An extension of two-Higgs-doublet model and the excesses of 750 GeV diphoton, muon g-2 and h µτ → Xiao-Fang Han1, Lei Wang2,1, Jin Min Yang3,4 6 1 1 Department of Physics, Yantai University, Yantai 264005, P. R. China 0 2 2 IFIC, Universitat de Vale`ncia-CSIC, n a J Apt. Correus 22085, E-46071 Vale`ncia, Spain 1 3 Institute of Theoretical Physics, Academia Sinica, Beijing 100190, China 2 ] 4 Department of Physics, Tohoku University, Sendai 980-8578, Japan h p Abstract - p e Inthispaperwesimultaneouslyexplaintheexcessesofthe750GeVdiphoton,muong-2andh h → [ µτ in an extension of the two-Higgs-doublet model (2HDM) with additional vector-like fermions 2 v and a CP-odd scalar singlet (P) which is identified as the 750 GeV resonance. This 750 GeV 4 5 resonance has a mixingwith the CP-oddscalar (A) in 2HDM, which leads to a couplingbetween P 9 4 and the SM particles as well as a coupling between A and the vector-like fermions. Such a mixing 0 . 1 and couplings are strongly constrained by τ µγ, muon g-2 and the 750 GeV diphoton data. We 0 → 6 scan over the parameter space and find that such an extension can simultaneously account for the 1 : v observed excesses of 750 GeV diphoton, muon g-2 and h µτ. The 750 GeV resonance decays → i X in exotic modes, such as P hA, P HZ, P HA and P W H , and its width can be ± ∓ r → → → → a dozens of GeV and is sensitive to the mixing angle. PACS numbers: 12.60.Fr,14.80.Ec, 14.80.Bn 1 I. INTRODUCTION Very recently, the ATLAS and CMS collaborations have both reported an excess of 750 GeV diphoton resonance [1], with a local significance of 3.6σ and 2.6σ respectively. Combin- ing the 8 and 13 TeV data, the production cross section times the branching ratio is around 4.47 1.86 fb for CMS and 10.6 2.9 fb for ATLAS [2]. However, there are no excesses for ± ± dijet [3], tt¯[4], diboson or dilepton channels, which gives a challenge to possible new physics explanations of the 750 GeV diphoton resonance [2, 5–13]. In addition, the CMS has reported a 2.4σ excess in the lepton-flavor-violating (LFV) Higgs decay h µτ (here h is the 125 GeV SM-like Higgs), i.e., Br(h µτ) = (0.84+0.39)% → → −0.37 [14], whiletheATLASdataisBr(h µτ) = (0.7 0.62)%[15]. Thisexcess canbeexplained → ± in the general two-Higgs-doublet model (2HDM) with LFV Higgs interactions. Also such a model can give a sizable positive contribution to the muon anomalous magnetic moment (muon g-2) and accommodate the long-standing anomaly [16–18]. Attempting to simultaneously explain the excesses of 750 GeV diphoton, h µτ and → muon g-2, we in this work introduce additional vector-like fermions and a CP-odd scalar singlet (P) to the general 2HDM. The singlet P is identified as the 750 GeV resonance, which has a mixing with the CP-odd scalar (A) in the original 2HDM. Therefore, the model can lead to the P couplings to SM particles and the A couplings to vector-like fermions. In addition to the 125 GeV Higgs and 750 GeV resonance data, the LFV Higgs decay τ µγ can give strong constraints on the couplings and mixing. The dominant decays of → the 750 GeV resonance can be some exotic modes, such as P hA, P HA, P HZ → → → and P W H . Considering various relevant experimental constraints, we examine the ± ∓ → diphoton production and decay of the 750 GeV resonance, as well as muon g-2 and h µτ. → Our work is organized as follows. In Sec. II we introduce additional vector-like fermions and a CP-odd scalar singlet to the 2HDM. In Sec. III we perform numerical calculations and discuss the muon g-2, h µτ and the diphoton production and decay of the 750 GeV → resonance in the allowed parameter space. Finally, we give our conclusion in Sec. IV. 2 II. MODEL We introduce a CP-odd scalar singlet field P to the general 2HDM with the assumption 0 that P does not develop a vacuum expectation value (VEV). The Higgs potential is given 0 by [18, 19] 1 λ V = V2HDM + 2m2P0P02 + 4P0P04 −iµP0Φ†1Φ2 +h.c., (1) with V2HDM = µ1(Φ†1Φ1)+µ2(Φ†2Φ2)+ µ3Φ†1Φ2 +h.c. h i +λ1(Φ†1Φ1)2 +λ2(Φ†2Φ2)2 +λ3(Φ†1Φ1)(Φ†2Φ2)+λ4(Φ†1Φ2)(Φ†2Φ1) + λ5(Φ†1Φ2)2 +h.c. + λ6(Φ†1Φ1)(Φ†1Φ2)+h.c. h i h i + λ7(Φ†2Φ2)(Φ†1Φ2)+h.c. . (2) h i In the Higgs basis, the Φ field has a VEV v =246 GeV, and the VEV of Φ field is zero. 1 2 The two complex scalar doublets with hypercharge Y = 1 can be expressed as G+ H+ Φ =   , Φ =  . (3) 1 2 1 (v+ρ +iG ) 1 (ρ +iA ) √2 1 0 √2 2 0     The Nambu-Goldstone bosons G0 and G+ are eaten by the gauge bosons. The physical CP-even Higgs bosons h and H are the linear combinations of ρ and ρ : 1 2 ρ cosα sinα h 1   =    , (4) ρ sinα cosα H 2   −    where tan2α = 2λ v2/(m2 m2 ) with 6 h22 − h11 1 m2 = 2λ v2, m2 = m2 +v2( λ +λ ). (5) h11 1 h22 H± 2 4 5 The masses of two CP-even Higgs bosons are given as 1 m2 = m2 +m2 (m2 m2 )2 +4λ2v4 . (6) h,H 2 (cid:20) h11 h22 ∓q h11 − h22 6 (cid:21) The field H+ is the mass eigenstate of the charged Higgs boson, and the CP-odd Higgs field A has a mixing with P : 0 0 A cosθ sinθ A 0   =  −   , (7) P sinθ cosθ P 0       3 where tan2θ = 2µv/(m2 m2 ) with A0 − P0 1 m2 = m2 +v2( λ λ ). (8) A0 H± 2 4 − 5 The masses of two CP-odd scalars are given as 1 m2 = m2 +m2 (m2 m2 )2 +4µ2v2 . (9) A,P 2 h A0 P0 ∓q A0 − P0 i The 750 GeV Higgs boson P couplings to other Higgs bosons and gauge bosons as 1 PAh : c s v[(λ +λ 2λ )c λ s ] (m2 m2)s c , θ θ 3 4 − 5 α − 7 α − 4v A − P 4θ α 1 PAH : c s v[(λ +λ 2λ )s +λ c ] (m2 m2)s s , θ θ 3 4 − 5 α 7 α − 4v A − P 4θ α e PhZ : s s (p p )µ, α θ 1 2 − 2s c − W W e PHZ : c s (p p )µ, α θ 1 2 2s c − W W e PH W : s (p p )µ. (10) ± ∓ θ 2 1 2s − W The general Yukawa interactions of the SM fermions are given by = y Q Φ˜ u + y Q Φ d + y L Φ e u L 1 R d L 1 R ℓ L 1 R −L +ρuQ Φ˜ u + ρdQ Φ d + ρℓL Φ e + h.c., (11) L 2 R L 2 R L 2 R where QT = (u ,d ), LT = (ν ,l ), Φ = iτ Φ , and y , y , y , ρu, ρd and ρℓ are 3 3 L L L L L L 1,2 2 ∗1,2 u d ℓ × matrices in family space. e Also, we introduce a singlet quark with 2 electric charge and multiple singlet leptons. 3 The Yukawa interactions of vector-like fermions are written as = m T¯ T +i y P T¯ γ T + m L¯ L +i y P L¯ γ L . (12) T T 0 5 Li i i Li 0 i 5 i −L Xi (cid:0) (cid:1) Then we obtain the Yukawa couplings of the neutral Higgs bosons: mf ρf mf ρf y = i c δ ij s , y = i s δ + ij c , hij α ij α Hij α ij α v − √2 v √2 ρf ρf ij ij y = i c (for u), y = i c (for d, ℓ), Aij θ Aij θ − √2 √2 ρf ρf ij ij y = i s (for u), y = i s (for d, ℓ), Pij θ Pij θ √2 − √2 y = iy s , y = iy s , ATT T θ ALiLi Li θ y = iy c , y = iy c . (13) PTT T θ PLiLi Li θ 4 For the diagonal matrix elements of ρu, ρd and ρℓ, we take √2mu √2md √2mℓ ρu = i κ , ρd = i κ , ρℓ = iκ , (14) ii v u ii v d ii v ℓ which corresponds to the aligned 2HDM [20]. We assume that ρℓ and ρℓ are nonzero, and µτ τµ other nondiagonal matrix elements of ρu, ρd and ρℓ are zero. The vector-like quark is introduced to make the 750 GeV Higgs singlet to beproduced via the gluon-gluon fusion process. However, the vector-like quark can also enhance the cross section of gg A, which will be constrained by the experimental data from the ATLAS → and CMS searches. Therefore, we expect that the vector-like leptons play the main role in enhancing the 750 GeV diphoton production rate. The decay P γγ can be enhanced by → the vector-like leptons, and its amplitude is proportional to the couplings and the square of electric charge. Here we do not discuss the electric charge and coupling of every vector-like lepton as well as the quantity of vector-like leptons in detail; instead we focus on the total contribution of vector-like leptons, which depends on Y = y Q2 , (15) L Li Li Xi where L denotes the i-th vector-like lepton. i III. NUMERICAL CALCULATIONS AND DISCUSSIONS A. Numerical calculations In our calculations, we scan over the parameters in the following range 0.06 < s < 0.06, 0.3 < s < 0.3, α θ − − 0.05 < ρ = ρ < 1, 50 < κ < 50, µτ τµ ℓ − 0 < Y < 50, 0 < λ , λ < 4π, L 3 7 200 GeV < m < 450 GeV, (16) H and fix mh = 125.5 GeV mP = 750 GeV, mH± = mA = 500 GeV, m = 400 GeV, m = 700 GeV, y = 2.0, Li T T κ = κ = 0. (17) u d 5 During the scan, we consider the following experimental constraints and observables: (1) Precision electroweak data. According to the expressions for the oblique parameters S, T and U in the 2HDM [21], for 0.06 < s < 0.06 and c 1, the expressions in α α − ≃ this model are approximately given as 1 S = c2c2F (m2,m2 ,m2)+c2s2F (m2,m2 ,m2) F (m2,m2 ,m2 ) , πm2 α θ S Z H A α θ S Z H P − S Z H± H± Z (cid:2) (cid:3) 1 T = c2c2F (m2 ,m2) c2s2F (m2 ,m2)+c2F (m2 ,m2 ) 16πm2 s2 − α θ T H A − α θ T H P α T H± H W W (cid:2) + c2F (m2 ,m2)+s2F (m2 ,m2) , θ T H± A θ T H± P 1 (cid:3) U = c2F (m2 ,m2 ,m2 ) 2F (m2 ,m2 ,m2 ) πm2 α S W H± H − S W H± H± W (cid:2) +c2F (m2 ,m2 ,m2)+s2F (m2 ,m2 ,m2) θ S W H± A θ S W H± P 1 (cid:3) c2c2F (m2,m2 ,m2)+c2s2F (m2,m2 ,m2) −πm2 α θ S Z H A α θ S Z H P Z (cid:2) F (m2,m2 ,m2 ) , (18) − S Z H± H± (cid:3) where 1 ab a F (a,b) = (a+b) log( ), F (a,b,c) = B (a,b,c) B (0,b,c) (19) T S 22 22 2 − a b b − − with 1 1 1 1 B (a,b,c) = b+c a dx Xlog(X iǫ), 22 4 (cid:20) − 3 (cid:21)− 2 Z − 0 X = bx+c(1 x) ax(1 x). (20) − − − Here we require [22] S = 0.03 0.1, T = 0.01 0.12, U = 0.05 0.1 (21) − ± ± ± (2) The 125 GeV Higgs data. For 0.06 < s < 0.06, κ = κ = 0 and 50 < κ < 50, α u d ℓ − − the 125 GeVHiggs couplings to the gauge bosons, up-type quark and down-type quark are very close to the SM values, but the coupling to τ¯τ can have a sizable deviation from the SM value. The signal strength of τ¯τ channel is µˆ = 1.41+0.4 from ATLAS ττ 0.35 − [23] and µˆ = 0.89+0.31 from CMS [24]. We require 0.33 < µˆ < 2.21 and such a ττ 0.28 ττ − bound will give strong constrains on s and κ for which the absolute value of the α ℓ coupling of the 125 GeV Higgs and τ¯τ is around the SM value. 6 (3) Non-observation of additional Higgs bosons. For 0.06 < s < 0.06 and κ = κ = 0, α u d − the cross sections of H and H at the collider are very small, and hence H and H can ± ± be hardly constrained by the current experimental data from the ATLAS and CMS searches. The pseudoscalar A can be produced via the gluon-gluon fusion process with vector-like quark loop, and the decay A γγ, A γZ and A ZZ can be enhanced → → → by the vector-like quark and leptons at one-loop level. For m =500 GeV, we impose A the following relevant bounds at the 8 TeV LHC [25–29] R < 6 fb, R < 45 fb, R < 6.8 fb, R < 60 fb, R < 26 fb. (22) γγ ZZ Zγ hZ τ¯τ (4) The 750 GeV resonance data. The 750 GeV Higgs singlet P can be produced via the gluon-gluon fusion process with vector-like quark loop, and the decays A γγ, → A γZ and A ZZ can be enhanced by the vector-like quark and leptons at one- → → loop level. Due to the mixing with A, the 750 GeV singlet can decay into the SM particles, such as hZ and τ¯τ. For the 750 GeV Higgs singlet, we impose the following bounds at the 8 TeV LHC [25–29] R < 2 fb, R < 12 fb, R < 4 fb, R < 19 fb, R < 12 fb. (23) γγ ZZ Zγ hZ τ¯τ At the 13 TeV LHC, we require the 750 GeV diphoton production rate as 2 fb < R < 10 fb. (24) γγ (5) The data of Br(h µτ). The branching ratio of h µτ is given by → → s2(ρ2 +ρ2 )m α µτ τµ h Br(h µτ) = , (25) → 16πΓ h where Γ is the total width of 125 GeV Higgs. To explain the h µτ excess reported h → by CMS within 2σ range, we require 0.1% < Br(h µτ) < 1.62%. (26) → (6) Themuong-2data. Thedominantcontributionstothemuong-2arefromtheone-loop diagrams with the Higgs LFV coupling [30], δa = mµmτρµτρτµ s2α(log mm2h2τ − 23) + c2α(log mm2H2τ − 32) µ1 16π2 m2 m2 h H  c2log(m2A 3) s2log(m2P 3) θ m2τ − 2 θ m2τ − 2 . (27) − m2 − m2 A P  7 The muon g-2 can be also corrected by the two-loop Barr-Zee diagrams with the fermions loops, W and Goldstone loops. Using the well-known classical formulates [31], the main contributions of two-loop Barr-Zee diagrams in this model are given as αm δa = µ Nc Q2 y y F (x ) µ2 −4π3m f f φµµ φff φ fφ f φ=h,H,A,PX;f=t,b,τ,T,Li αm 23 µ + y g 3F (x )+ F (x ) 8π3v φµµ φWW (cid:20) H Wφ 4 A Wφ φX=h,H 3 m2 φ + G(x )+ F (x ) F (x ) , (28) 4 Wφ 2m2 { H Wφ − A Wφ }(cid:21) W where x = m2/m2, x = m2 /m2, g = s , g = c and fφ f φ Wφ W φ HWW α hWW α y 1 1 2x(1 x) x(1 x) F (y) = F (y) = dx − − log − (for φ = h, H) (29) φ H 2 Z x(1 x) y y 0 − − y 1 1 x(1 x) F (y) = F (y) = dx log − (for φ = A, P) (30) φ A 2 Z x(1 x) y y 0 − − y 1 1 y x(1 x) G(y) = dx 1 log − . (31) −2 Z x(1 x) y (cid:20) − x(1 x) y y (cid:21) 0 − − − − The experimental value of muon g-2 excess is [32] δa = (26.2 8.5) 10 10. (32) µ − ± × (7) The data of Br(τ µγ). The LFV coupling of the Higgs boson gives the dominant → contributions to the decay τ µγ. The branching ratio of τ µγ is given by → → BR(τ µγ) 48π3α( A +A +A 2 + A +A +A 2) 1L0 1Lc 2L 1R0 1Rc 2R → = | | | | , (33) BR(τ µν¯ ν ) G2 → µ τ F whereA ,A , A andA arefromtheone-loopdiagramswiththeHiggsbosons 1L0 1Lc 1R0 1Rc and tau lepton [17], and y m2 3 y φ∗τµ φ φττ A = y log + , (34) 1L0 16π2m2 (cid:20) φ∗ττ (cid:18) m2 − 2(cid:19) 6 (cid:21) φ=h,XH, A, P φ τ (ρe ρe)µτ † A = , (35) 1Lc −192π2m2 H− A = A y y , y y , (36) 1R0 1L0 φ∗τµ → φµτ φττ ↔ φ∗ττ (cid:0) (cid:1) A = 0. (37) 1Rc 8 Here A and A are from the two-loop Barr-Zee diagrams with the third-generation 2L 2R fermions loops, vector-like fermions loops and W loops [17]: N Q α y C f φ∗τµ A = [Q Re(y )F (x ) iIm(y )F (x ) 2L − 8π3 m m f { φff H fφ − φff A fφ } φ=h,H,A,PX;f=t,b,τ,T,Li τ f (1 4s2 )(2T 4Q s2 ) + − W 3f − f W Re(y )F˜ (x ,x ) iIm(y )F˜ (x ,x ) 16s2 c2 φff H fφ fZ − φff A fφ fZ (cid:21) W W n o α g y 23 φWW φ∗τµ + 3F (x )+ F (x ) 16π3 m v (cid:20) H Wφ 4 A Wφ φX=h,H τ 3 m2 φ + G(x )+ F (x ) F (x ) 4 Wφ 2m2 { H Wφ − A Wφ } W 1 4s2 1 t2 + − W 5 t2 + − W F˜ (x ,x ) 8s2 (cid:26)(cid:18) − W 2x (cid:19) H Wφ WZ W Wφ 1 t2 3 + 7 3t2 − W F˜ (x ,x )+ F (x )+G(x ) , (38) (cid:18) − W − 2x (cid:19) A Wφ WZ 2 { A Wφ Wφ }(cid:27)(cid:21) Wφ A = A y y , i i , (39) 2R 2L φ∗τµ → φµτ → − (cid:0) (cid:1) where T denotes the isospin of the fermion, t2 = s2 /c2 , x = m2/m2 and 3f W W W fZ f Z x = m2 /m2, and WZ W Z xF (y) yF (x) F˜ (x,y) = H − H , (40) H x y − xF (y) yF (x) F˜ (x,y) = A − A . (41) A x y − The terms in the first two lines of Eq. (38) come from the effective φγγ vertex and φZγ vertex induced by the third-generation fermion loop and vector-like fermion loop. Other terms arefromtheeffective φγγ vertex and φZγ vertex induced by the W-boson loop. The current upper bound of Br(τ µγ) is [33, 34] → Br(τ µγ) < 4.4 10 8. (42) − → × B. Results and discussions In Fig. 1, we project the surviving samples on the planes of κ versus s , ρ versus ℓ α µτ m , Y versus κ , and ρ versus s . The upper-left panel shows that there is a strong H L ℓ µτ α correlation between s and κ due to the experimental constraints of the τ¯τ channel data of α ℓ 125 GeV Higgs. The surviving samples have two different 125 GeV Higgs couplings to τ¯τ, 9 0.7 40 0.6 20 0.5 0.4 kl0 mt r 0.3 -20 0.2 -40 0.1 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 200 250 300 350 400 450 s m (GeV) a H 50 0.7 0.6 40 0.5 30 L 0.4 Y mt r 20 0.3 0.2 10 0.1 0 -40 -20 k0 20 40 -0.06 -0.04 -0.02 s0 0.02 0.04 0.06 l a FIG. 1: Under the constrains of the oblique parameters and the LHC Higgs data, the surviving samples projected on the planes of κ versus s , ρ versus m , Y versus κ and ρ versus s . ℓ α µτ H L ℓ µτ α The circles (green) are allowed by the muon g-2, the pluses (red) allowed by the muon g-2 and Br(τ µγ), and the bullets (black) allowed by the muon g-2, Br(τ µγ) and Br(h µτ). → → → and their absolute values are around the SM value. One is the SM-like Higgs coupling with the same sign as the coupling of the gauge boson, and the other is the Yukawa coupling with the opposite sign to the coupling of the gauge boson for a relative large κ . ℓ From the upper-right panel of Fig. 1, we see that the muon g-2 favors ρ to increase µτ with m . As shown in Eq. (27), the muon g-2 can obtain positive contributions from the H H loop and negative contributions from A and P loops for ρ = ρ . With the decreasing µτ τµ of the mass splitting of H and A, the cancelation between the contributions of H and A 10