Lightest Higgs boson decays h\rightarrow MZ in the μ from ν supersymmetric standard model
Chang-Xin Liu, Xiao-Ya Wang, Hai-Bin Zhang, Jin-Lei Yang, Shu-Min Zhao, Tai-Fu Feng
LLightest Higgs boson decays h → M Z in the µ from ν supersymmetric standard model Chang-Xin Liu a,b ∗ , Xiao-Ya Wang a,b † , Hai-Bin Zhang a,b ‡ ,Jin-Lei Yang a,b,c § , Shu-Min Zhao a,b ¶ , Tai-Fu Feng a,b,e ∗∗ a Department of Physics, Hebei University, Baoding, 071002, China b Key Laboratory of High-precision Computation and Application ofQuantum Field Theory of Hebei Province, Baoding, 071002, China c Institute of theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China e College of Physics, Chongqing University, Chongqing, 400044, China
Abstract
We study the lightest Higgs boson decays h → M Z in the framework of the µ from ν super-symmetric standard model ( µν SSM), where M is a vector meson ( ρ, ω, φ, J/ Ψ , Υ). Compared tothe minimal supersymmetric standard model (MSSM), the µν SSM introduces three right-handedneutrino superfields, which lead to the mixing of the Higgs doublets with the sneutrinos. Themixing affects the lightest Higgs boson mass and the Higgs couplings. Compared to the standardmodel, the µν SSM can give large new physics contributions to the decay width of h → M Z insuitable parameter space, which may be detected by the HL-LHC or the other future high energycolliders.
PACS numbers: 12.60.Jv, 14.80.DaKeywords: Supersymmetry, Higgs boson decay ∗ [email protected] † ‡ Corresponding [email protected] § [email protected] ¶ [email protected] ∗∗ Corresponding [email protected] a r X i v : . [ h e p - ph ] D ec . INTRODUCTION Since the Higgs boson was discovered by the Large Hadron Collider (LHC) [1, 2], themeasured mass of the Higgs boson now is [3] m h = 125 . ± .
14 GeV . (1)Therefore, the accurate Higgs boson mass gives most stringent constraint on parameter spacefor the standard model and its various extensions. The next step is focusing on searchingfor the properties of the Higgs boson both experimentally and theoretically.As one of the extensions of the standard model, the µ from ν supersymmetric standardmodel ( µν SSM) [4–10] can solve the µ problem [11] of the minimal supersymmetric standardmodel (MSSM) [12–16] through introducing three singlet right-handed neutrino superfieldsˆ ν ci ( i = 1 , , h → γγ , h → V V ∗ ( V = Z, W ), h → f ¯ f ( f = b, τ, µ ), h → µτ , h → Zγ , the masses of the Higgs bosons in the µν SSMhave been researched [17–21]. In this work, we study the 125 GeV Higgs boson rare decays h → M Z in the framework of the µν SSM, where M is a vector meson ( ρ, ω, φ, J/ψ, Υ).For the processes h → M Z , there are two types of decay topologies, one is the directcontributions which the Higgs boson couples to the quarks forming the meson, another isindirect contributions which the meson is converted by an off-shell vector boson through thelocal matrix element [22–24]. Actually, the effective hγZ vertex is very important for theindirect contributions of the decays h → M Z , so the indirect contributions is more obviousthan the direct contributions. In this work, the QCD factorization [25–28] is used for therare Higgs boson decays h → M Z .The paper is organized as follows. In Sec. II, we introduce the µν SSM briefly, about thesuperpotential and the soft SUSY-breaking terms. In Sec. III, we study the decay processes h → M Z in the µν SSM. Sec. IV and Sec. V, We show the numerical analysis and theconclusion. 2
I. THE µν SSM
In addition to the MSSM Yukawa couplings for quarks and charged leptons, the super-potential of the µν SSM contains Yukawa couplings for neutrinos, two additional types ofterms involving the Higgs doublet superfields ˆ H u and ˆ H d , and the right-handed neutrinosuperfields ˆ ν ci , [4] W = (cid:15) 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 (cid:17) + (cid:15) ab Y ν ij ˆ H bu ˆ L ai ˆ ν cj − (cid:15) 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) (the index T denotesthe transposition) represent SU (2) doublet superfields, and ˆ u ci , ˆ d ci , and ˆ e ci are the singletup-type quark, down-type quark and charged lepton superfields, respectively. In addition, Y u,d,e,ν , λ , and κ are dimensionless matrices, a vector, and a totally symmetric tensor. a, b =1 , (cid:15) = 1, and i, j, k = 1 , , ν ci ) of the singlet neutrino superfields ˆ ν ci are induced, the effective bilinearterms (cid:15) ab ε i ˆ H bu ˆ L ai and (cid:15) ab µ ˆ H ad ˆ H bu are generated, with ε i = Y ν ij (cid:68) ˜ ν cj (cid:69) and µ = λ i (cid:104) ˜ ν ci (cid:105) , once theelectroweak symmetry is broken. The last term generates the effective Majorana masses forneutrinos at the electroweak scale. Therefore, the µν SSM can generate three tiny neutrinomasses at the tree level through TeV scale seesaw mechanism [5, 29–35].In supersymmetric (SUSY) extensions of the standard model, the R-parity of a particleis defined as R = ( − L +3 B +2 S [12–16]. R-parity is violated if either the baryon number ( B )or lepton number ( L ) is not conserved, where S denotes the spin of concerned componentfield. The last two terms in Eq. (2) explicitly violate lepton number and R-parity. R-paritybreaking implies that the lightest supersymmetric particle (LSP) is no longer stable. In thiscontext, the neutralino or the sneutrino are no longer candidates for the dark matter (DM).However, other SUSY particles such as the gravitino or the axino can still be used as darkmatter candidates [5, 6, 32, 36–41].The general soft SUSY-breaking terms of the µν SSM are given by −L soft = m Q ij ˜ Q a ∗ i ˜ Q aj + m u cij ˜ u c ∗ i ˜ u cj + m d cij ˜ d c ∗ i ˜ d cj + m L ij ˜ L a ∗ i ˜ L aj m e cij ˜ e c ∗ i ˜ e cj + m H d H a ∗ d H ad + m H u H a ∗ u H au + m ν cij ˜ ν c ∗ i ˜ ν cj + (cid:15) ab (cid:104) ( A u Y u ) ij H bu ˜ Q ai ˜ u cj + ( A d Y d ) ij H ad ˜ Q bi ˜ d cj + ( A e Y e ) ij H ad ˜ L bi ˜ e cj + H . c . (cid:105) + (cid:20) (cid:15) ab ( A ν Y ν ) ij H bu ˜ L ai ˜ ν cj − (cid:15) ab ( A λ λ ) i ˜ ν ci H ad H bu + 13 ( A κ κ ) ijk ˜ ν ci ˜ ν cj ˜ ν ck + H . c . (cid:21) − (cid:16) M ˜ λ ˜ λ + M ˜ λ ˜ λ + M ˜ λ ˜ λ + H . c . (cid:17) . (3)Here, the first two lines contain mass squared terms of squarks, sleptons, and Higgses. Thenext two lines consist of the trilinear scalar couplings. In the last line, M , M , and M denote Majorana masses corresponding to SU (3), SU (2), and U (1) gauginos ˆ λ , ˆ λ , and ˆ λ ,respectively. In addition to the terms from L soft , the tree-level scalar potential receives theusual D - and F -term contributions [5, 6].Once the electroweak symmetry is spontaneously broken, the neutral scalars develop ingeneral the VEVs: (cid:104) H d (cid:105) = υ d , (cid:104) H u (cid:105) = υ u , (cid:104) ˜ ν i (cid:105) = υ ν i , (cid:104) ˜ ν ci (cid:105) = υ ν ci . (4)One can define the neutral scalars as H d = h d + iP d √ υ d , ˜ ν i = (˜ ν i ) (cid:60) + i (˜ ν i ) (cid:61) √ υ ν i ,H u = h u + iP u √ υ u , ˜ ν ci = (˜ ν ci ) (cid:60) + i (˜ ν ci ) (cid:61) √ υ ν ci , (5)and tan β = υ u υ d , υ = √ υ u + υ d . (6)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. In the following numerical section,we will use the FeynHiggs-2.13.0 [42–49] to calculate the radiative corrections for the Higgsboson mass about the MSSM part.
III. THE PROCESSES OF h → M Z
The lightest Higgs boson weak hadronic decays h → M Z are very interesting by the factthat the massive final-state gauge boson can be in a longitudinal polarization state [22].4 Z h Z h ZZ h Zγ FIG. 1: The dominating Feynman diagrams for h → M Z . In our work, in the case of h → M Z decays, M could only be a transversely polarizedvector meson [22, 50–52]. In Fig. 1, we showed the dominating Feynman diagrams for h → M Z . The first two graphs in Fig. 1 are the direct contributions, and the last twodiagrams represent the indirect contributions. In the last graph the crossed circle representsthe effective vertex h → Zγ ∗ from the one loop diagrams.The decay width of h → M Z can be given asΓ( h → M Z ) = m h πυ λ / (1 , r Z , r M )(1 − r Z − r M ) × (cid:34) | F MZ (cid:107) | + 8 r M r Z (1 − r Z − r M ) ( | F MZ ⊥ | + | ˜ F MZ ⊥ | ) (cid:35) , (7)here λ ( x, y, z ) = ( x − y − z ) − yz , r Z = m Z m h , and r M = m M m h , m M is the mass of vectormeson. We notice that the mass ratio r M is very small for all mesons, but it can makethe contributions to the transverse polarization states to the h → M Z rates which aresignificant, so we still keep the mass ratio r M in our analysis [22, 23].There are two parts of the form factor of Eq. (7), the direct and the indirect contribu-tions. We analyse the indirect contributions firstly, which may make the dominant effectsmore improvement. It involves hadronic matrix elements of local currents, hence it can becalculated to all orders in QCD [22]. So we can obtain F MZ || indirect = κ Z − r M /r Z (cid:88) q f qM υ q + C γZ α ( m M )4 π r Z − r Z − r M (cid:88) q f qM Q q ,F MZ ⊥ indirect = κ Z − r M /r Z (cid:88) q f qM υ q + C γZ α ( m M )4 π − r Z − r M r M (cid:88) q f qM Q q , (cid:101) F MZ ⊥ indirect = (cid:101) C γZ α ( m M )4 π λ / (1 , r Z , r M ) r M (cid:88) q f qM Q q , (8)where υ q = T q / − Q q s W is the vector coupling of the Z boson to the quark q . The flavor-specific decays constants f qM is defined in terms of the local matrix elements [22, 23, 52] (cid:104) M ( k, ε ) | ¯ q γ µ q | (cid:105) = − if qM m V ε ∗ µ . (9)5 esons M m M /GeV f M /GeV Q M υ M f ⊥ M /f M = f q ⊥ M /f qM ρ √ √ ( − s W ) 0.72 ω √ − s W √ φ − − + s W J/ψ
23 14 - s W − − + s W f M , Q M , υ M will be used in the numerical analysis. ZγFh ZγSh
FIG. 2: The one loop diagrams for h → γZ in the µν SSM, with F = χ denoting charged fermionsand S = U + I , D − I , S ± denoting squarks and charged scalars. We use the relations to simplify our calculation (cid:88) q f qM Q q = f M Q M , (cid:88) q f qM υ q = f M υ M . (10)The mesons decay constants f M , Q M , υ M for the vector meson M = ( ρ, ω, φ, J/ Ψ , Υ) can beseen in Table I.The concrete forms of C γZ and (cid:101) C γZ in Eq. (8) are given by [22, 23, 53] C γZ = C SMγZ + C NPγZ , (cid:101) C γZ = (cid:101) C SMγZ + (cid:101) C NPγZ , (11) C SMγZ = (cid:88) q N c Q q υ q A f ( τ q , r Z ) + (cid:88) l Q l υ l A f ( τ l , r Z ) − A γZW ( τ W , r Z ) , (cid:101) C SMγZ = (cid:88) q ˜ κ q N c Q q υ q B f ( τ q , r Z ) + (cid:88) l ˜ κ l Q l υ l B f ( τ l , r Z ) , (12)where τ i = 4 m i /m h . C SMγZ and (cid:101) C SMγZ is the standard model (SM) contribution to h → γZ .We can find the loop functions A f , A γZW and B f in Refs. [54–56].In Fig. 2, we show the one loop diagrams of h → γZ in the µν SSM, where the newphysics contributions of C γZ originate from the charginos, squarks and charged scalars. TheQCD corrections to the process h → γZ has been discussed in Ref. [57], the corrections areonly about 0.1%, which are too small to be considered.6n the standard model, the CP-odd coupling (cid:101) C SMγZ is 0. We calculate the CP-odd couplingin the framework of µν SSM, but the result is very close to 0. That is to say, the effect of CP-odd coupling (cid:101) C γZ can be neglected. Here, we can give the expression of CP-even coupling C NPγZ in the µν SSM C NPγZ = c W c W − g hS + α S − α m Z m S ± α A ( x S ± α , λ S ± α )+ (cid:88) ˜ f = U + I ,D − I N c Q ˜ f ˆ υ ˜ f g h ˜ f ˜ f m Z m f A ( x ˜ f , λ ˜ f )+ (cid:88) m,n = L,R g mhχ i χ i g nZχ i χ i m W m χ i A / ( x χ i , λ i )] , (13)where x i = 4 m i /m h , λ i = 4 m i /m Z , ˆ v f = (2 I f − Q f s W ) /c W , ˆ v ˜ f = ( I f cos θ f − Q f s W ) /c W ,ˆ v ˜ f = ( I f sin θ f − Q f s W ) /c W , θ f is the mixing angle of sfermions ˜ f , . The form factors A , A / , A and the concrete expressions of couplings g hS + α S − α , g h ˜ f ˜ f , g L,Rhχ i χ j , g L,RZχ i χ j , can be seenin Refs. [17–20, 58].Compared to the indirect contributions, the direct contributions to the decay amplitudescan be calculated in a power series in (Λ QCD /m h ) or ( m q /m h ) [22, 23]. Here, the Λ QCD is ahadronic scale, m q is the masses of the constituent quarks of a given meson. The asymptoticfunction φ ⊥ V ( x ) = 6 x (1 − x ) [23, 59–61] is needed, the direct contributions are as follow, F V Z ⊥ direct = Σ q f q ⊥ V υ q κ q m q m V − r Z + 2 r Z ln r Z (1 − r Z ) , (14)˜ F V Z ⊥ , direct = Σ q f q ⊥ V υ q ˜ κ q m q m V − r Z + 2 r Z ln r Z (1 − r Z ) . (15)Here, f q ⊥ M are the flavor-specific transverse decay constants of the meson, and f ⊥ M is thetransverse decay constant, as the authors define in Ref. [52]. In our calculations, we foundthat comparing with indirect contributions, the direct contribution is small. This type ofdirect contributions are strongly suppressed, and there is no new physics windows. Therefore,the indirect contributions are more important than the direct contributions in our study. IV. NUMERICAL RESULTS
In this section, we will discuss the numerical results. Firstly, we make the minimal flavorviolation (MFV) assumptions for some parameters, which assume κ ijk = κδ ij δ jk , ( A κ κ ) ijk = A κ κδ ij δ jk , λ i = λ, A λ λ ) i = A λ λ, Y e ij = Y e i δ ij , ( A e Y e ) ij = A e Y e i δ ij ,Y ν ij = Y ν i δ ij , ( A ν Y ν ) ij = a ν i δ ij , m ν cij = m ν ci δ ij ,m Q ij = m Q i δ ij , m u cij = m u ci δ ij , m d cij = m d ci δ ij ,m L ij = m L δ ij , m e cij = m e c δ ij , υ ν ci = υ ν c , (16)where i, j, k = 1 , , m ν ci can be constrained by the minimization conditions of theneutral scalar potential seen in Ref. [19]. To agree with experimental observations on quarkmixing, one can have Y u ij = Y u i V uL ij , ( A u Y u ) ij = A u i Y u ij ,Y d ij = Y d i V dL ij , ( A d Y d ) ij = A d Y d ij , (17)and V = V uL V d † L denotes the CKM matrix. Y u i = m u i υ u , Y d i = m d i υ d , Y e i = m l i υ d , (18)where the m u i , m d i and m l i stands for the up-quark, down-quark and charged lepton masses,and we can found the value of the masses from PDG [3]. Through our previous work [35],we have discussed in detail how the neutrino oscillation data constrain neutrino Yukawacouplings Y ν i ∼ O (10 − ) and left-handed sneutrino VEVs υ ν i ∼ O (10 − GeV) via the seesawmechanism.For a better description, we define σ M = Γ µν SSM ( h → M Z ) / Γ SM ( h → M Z ) , (19)to show the ratio of Γ µν SSM ( h → M Z ) to Γ SM ( h → M Z ). Here, M = ρ, ω, φ, J/ψ, Υ.Through analysis of the parameter space of the µν SSM in Ref. [5], we take reasonableparameter values to be κ = 0 . A λ = 500 GeV, A κ = −
300 GeV and A u , = A d = A e = 1 TeV in the following. In terms of experimental observations, the first and secondgenerations of squarks are strongly constrained by direct searches at the LHC [3], small-masssquarks are easily eliminated by experiments. Therefore, we take m ˜ Q , = m ˜ u c , = m ˜ d c , , =3 TeV. The sleptons do not have strong experimental constraints like squarks, we take m ˜ L = m ˜ e c = 1 TeV. And we will choose the gauginos’ Majorana masses M = M = µ = 3 λυ ν c forsimplicity. The gluino mass m ˜ g ≈ M , is larger than about 2.26 TeV [3]. So, here we take M = 2 . A u = A t , m ˜ u c and tan β can affect the lightest Higgs boson8 .03 0.04 0.05 0.06 0.07 0.08 0.09 0.101.001.051.101.151.20 λσ M σ ω σ ρ σ J / ψ σ ϕ σ Υ FIG. 3: (color online) The ratio σ M varies with λ .Parameters Min Max Steptan β m ˜ u c / TeV 1 4 0.2 υ ν c / TeV 3 10 0.2 λ A t / TeV 1 4 0.2TABLE II: Scanning parameters for the decay h → M Z . mass, and the parameter λ can affect the charginos mass and Higgs couplings. So, we willconsider the following parameters tan β , A u = A t , m ˜ u c , υ ν c and λ in our next analysis.The contributions of charginos account for a large proportion in the new physics. Wefocus on the impact of the new parameters λ and υ ν c , which can change charginos massand the couplings of Higgs and charginos. In Fig. 3, we plot the ratio σ M varying with λ ,the range of λ is from 0.03 to 0.1. Here we make tan β = 5, υ ν c = 3 . A t = 4 TeV, m ˜ u c = 2 . σ experimental error.If the λ is small, then it will cause the masses of chargino become small. In orderto comply with experimental observations, we excluded the low-mass charginos, keep thechargino mass at least greater than 100 GeV. Besides, λ can affect the couplings of Higgsand charginos. When the λ becomes large, the couplings will be small and the charginomasses will be big, then the new physics contributions will be decreased.In Fig. 3. We can see that the σ M become smaller and tends to 1, when the parameter λ grows. The numerical results show that σ ω is slightly big than σ ρ , because the new physics9 IG. 4: (a) σ ρ versus the parameter υ ν c /GeV and (b) σ ω versus the parameter λ . corrections for | F ωZ || | is bigger than | F ρZ || | . In Table I, we have showed the meson decayconstants, the meson decay constants of ρ (flavor eigenstates ρ = ( u ¯ u − d ¯ d ) / √
2) and ω (flavor eigenstates ω I = ( u ¯ u + d ¯ d ) / √
2) are very similar and the mass are very light andclose. Although the branch ratio of these two processes is very different in the standardmodel, the corrections of the new physics for these two processes relative to the SM are veryclose.In Fig. 3, if the mass of the meson is smaller, the new physics contributions relative tothe SM will be bigger. The larger the mass of the meson, the smaller the contribution ofnew physics relative to the SM. The mass of
J/ψ is bigger than φ , but the σ J/ψ is still biggerthan σ φ in Fig. 3, because of the effect of the meson decay constant. For the last process h → Υ Z , σ Υ is always around 1 in Fig. 3, due that the mass m Υ is so big.For better analysis, we scanned the parameters for the Higgs boson decays h → M Z .Through scanning the parameter space seen in Table II, we plot Fig. 4 and Fig. 5, wherethe dots are the corresponding physical quantity’s values of the remaining parameters afterbeing constrained by the lightest Higgs boson mass in the µν SSM with 124 .
68 GeV ≤ m h ≤ .
52 GeV, where a 3 σ experimental error is considered.In Fig. 4 (a), we can see that the σ ρ can reach about 1.18, that is to say the new physicscorrections relative to the SM can reach about 18%. As the υ ν c grows, the σ ρ tends to 1,because the big υ ν c can make the chargino masses larger, which can suppress the new physicscorrections. The effect of the new physics contributions to the process h → ωZ is a littlebigger than h → ρZ . In Fig. 4 (b), low λ can improve the new physics corrections, andthe maximum value of the is about 1.19, which means that the contributions of new physicsrelative to the SM can reach 19%. 10 IG. 5: (a) σ φ versus the parameter tan β and (b) σ J/ψ versus the parameter υ ν c /GeV. Then, we study the process h → φZ . In Fig. 5 (a), we plot σ φ varying with tan β . Inreasonable parameters space, lower tan β can enhance the new physics corrections. Fig. 5 (a)shows that the maximum value of σ φ is about 1.09. It means that Γ( h → φZ ) including thenew physics corrections can over the standard model value by 9%, which is a little smallerthan the first two processes.In standard model, the authors [22] have showed the branching ratios for the processes h → φZ and h → J/ψZ are very close. In our study, the new physics results for the twoprocesses are also very close. Fig. 5 (b) pictures σ J/ψ varying with υ ν c . The results showthat the maximum value of σ J/ψ is about 1.11, that is to say the new physics correctionscan reach to 11%.Last, we discuss the process h → Υ Z . We scanned the parameter space, but the resultsare all around 1. The SM branching ratio of h → Υ Z is bigger than the other process,but compared to other mesons we have discussed, the mass of Υ is bigger. The new physicscontributions are depressed by the heavy mass. So, the new physics to h → γZ contributionscan be negligible. V. CONCLUSION
In this paper, we have discussed the lightest Higgs boson rare decays h → M Z with themeson M = ρ, ω, φ, J/ψ, Υ in the framework of the µν SSM. The numerical results show thatthe new physics contributions to the processes h → ρZ and h → ωZ are more considerable.Compared the new physics corrections to the SM values, the maximum value of the newphysics corrections for the two processes are around 18%. The new physics corrections for11he processes h → φZ and h → J/ψZ are not great as the first two process, the maximumvalue of new physics corrections are around 10%. The SM branching ratio of h → Υ Z isbigger than the other process, the new physics corrections relative to the SM are very tinywhich can be neglected. If the final state meson is light, the new physics contributions willbe more great, the heavy final state meson will suppress the contributions of new physics.We have showed the theoretically calculable of the decay h → M Z in the µν SSM, this decayprocess is experimentally promising, which is accessible at HL-LHC or the other future highenergy colliders [24, 50].
Acknowledgments
The work has been supported by the National Natural Science Foundation of China(NNSFC) with Grants No. 11705045, No. 11535002, No. 12075074, the youth top-notchtalent support program of the Hebei Province, and Midwest Universities ComprehensiveStrength Promotion project. [1] G. Aad et al. (ATLAS Collaboration),
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