f N f m f (cid:16) log m f Q − (cid:17) − X f = t,b,τ N f m f (cid:16) log m f Q − (cid:17)o , (9)where, Q denotes the renormalization scale, N t = N b = 3 and N τ = 1, ˜ f = ˜ t , , ˜ b , , ˜ τ , .The masses of the third fermions f = t, b, τ and corresponding supersymmetric partners˜ f = ˜ t , , ˜ b , , ˜ τ , in the µν SSM are collected in Appendix A. Including the one-loop effectivepotential, the Higgs potential is written as V = V + V . (10)Through the Higgs potential, we will calculate the minimization conditions of the potentialand the Higgs masses in the following.Minimizing the Higgs potential, we can obtain the minimization conditions of the poten-tial, linking the soft mass parameters to the VEVs of the neutral scalar fields: m H d = − ∆ T H d + (( A λ λ ) i υ ν ci + λ j κ ijk υ ν ci υ ν ck ) tan β − ( λ i λ j υ ν ci υ ν cj + λ i λ i υ u ) + G υ u − υ d ) , (11) m H u = − ∆ T H u + (( A λ λ ) i υ ν ci + λ j κ ijk υ ν ci υ ν ck ) cot β − ( λ i λ j υ ν ci υ ν cj + λ i λ i υ d ) + G υ d − υ u ) , (12) m ν cij υ ν cj = − ∆ T ˜ ν cij υ ν cj + ( A λ λ ) i υ d υ u − ( A κ κ ) ijk υ ν cj υ ν ck + 2 λ j κ ijk υ ν ck υ d υ u − κ lim κ ljk υ ν cm υ ν cj υ ν ck − λ i λ j υ ν cj ( υ d + υ u ) , ( i = 1 , ,
3) (13)where, as usual, tan β = υ u /υ d . ∆ T H d , ∆ T H u , and ∆ T ˜ ν cij υ ν cj come from one-loop correctionsto the minimization conditions, which are taken in Appendix B. Here, neglecting the small4erms including Y ν or υ ν i in the Higgs sector, we do not give the minimization conditionsof the potential about the left-handed sneutrino VEVs, which can be used to constrain υ ν i [17, 22].From the Higgs potential, one can derive the 8 × S ′ T = ( h d , h u , (˜ ν ci ) ℜ , (˜ ν i ) ℜ ) and the CP-odd neutral scalars P ′ T = ( P d , P u , (˜ ν ci ) ℑ , (˜ ν i ) ℑ )in the unrotated basis. Ignoring the small terms including Y ν or υ ν i , the 5 × × × (cid:16) m L ij + G ( υ d − υ u ) δ ij (cid:17) × , which is dominated by the soft mass m L ij . Through the Higgspotential, the 5 × M S = M H M X (cid:16) M X (cid:17) T M R , (14)where M H denotes the 2 × M R is the 3 × M X represents the 2 × × M H can be written by M H = M h d h d + ∆ M h d h u + ∆ M h d h u + ∆ M h u h u + ∆ , (15)with the tree-level contributions as M h d h u = − h m A + (cid:16) − λ i λ i s W c W /e (cid:17) m Z i sin β cos β, (16) M h d h d = m A sin β + m Z cos β, (17) M h u h u = m A cos β + m Z sin β, (18)and the neutral pseudoscalar mass squared as m A ≃ β h ( A λ λ ) i υ ν ci + λ k κ ijk υ ν ci υ ν cj i . (19)Comparing with the MSSM, M h d h u has an additional term (4 λ i λ i s W c W /e ) m Z sin β cos β ,which can give a new contribution to the lightest Higgs boson mass. The radiative corrections5 , ∆ , and ∆ from the third fermions f = t, b, τ and their superpartners can be foundin Ref. [9], which agree with the results of the MSSM [24–37]. Here, the radiative correctionsfrom the top quark and its superpartners include the two-loop leading-log effects, which canobviously affect the mass of the lightest Higgs boson.Furthermore, the 2 × M X is M X = (cid:16) M h d (˜ ν ci ) ℜ + ∆ i ) (cid:17) × (cid:16) M h u (˜ ν ci ) ℜ + ∆ i ) (cid:17) × , (20)where M h d (˜ ν ci ) ℜ = h λ i λ j υ ν cj cot β − (cid:16) ( A λ λ ) i + 2 λ k κ ijk υ ν cj (cid:17)i υ u , (21) M h u (˜ ν ci ) ℜ = h λ i λ j υ ν cj tan β − (cid:16) ( A λ λ ) i + 2 λ k κ ijk υ ν cj (cid:17)i υ d , (22)and the radiative corrections from the third fermions f = t, b, τ and their superpartners are∆ i ) = λ i υ u ∆ R , ∆ i ) = λ i υ d ∆ R , (23)∆ R = G F √ π n m t sin β µ ( A t − µ cot β ) tan β ( m t − m t ) g ( m t , m t )+ 3 m b cos β ( − A b + µ tan β )( m b − m b ) h log m b m b + A b ( A b − µ tan β )( m b − m b ) g ( m b , m b ) i + m τ cos β ( − A τ + µ tan β )( m τ − m τ ) h log m τ m τ + A τ ( A τ − µ tan β )( m τ − m τ ) g ( m τ , m τ ) io , (24)∆ R = G F √ π n m t sin β ( − A t + µ cot β )( m t − m t ) h log m t m t + A t ( A t − µ cot β )( m t − m t ) g ( m t , m t ) i + 3 m b cos β µ ( A b − µ tan β ) cot β ( m b − m b ) g ( m b , m b )+ m τ cos β µ ( A τ − µ tan β ) cot β ( m τ − m τ ) g ( m τ , m τ ) o , (25)with µ = λ i υ ν ci , g ( m , m ) = 2 − m + m m − m log m m . Here, we can know that the radiativecorrections to the mixing are proportional to the parameters λ i .Similarly, one can derive the 3 × M R = (cid:18) M ν ci ) ℜ (˜ ν cj ) ℜ + ∆ (2+ i )(2+ j ) (cid:19) × , (26)6ith M ν ci ) ℜ (˜ ν cj ) ℜ = m ν cij + 2( A κ κ ) ijk υ ν ck − λ k κ ijk υ d υ u + λ i λ j ( υ d + υ u )+ (2 κ ijk κ lmk + 4 κ ilk κ jmk ) υ ν cl υ ν cm , (27)and the corrections from the third fermions and their superpartners are∆ (2+ i )(2+ j ) = λ i λ j ∆ RR , (28)∆ RR = G F √ π n m t sin β υ d ( A t − µ cot β ) ( m t − m t ) g ( m t , m t )+ 3 m b cos β υ u ( A b − µ tan β ) ( m b − m b ) g ( m b , m b )+ m τ cos β υ u ( A τ − µ tan β ) ( m τ − m τ ) g ( m τ , m τ ) o . (29)Here, the radiative corrections to the mass submatrix for right-handed sneutrinos are pro-portional to λ i λ j . III. DIAGONALIZATION OF THE MASS MATRIX
The mass squared matrix M H which contains the radiative corrections can be diagonalizedas U TH M H U H = diag (cid:16) m H , m H (cid:17) , (30)by the 2 × U H , U H = − sin α cos α cos α sin α . (31)Here, the neutral doubletlike Higgs mass squared eigenvalues m H , can be derived, m H , = 12 h Tr M H ∓ q (Tr M H ) − M H i , (32)where Tr M H = M H + M H , Det M H = M H M H − ( M H ) . The mixing angle α canbe determined by [32] sin 2 α = 2 M H q (Tr M H ) − M H , α = M H − M H q (Tr M H ) − M H , (33)which reduce to − sin 2 β and − cos 2 β , respectively, in the large m A limit. The conventionis that π/ ≤ β < π/ β ≥
1, while − π/ < α <
0. In the large m A limit, α = − π/ β .In the large m A limit, the light neutral doubletlike Higgs mass is approximately given as m H ≃ m Z cos β + 2 λ i λ i s W c W e m Z sin β + △ m H . (34)Comparing with the MSSM, the µν SSM gets an additional term λ i λ i s W c W e m Z sin β [5].Here, the radiative corrections △ m H can be computed more precisely by some public tools,for example, FeynHiggs [40–47], SOFTSUSY [48–50], SPheno [51, 52], and so on. In thefollowing numerical section, we will use the FeynHiggs-2.13.0 to calculate the radiative cor-rections for the Higgs boson mass about the MSSM part.To further deal with the mass submatrix M R and M X , in the following we choose theusual minimal scenario for the parameter space: λ i = λ, ( A λ λ ) i = A λ λ, υ ν ci = υ ν c ,κ ijk = κδ ij δ jk , ( A κ κ ) ijk = A κ κδ ij δ jk , m ν cij = m ν ci δ ij , (35)Then, the 3 × M R = X R y R y R y R X R y R y R y R X R , (36)with X R = ( A κ + 4 κυ ν c ) κυ ν c + A λ λυ d υ u /υ ν c + λ ∆ RR , (37) y R = λ ( υ + ∆ RR ) , (38)where υ = υ d + υ u . Here the radiative corrections keep the dominating contributions whichare proportional to m f ( f = t, b, τ ). Through the 3 × U R , U R = √ − √ √ − √ √ √ √ √ , (39)8he mass squared matrix M R can be diagonalized as U TR M R U R = diag (cid:16) m R , m R , m R (cid:17) , (40)with m R = X R + 2 y R = ( A κ + 4 κυ ν c ) κυ ν c + A λ λυ d υ u /υ ν c + λ (2 υ + 3∆ RR ) , (41) m R = m R = X R − y R = ( A κ + 4 κυ ν c ) κυ ν c + A λ λυ d υ u /υ ν c − λ υ . (42)The radiative corrections are proportional to λ , which will be tamped down as λ ∼ O (0 . m S R ≈ m R ≈ m R = m R ≈ ( A κ + 4 κυ ν c ) κυ ν c + A λ λυ d υ u /υ ν c . (43)Due to υ ν c ≫ υ u,d , the main contribution to the mass squared is the first term as κ islarge. Additionally, the masses squared of the CP-odd right-handed sneutrinos m P R can beapproximated as m P R ≈ − A κ κυ ν c + (4 κ + A λ /υ ν c ) λυ d υ u , (44)where the first term is the leading contribution. Therefore, one can use the approximaterelation, − κυ ν c < ∼ A κ < ∼ , (45)to avoid the tachyons.In the minimal scenario for the parameter space presented in Eq. (35), the 2 × M X is simplified as M X = M X M X M X M X M X M X , (46)where M X = λυ sin β h υ ν c (3 λ cot β − κ ) − A λ + ∆ R i , (47) M X = λυ cos β h υ ν c (3 λ tan β − κ ) − A λ + ∆ R i . (48)9hen, we do the calculation: U TH U TR M H M X (cid:16) M X (cid:17) T M R U H U R = H ⊕ m R m R , (49)with H = m H A X m H A X A X A X m R , (50)where A X = √ − M X sin α + M X cos α ) , (51) A X = √ M X cos α + M X sin α ) . (52)In the large m A limit, α = − π/ β . Then, one can have the following approximateexpressions: A X ≃ √ λυ sin 2 β h υ ν c (cid:16) λ sin 2 β − κ (cid:17) − A λ + 12 (∆ R + ∆ R ) i , (53) A X ≃ √ λυ h (2 κυ ν c + A λ ) cos 2 β + ∆ R sin β − ∆ R cos β i . (54)If A X = 0, the mixing of Higgs doublets and right-handed sneutrinos will not affect thelightest Higgs boson mass [5]; namely, one can adopt the relation A λ = 2 υ ν c (cid:16) λ sin 2 β − κ (cid:17) + 12 (∆ R + ∆ R ) , (55)which is analogous to the NMSSM [53, 54]. To relax the conditions, if A λ is around thevalue in Eq. (55), the contribution to the lightest Higgs boson mass from the mixing couldalso be neglected approximately. In the case A X ≈
0, the mass of the lightest Higgs bosonis just m H , which shows, approximately, in Eq. (34).If A X = 0, we need to diagonalize the 3 × H further: U TX H U X = diag (cid:16) m h , m H , m S (cid:17) , (56)where the eigenvalues m h , m H , m S and the unitary matrix U X can be concretely seen inAppendix C. Then, the lightest Higgs boson mass is exactly m h . In the large m A limit,10 H ≃ m A , one can have the lightest Higgs boson mass squared approximately as m h ≃ n m H + m R − ( A X ) m H − vuuth m R − m H − ( A X ) m H i + 4( A X ) o . (57)The approximate expression works well, which can be easily checked in the numerical cal-culation. When m H and m R are all large, Eq. (57) could be approximated by m h ≈ m H − ( A X ) m R = m H h − ( A X ) m R m H i . (58)In the numerical analysis, we can define the quantity ξ h = ( A X ) m R m H (59)to analyze how the mixing affects the mass of the lightest Higgs boson.One can diagonalize the 5 × R TS M S R S = diag (cid:16) m S , m S , m S , m S , m S (cid:17) , (60)with m S = m h , m S = m H , m S = m R = m S = m R , and the 5 × R S R S = U H U R U X I × , (61)where I × denotes the 2 × IV. NUMERICAL ANALYSIS
In this section, we will do the numerical analysis for the masses of the Higgs bosons.First, we choose the values of the parameter space. For the relevant parameters in the SM,we choose α s ( m Z ) = 0 . , m Z = 91 .
188 GeV , m W = 80 .
385 GeV ,m t = 173 . , m b = 4 .
66 GeV , m τ = 1 .
777 GeV . (62)The other SM parameters can be seen in Ref. [55] from the Particle Data Group. Here,we choose a suitable A κ = −
500 GeV to avoid the tachyons easily, through Eq. (45).11 .06 0.08 0.10 0.12 0.14 0.16 0.18 0.20110115120125130135 Λ H a L m h (cid:144) G e V Λ I b M Ξ h FIG. 1: (a) m h varies with λ , the solid line and dash-dot line denote m h and m H as tan β = 20,the dash line and dot line denote m h and m H as tan β = 6. (b) ξ h varies with λ , the solid lineand dash line represent as tan β = 20 and tan β = 6, respectively. When κ = 0 . A λ = 500 GeVand υ ν c = 2 TeV. Considering the direct search for supersymmetric particles [55], we could reasonably choose M = 2 M = 800 GeV, M = 2 TeV, m ˜ Q = m ˜ U = m ˜ D = 2 TeV, m ˜ L = m ˜ E = 1 TeV, A b = A τ = 1 TeV, and A t = 2 . m ˜ Q , m ˜ U , A t and the gaugino mass parameters affect the radiative corrections to the lightest Higgs mass.Therefore, one can take the proper values for m ˜ Q , m ˜ U , A t and the gaugino mass parametersto keep the lightest Higgs mass around 125 GeV.In the following, we will analyze how the mixing of Higgs doublets and right-handedsneutrinos affects the lightest Higgs boson mass. Through A X in Eq. (53), one knowsthat the parameters which affect the lightest Higgs boson mass from the mixing will be λ, tan β, κ, A λ , and υ ν c . Here, we specify that the parameter µ = 3 λυ ν c , which is dominatedby the parameters λ and υ ν c .When κ = 0 . A λ = 500 GeV, and υ ν c = 2 TeV, we plot the lightest Higgs boson mass m h , varying with the parameter λ in Fig. 1(a), where the solid line and dash-dot line denote m h and m H as tan β = 20, the dash line and dot line denote m h and m H as tan β = 6,respectively. The mass m H denotes the lightest Higgs boson mass if we do not considerthe mixing of Higgs doublets and right-handed sneutrinos, and the mass m h is exactly thelightest Higgs boson mass considering the mixing. The numerical results indicate that themixing could have significant effects on the lightest Higgs boson mass, as the parameter λ is12 .2 0.3 0.4 0.5 0.6110115120125130135 Κ H a L m h (cid:144) G e V Κ I b M Ξ h FIG. 2: (a) m h varies with κ , the solid line and dash-dot line denote m h and m H as tan β = 20,the dash line and dot line denote m h and m H as tan β = 6. (b) ξ h varies with κ , the solid lineand dash line represent as tan β = 20 and tan β = 6, respectively. When λ = 0 . A λ = 500 GeVand υ ν c = 2 TeV. large. With an increase of λ , the lightest Higgs boson mass m h drops down quickly, whichdeviates from the mass m H . For large tan β , the lightest Higgs boson mass m h decreasesmore quickly with increasing λ .To see the reason more clearly, we also plot the quantity ξ h , varying with λ in Fig. 1(b),where the solid line and dash line, respectively, represent as tan β = 20 and tan β = 6.The quantity ξ h is defined in Eq. (59) to quantify the effect on the lightest Higgs bosonmass from the mixing of Higgs doublets and right-handed sneutrinos. The figure showsthat ξ h increases quickly with an increase of λ , and ξ h for large tan β is larger than it isfor small tan β . When λ is small, ξ h is also small, and then m h is close to m H because A X in Eq. (53) is in proportion to the parameter λ . Additionally, in this parameter space, m H ≈ m A ≈ . m S R ≈ . m P R ≈ . β = 6 and λ = 0 . m A ∼ O (TeV), we can believe that the parameter space is in the large m A limit, and accordingly the approximate expressions Eq. (34) and Eq. (57) will work well.Meanwhile m S R ∼ O (TeV), and the approximate expression Eq. (58) is also consistent withthe exact one.We also picture the lightest Higgs boson mass m h varying with the parameter κ inFig. 2(a), where the solid line and dash-dot line denote m h and m H as tan β = 20, thedash line and dot line denote m h and m H as tan β = 6. And the quantity ξ h varies with13 A Λ (cid:144) TeV H a L m h (cid:144) G e V A Λ (cid:144) TeV I b M Ξ h FIG. 3: (a) m h varies with A λ , the solid line and dash-dot line denote m h and m H as tan β = 20,the dash line and dot line denote m h and m H as tan β = 6. (b) ξ h varies with A λ , the solid lineand dash line represent as tan β = 20 and tan β = 6, respectively. When κ = 0 . λ = 0 . υ ν c = 2 TeV. the parameter κ in Fig. 2(b), where the solid line and dash line represent as tan β = 20 andtan β = 6, respectively. Here, we take λ = 0 . A λ = 500 GeV and υ ν c = 2 TeV. We cansee that the lightest Higgs boson mass m h deviates from the mass m H largely, when theparameter κ is small. Of course, for small κ , the quantity ξ h is large. Constrained by theLandau pole condition [5], we choose the parameter κ ≤ . κ = 0 . λ = 0 . υ ν c = 2 TeV, we draw the lightest Higgs bosonmass m h , varying with the parameter A λ , where the solid line and dash-dot line denote m h and m H as tan β = 20, the dash line and dot line denote m h and m H as tan β = 6.And Fig. 3(b) shows the quantity ξ h versus A λ , where the solid line and dash line representas tan β = 20 and tan β = 6, respectively. The numerical results show that m h ≃ m H and ξ h ≃ A λ ≈ β = 6, and as A λ ≈
10 TeV for tan β = 20, which isin accordance with Eq. (55). Comparing with the large tree-level contributions, the smallone-loop contributions can be ignored, then Eq. (55) can be approximated as A λ ≃ υ ν c (cid:16) λ sin 2 β − κ (cid:17) . (63)Therefore, when A λ is around 2 υ ν c (cid:16) λ/ sin 2 β − κ (cid:17) , we could regard the lightest Higgs bosonmass as m h ≈ m H . If A λ drifts off the value of 2 υ ν c (cid:16) λ/ sin 2 β − κ (cid:17) significantly, the lightestHiggs boson mass m h will deviate from the mass m H .14 Υ Ν c (cid:144) TeV H a L m h (cid:144) G e V Υ Ν c (cid:144) TeV I b M Ξ h FIG. 4: (a) m h varies with υ ν c , the solid line and dash-dot line denote m h and m H as tan β = 20,the dash line and dot line denote m h and m H as tan β = 6. (b) ξ h varies with υ ν c , the solid lineand dash line represent as tan β = 20 and tan β = 6, respectively. When κ = 0 . λ = 0 . A λ = 500 GeV. Finally, for κ = 0 . λ = 0 .
1, and A λ = 500 GeV, we plot the lightest Higgs boson mass m h versus the parameter υ ν c in Fig. 4(a), where the solid line and dash-dot line denote m h and m H as tan β = 20, and the dash line and dot line denote m h and m H as tan β = 6.Fig. 4(b) shows ξ h varying with υ ν c , where the solid line and dash line represent as tan β = 20and tan β = 6, respectively. We can see that the lightest Higgs boson mass m h is parallel tothe mass m H with increasing of υ ν c . Through Eq. (41), m R ∼ O ( υ ν c ), and A X ∼ O ( υ ν c )as shown in Eq. (53). Therefore, the quantity ξ h = ( A X ) m R m H defined in Eq. (59) becomes flatwith an increase of υ ν c , which can be seen in Fig. 4(b). In addition, Fig. 4(a) indicates that m h and m H are decreasing slowly, with an increase of υ ν c , because the parameter µ = 3 λυ ν c ,which can affect the radiative corrections for the lightest Higgs boson mass. V. SUMMARY
In the framework of the µν SSM, the three singlet right-handed neutrino superfields ˆ ν ci are introduced to solve the µ problem of the MSSM. Correspondingly, the right-handedsneutrino VEVs lead to the mixing of the neutral components of the Higgs doublets withthe sneutrinos, which produce a large CP-even neutral scalar mass matrix. Therefore, themixing would affect the lightest Higgs boson mass. In this work, we consider the Higgs15oson mass radiative corrections with effective potential methods and then analyticallydiagonalize the CP-even neutral scalar mass matrix. Meanwhile, in the large m A limit,we give an approximate expression for the lightest Higgs boson mass seen in Eq. (57). Innumerical analysis, we analyze how the key parameters λ, tan β, κ, A λ , and υ ν c affect thelightest Higgs boson mass. Acknowledgments
The work has been supported by the National Natural Science Foundation of China(NNSFC) with Grants No. 11535002, No. 11605037 and No. 11647120, Natural ScienceFoundation of Hebei province with Grants No. A2016201010 and No. A2016201069, Foun-dation of Department of Education of Liaoning province with Grant No. 2016TSPY10,Youth Foundation of the University of Science and Technology Liaoning with Grant No.2016QN11, Hebei Key Lab of Optic-Eletronic Information and Materials, and the MidwestUniversities Comprehensive Strength Promotion project.
Appendix A: The masses for the third fermions and their superpartners
The masses for the third fermions f = t, b, τ are m t = Y t | H u | , m b = Y b | H d | , m τ = Y τ | H d | . (A1)The corresponding 2 × f L − ˜ f R ( ˜ f = ˜ t, ˜ b, ˜ τ ) mass squared matrices are M f = M f L M X f M ∗ X f M f R , ( ˜ f = ˜ t, ˜ b, ˜ τ ) (A2)where the concrete expressions for matrix elements can be given as M t L = m Q + 3 g − g
12 ( | H d | − | H u | ) + Y t | H u | , (A3) M t R = m U + g | H d | − | H u | ) + Y t | H u | , (A4) M X t = Y t ( A t | H u | − λ i ˜ ν c ∗ i H d ) , (A5) M b L = m Q − g + g
12 ( | H d | − | H u | ) + Y b | H d | , (A6)16 b R = m D − g | H d | − | H u | ) + Y b | H d | , (A7) M X b = Y b ( A b H d − λ i ˜ ν c ∗ i H ∗ u ) , (A8) M τ L = m L + g − g | H d | − | H u | ) + Y τ | H d | , (A9) M τ R = m E − g | H d | − | H u | ) + Y τ | H d | , (A10) M X τ = Y τ ( A τ H d − λ i ˜ ν c ∗ i H ∗ u ) . (A11)Here we ignore the small terms including Y ν or | ˜ ν i | . The eigenvalues m f , of the ˜ f = ˜ t, ˜ b, ˜ τ mass squared matrices can be given by m f , = M f L + M f R ± vuut(cid:16) M f L − M f R (cid:17) + | M X f | . (A12)If substituting the VEVs for the corresponding neutral scalars, the masses of the thirdfermions f = t, b, τ and their superpartners are manifestly obtained. Appendix B: The corrections to the minimization conditions
Considering one-loop corrections to the minimization conditions from the third fermions f = t, b, τ and their superpartners, ∆ T H d , ∆ T H u , and ∆ T ˜ ν cij υ ν cj are given below:∆ T H d = 3(4 π ) n G h f ( m t ) + f ( m t ) i − h Y t µ ( A t tan β − µ ) − g − g
24 ( m t L − m t R ) i f ( m t ) − f ( m t ) m t − m t + (cid:16) Y b − G (cid:17)h f ( m b ) + f ( m b ) i − Y b f ( m b )+ h Y b A b ( A b − µ tan β ) − g − g
24 ( m b L − m b R ) i f ( m b ) − f ( m b ) m b − m b o + 1(4 π ) n(cid:16) Y τ − G (cid:17)h f ( m τ ) + f ( m τ ) i − Y τ f ( m τ )+ h Y τ A τ ( A τ − µ tan β ) − g − g m τ L − m τ R ) i f ( m τ ) − f ( m τ ) m τ − m τ o , (B1)∆ T H u = 3(4 π ) n(cid:16) Y t − G (cid:17)h f ( m t ) + f ( m t ) i − Y t f ( m t )17 h Y t A t ( A t − µ cot β ) − g − g
24 ( m t L − m t R ) i f ( m t ) − f ( m t ) m t − m t + G h f ( m b ) + f ( m b ) i − h Y b µ ( A b cot β − µ ) − g − g
24 ( m b L − m b R ) i f ( m b ) − f ( m b ) m b − m b o + 1(4 π ) n G h f ( m τ ) + f ( m τ ) i − h Y τ µ ( A τ cot β − µ ) − g − g m τ L − m τ R ) i f ( m τ ) − f ( m τ ) m τ − m τ o , (B2)∆ T ˜ ν cij υ ν cj = 3(4 π ) n λ i Y t υ d ( λ j υ ν cj − A t tan β ) f ( m t ) − f ( m t ) m t − m t + λ i Y b υ u ( λ j υ ν cj − A b cot β ) f ( m b ) − f ( m b ) m b − m b o + 1(4 π ) n λ i Y τ υ u ( λ j υ ν cj − A τ cot β ) f ( m τ ) − f ( m τ ) m τ − m τ o , (B3)with µ = λ i υ ν ci , f ( m ) = m (log m Q − Appendix C: The diagonalization of the × mass matrix The eigenvalues of the 3 × H are given as [21, 56] m = a − p (cos φ + √ φ ) , (C1) m = a − p (cos φ − √ φ ) , (C2) m = a p cos φ. (C3)To formulate the expressions in a concise form, one can define the notations, p = √ a − b, (C4) φ = 13 arccos( 1 p ( a − ab + 272 c )) , (C5)with a = Tr( H ) , (C6)18 = H H + H H + H H − H − H − H , (C7) c = Det( H ) . (C8)In a general way, m ≤ m ≤ m . So, one can have two possibilities on the mass spectrum:(i) spectrum with m h < m H ≤ m S : m h = m , m H = m , m S = m , (C9)(ii) spectrum with m h < m S < m H : m h = m , m H = m , m S = m . (C10)The normalized eigenvectors for the mass squared matrix H are given by U X U X U X = 1 q | X | + | Y | + | Z | X Y Z , (C11) U X U X U X = 1 q | X | + | Y | + | Z | X Y Z , (C12) U X U X U X = 1 q | X | + | Y | + | Z | X Y Z , (C13)with X = ( H − m h )( H − m h ) − H , (C14) Y = H H − H ( H − m h ) , (C15) Z = H H − H ( H − m h ) , (C16) X = H H − H (cid:16) H − m H (cid:17) , (C17) Y = ( H − m H )( H − m H ) − H , (C18) Z = H H − H (cid:16) H − m H (cid:17) , (C19)19 = H H − H (cid:16) H − m S (cid:17) , (C20) Y = H H − H (cid:16) H − m S (cid:17) , (C21) Z = ( H − m S )( H − m S ) − H . (C22) [1] G. Aad et al. (ATLAS Collaboration), Phys. Lett. B 716 (2012) 1, arXiv:1207.7214 [hep-ex].[2] S. Chatrchyan et al. (CMS Collaboration), Phys. Lett. B 716 (2012) 30, arXiv:1207.7235[hep-ex].[3] G. Aad et al. (ATLAS and CMS Collaborations), Phys. Rev. Lett. 114 (2015) 191803,arXiv:1503.07589 [hep-ex].[4] D.E. L´opez-Fogliani and C. Mu˜noz, Phys. Rev. Lett. 97 (2006) 041801, hep-ph/0508297.[5] N. Escudero, D.E. L´opez-Fogliani, C. Mu˜noz, and R. Ruiz de Austri, JHEP 12 (2008) 099,arXiv:0810.1507.[6] J. Fidalgo, D.E. L´opez-Fogliani, C. Mu˜noz, and R. Ruiz de Austri, JHEP 10 (2011) 020,arXiv:1107.4614.[7] H.-B. Zhang, T.-F. Feng, G.-F. Luo, Z.-F. Ge, and S.-M. Zhao, JHEP 07 (2013) 069 [Erratum-ibid. 10 (2013) 173], arXiv:1305.4352.[8] H.-B. Zhang, T.-F. Feng, S.-M. Zhao, and F. Sun, Int. J. Mod. Phys. A 29 (2014) 1450123,arXiv:1407.7365.[9] H.-B. Zhang, T.-F. Feng, F. Sun, K.-S. Sun, J.-B. Chen, and S.-M. Zhao, Phys. Rev. D 89(2014) 115007, arXiv:1307.3607.[10] H.-B. Zhang, T.-F. Feng, S.-M. Zhao, Y.-L. Yan, and F. Sun, Chin. Phys. C 41 (2017) 043106,arXiv:1511.08979.[11] H.P. Nilles, Phys. Rept. 110 (1984) 1.[12] H.E. Haber and G.L. Kane, Phys. Rept. 117 (1985) 75.[13] H.E. Haber, hep-ph/9306207.[14] S.P. Martin, hep-ph/9709356.[15] J. Rosiek, Phys. Rev. D 41 (1990) 3464, hep-ph/9511250.
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