h → μτ and muon g − 2 in the alignment limit of two-Higgs-doublet model

Available online at www.sciencedirect.com

ScienceDirect Nuclear Physics B 919 (2017) 123–141 www.elsevier.com/locate/nuclphysb

h → μτ and muon g − 2 in the alignment limit of two-Higgs-doublet model Lei Wang a , Shuo Yang b,∗ , Xiao-Fang Han a,∗ a Department of Physics, Yantai University, Yantai 264005, PR China b Department of Physics, Dalian University, Dalian 116622, PR China

Received 28 September 2016; received in revised form 19 February 2017; accepted 14 March 2017 Available online 18 March 2017 Editor: Hong-Jian He

Abstract We examine the h → μτ and muon g − 2 in the exact alignment limit of two-Higgs-doublet model. In this case, the couplings of the SM-like Higgs to the SM particles are the same as the Higgs couplings in the SM at the tree level, and the tree-level lepton–flavor–violating coupling hμτ is absent. We assume the lepton–flavor–violating μτ excess observed by CMS to be respectively from the other neutral Higgses, H and A, which almost degenerates with the SM-like Higgs at the 125 GeV. After imposing the relevant theoretical constraints and experimental constraints from the precision electroweak data, B-meson decays, τ decays and Higgs searches, we find that the muon g − 2 anomaly and μτ excess favor the small lepton Yukawa coupling and top Yukawa coupling of the non-SM-like Higgs around 125 GeV, and the lepton– flavor–violating coupling is sensitive to another heavy neutral Higgs mass. In addition, if the μτ excess is from H around 125 GeV, the experimental data of the heavy Higgs decaying into μτ favor mA > 230 GeV for a relatively large H t¯t coupling. © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

1. Introduction The ATLAS and CMS Collaborations have probed the lepton–flavor–violating (LFV) Higgs decay h → μτ around 125 GeV at the LHC run-I [1–3] and early run-II [4]. By the analysis of * Corresponding authors.

E-mail addresses: [email protected] (S. Yang), [email protected] (X.-F. Han). http://dx.doi.org/10.1016/j.nuclphysb.2017.03.013 0550-3213/© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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√ data sample corresponding to an integrated luminosity of 20.3 fb−1 at the s = 8 TeV LHC, the ATLAS Collaboration found a mild deviation of 1σ significance in the h → μτ channel and set an upper limit of Br(h → μτ ) < 1.43% at 95% confidence level with a best fit Br(h → μτ ) = (0.53 ± 0.51)% √ [2]. Based on the data sample corresponding to an integrated luminosity of 19.7 fb−1 at the s = 8 TeV LHC, the CMS Collaboration imposed an upper limit of Br(h → μτ ) < 1.51% at 95% confidence level, while the best fit value is Br(h → μτ ) = (0.84+0.39 −0.37 )% √ with a small excess of 2.4σ [3]. At the s = 13 TeV LHC run-II with an integrated luminosity of 2.3 fb−1 , the CMS Collaboration did not observe the excess and imposed an upper limit of Br(h → μτ ) < 1.2% [4]. However, the CMS search result at the early LHC run-II can not definitely kill the excess of h → μτ due to the low integrated luminosity. If the h → μτ excess is not a statistical fluctuation, the new physics with the LFV interactions can give a simple explanation for the excess. On the other hand, the long-standing anomaly of the muon anomalous magnetic moment (muon g − 2) implies that the new physics is connected to muons. The two excesses can be simultaneously explained by the LFV Higgs interactions, such as the general two-Higgs-doublet model (2HDM) with the LFV Higgs interactions. There have been many studies on the h → μτ excess in the framework of 2HDM [5–7] and some other new physics models [8]. In this paper, we discuss the excesses of h → μτ and muon g − 2 in the exact alignment limit of the general 2HDM where one of the neutral Higgs mass eigenstates is aligned with the direction of the scalar field vacuum expectation value (VEV) [9]. In the interesting scenario, the SM-like Higgs couplings to the SM particles are the same as the Higgs couplings in the SM at the tree level, and the tree-level LFV coupling hμτ is absent. We assume the μτ excess observed by CMS to be respectively from the other neutral Higgses, H and A, which almost degenerates with the SM-like Higgs at the 125 GeV. In our discussions, we impose the relevant theoretical constraints from the vacuum stability, unitarity and perturbativity as well as the experimental constraints from the precision electroweak data, B-meson decays, τ decays and Higgs searches. Our work is organized as follows. In Sec. 2 we recapitulate the alignment limit of 2HDM. In Sec. 3 we perform the numerical calculations and discuss the muon g − 2 anomaly and the μτ excess around 125 GeV after imposing the relevant theoretical and experimental constraints. Finally, we give our conclusion in Sec. 4. 2. Two-Higgs-doublet model and the alignment limit The alignment limit of 2HDM is defined as the limit in which one of the two neutral CP-even Higgs mass eigenstates aligns with the direction of the scalar field VEV [9]. The alignment limit can be easily realized in the decoupling limit [10], namely that all the non-SM-like Higgses are very heavy. The possibility of alignment without decoupling limit was first noted in [10], “reinvented” in [11–13] and further studied in [9,14–17]. The alignment limit is basis-independent, and clearly exhibited in the Higgs basis. The alignment limit also exists in the Minimal Supersymmetric Standard Model which is a constrained incarnation of the general 2HDM. There are some detailed discussions in [18,19] and a very recent study in [20]. 2.1. Two-Higgs-doublet model in the Higgs basis The general Higgs potential is written as [21]

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  V = μ1 (H1† H1 ) + μ2 (H2† H2 ) + μ3 (H1† H2 + h.c.) k2 k1 † (H1 H1 )2 + (H2† H2 )2 + k3 (H1† H1 )(H2† H2 ) + k4 (H1† H2 )(H2† H1 ) 2 2     k5 † (H1 H2 )2 + h.c. + k6 (H1† H1 )(H1† H2 ) + h.c. + 2   + k7 (H2† H2 )(H1† H2 ) + h.c. .

+

(1)

All μi and ki are real in the CP-conserving case. In the Higgs basis, the H1 field has a VEV v = 246 GeV, and the VEV of H2 field is zero. The two complex scalar doublets have the hypercharge Y = 1,     G+ H+ H1 = √1 , H2 = √1 . (2) (v + ρ1 + iG0 ) (ρ2 + iA0 ) 2

2

The Nambu–Goldstone bosons G0 and G+ are eaten by the gauge bosons. The H ± and A are the mass eigenstates of the charged Higgs boson and CP-odd Higgs boson, and their masses are given by 1 m2A = m2H ± + v 2 (k4 − k5 ). 2 The physical CP-even Higgs bosons h and H are the linear combinations of ρ1 and ρ2 ,    ρ1 sθ h cθ = , ρ2 cθ −sθ H and their masses are given as  

1 m2h,H = m2A + (k1 + k5 )v 2 ∓ [m2A + (k5 − k1 )v 2 ]2 + 4k62 v 4 . 2

(3)

(4)

(5)

Where sθ ≡ sin θ and cθ ≡ cos θ , cos θ =

−k6 v 2

.

(6)

(m2H − m2h )(m2H − k1 v 2 )

In this paper we take the light CP-even Higgs h as the 125 GeV Higgs. For cos θ = 0, the mass eigenstates of CP-even Higgs bosons are obtained from the Eq. (4), h = ρ1 ,

H = −ρ2 ,

(7)

which is so called “alignment limit”. The Eq. (6) shows that the alignment limit can be realized in two ways: k6 = 0 or m2H  v 2 . The latter is called the decoupling limit. In this paper we focus on the former, which is the alignment without decoupling limit. In the alignment limit, the h couplings to gauge bosons are the same as the Higgs couplings in the SM, and the H has no couplings to gauge bosons. 2.2. The Higgs couplings We can rotate the Higgs basis by a mixing angle β,

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1 2



 =

cβ sβ

−sβ cβ



H1 H2

.

(8)

Where sβ ≡ sin β, cβ ≡ cos β, and tan β = v2 /v1 with v2 and v1 being the VEVs of 2 and 1 and v 2 = v12 + v22 = (246 GeV)2 . The general Higgs potential is written as [21]   V = m211 (†1 1 ) + m222 (†2 2 ) − m212 (†1 2 + h.c.) λ1 † λ2 (1 1 )2 + (†2 2 )2 + λ3 (†1 1 )(†2 2 ) + λ4 (†1 2 )(†2 1 ) 2  2   λ5 † 2 + (1 2 ) + h.c. + λ6 (†1 1 )(†1 2 ) + h.c. 2   + λ7 (†2 2 )(†1 2 ) + h.c. . +

(9)

The parameters mij and λi are the linear combinations of the parameters in the Higgs basis: μi and ki . The detailed expressions are introduced in [9,22]. After spontaneous electroweak symmetry breaking, there are five physical Higgses: two neutral CP-even h and H , one neutral pseudoscalar A, and two charged scalar H ± . The general Yukawa interaction can be given as ˜ 1 uR + Yd1 QL 1 dR + Y 1 LL 1 eR −L = Yu1 QL  ˜ 2 uR + Yd2 QL 2 dR + Y 2 LL 2 eR + h.c. , + Yu2 QL 

(10)

1,2 = iτ2 ∗ , and Yu1,2 , Yd1,2 and Y 1,2 are 3 × 3 where QTL = (uL , dL ), LTL = (νL , lL ),  1,2 matrices in family space. To avoid the tree-level FCNC couplings of the quarks, we take Yu1 = cu ρu ,

Yu2 = su ρu ,

Yd1 = cd ρd ,

Yd2 = sd ρd ,

(11)

where cu ≡ cos θu , su ≡ sin θu , cd ≡ cos θd , sd ≡ sin θd and ρu (ρd ) is the 3 × 3 matrix. For this choice, the interaction corresponds to the aligned 2HDM [23,24]. For the Yukawa coupling matrix of the lepton, we take √ 2m i Xii = (sβ + cβ κ ), v Xτ μ = c β ρ τ μ , Xμτ = cβ ρμτ .

(12)

Where X = VL Y 2 VR† , and VL (VR ) is the unitary matrix which transforms the interaction eigenstates to the mass eigenstates of the left-handed (right-handed) lepton fields. The other nondiagonal matrix elements of X are zero. The Yukawa couplings of the neutral Higgs bosons are given as

mfi yhfi fi = sin(β − α) + cos(β − α)κf , v

mfi yHfi fi = cos(β − α) − sin(β − α)κf , v mf mf yAfi fi = −i i κf (for u), yAfi fi = i i κf (for d, ), v v

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ρτ μ ρμτ yhτ μ = cos(β − α) √ , yhμτ = cos(β − α) √ , 2 2 ρτ μ ρμτ yH τ μ = − sin(β − α) √ , yH μτ = − sin(β − α) √ , 2 2 ρτ μ ρμτ yAτ μ = i √ , yAμτ = i √ . 2 2

127

(13)

Where κu ≡ − tan(β − θu ), κd ≡ − tan(β − θd ). The κ is a free input parameter, which is used to parameterize the matrix element of the lepton Yukawa coupling, as shown in Eq. (12). In other words, the matrix elements of the lepton Yukawa coupling are taken as the Eq. (12) in order to obtain the Yukawa couplings of lepton in Eq. (13). The neutral Higgs bosons couplings to the gauge bosons normalized to the SM Higgs boson are given by yVh = sin(β − α),

yVH = cos(β − α),

(14)

where V denotes Z and W . In the exact alignment limit, namely cos(β − α) = 0, the Eq. (13) and Eq. (14) show that the 125 GeV Higgs (h) has the same couplings to the fermions and gauge bosons as the SM values, and the tree-level LFV couplings are absent. The heavy CP-even Higgs (H ) has no coupling to the gauge bosons, and there are the tree-level LFV couplings for the A and H . 3. Numerical calculations and discussions 3.1. Numerical calculations In the exact alignment limit, the SM-like Higgs has no tree-level LFV coupling. In order to explain the h → μτ excess reported by CMS, we assume the signal to be respectively from H and A, which almost degenerates with the SM-like Higgs at the 125 GeV. Here we take two scenarios simply: (i) mA = 126 GeV and (ii) mH = 126 GeV. In our calculations, the other involved parameters are randomly scanned in the following ranges: −(400 GeV)2 ≤ m212 ≤ (400 GeV)2 ,

0.1 ≤ tan β ≤ 10,

100 GeV ≤ mH ± ≤ 700 GeV, 0 ≤ κu ≤ 1.2,

−150 ≤ κ ≤ 150,

−0.3 ≤ ρτ μ ≤ 0.3

Scenario i : mA = 126 GeV,

150 GeV ≤ mH ≤ 700 GeV,

ρμτ = −ρτ μ ,

Scenario ii : mH = 126 GeV,

150 GeV ≤ mA ≤ 700 GeV,

ρμτ = ρτ μ .

(15)

In order to relax the constraints from the observables of down-type quarks, we take κd = 0. For the cases of mA = 126 GeV and mH = 126 GeV, we respectively take ρμτ = −ρτ μ and ρμτ = ρτ μ to produce a large positive contribution to the muon g − 2. The pseudoscalar A can give the positive contributions to the muon g − 2 via the two-loop Barr–Zee diagrams with the lepton–flavor-conserving (LFC) coupling. Therefore, we take |κ | < 150 to examine the possibility of explaining the muon g − 2. In the exact alignment limit, the hτ τ¯ coupling is independent on κ and equals the SM value. However, the Aτ τ¯ and H τ τ¯ couplings can reach 1.08 and be slightly larger than 1 for |κ | = 150, which can not lead to the problem on the perturbativity due to the suppression of the loop factor. In addition, for such large κ the Br(A → τ τ¯ ) and

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Br(H → τ τ¯ ) can reach 1. Due to κd = 0 and cos(β − α) = 0, the cross sections of A and H are equal to zero in the bb¯ associated production mode and vector boson fusion production mode. However, the searches for gg → A/H → τ τ¯ can give the constraints on κu . We will discuss the constraints in the following item (5). During the scan, we consider the following experimental constraints and observables: (1) Theoretical constraints and precision electroweak data. We use 2HDMC-1.6.5 [25] to implement the theoretical constraints from the vacuum stability, unitarity and coupling-constant perturbativity, as well as the constraints from the oblique parameters (S, T , U ) and δρ. (2) B-meson decays and Rb . Although the tree-level FCNCs in the quark sector are absent, they will appear at the one-loop level in this model. We consider the constraints of B-meson decays from mBs , mBd , B → Xs γ , and Bs → μ+ μ− , which are respectively calculated using the formulas in [26–28]. In addition, we consider the Rb constraints, which is calculated following the formulas in [29]. In fact, in this paper we take κd = 0 and 0 ≤ κu ≤ 1.2, which will relax the constraints from the bottom-quark observables sizably. (3) τ decays. In this model, the non-SM-like Higgses have the tree-level LFV couplings to τ lepton, and the LFC couplings to lepton can be sizably enhanced for −150 ≤ κ ≤ 150. Therefore, some τ decay processes can give very strong constraints on the model. (i) τ → 3μ. In the exact alignment limit, the LFV Aτ μ and H τ μ couplings generate the τ → μ+ μ− μ− process at the tree level, and the corresponding Feynman diagrams are shown in Fig. 1. The branching ratio of τ → 3μ is given as [30]  Br(τ → 3μ) I (φ1 , φ2 ) , (16) = Br(τ → μ¯ν ν) 64G2F φ ,φ =A,H 1

2

where I (φ1 , φ2 ) = 2

yφ1 μτ yφ∗1 μμ yφ∗2 μτ yφ2 μμ

+

m2φ1

∗

m2φ2

∗

yφ1 μτ yφ1 μμ yφ2 μτ yφ2 μμ m2φ1

+2

m2φ2

+

yφ1 τ μ yφ∗1 μμ yφ∗2 τ μ yφ2 μμ m2φ1

∗

m2φ2 ∗

yφ1 τ μ yφ1 μμ yφ2 τ μ yφ2 μμ m2φ1

m2φ2

.

(17)

The current experimental upper bound of Br(τ → 3μ) is [31], Br(τ → 3μ) < 2.1 × 10−8 .

(18)

From the Eq. (17), we can find that the experimental data of Br(τ → 3μ) will give the very strong constraints on the products, ρτ μ × κ and ρμτ × κ . (ii) τ → μγ . The main Feynman diagrams of τ → μγ in the model are shown in Fig. 2. In the exact alignment limit, the SM-like Higgs has no tree-level LFV coupling, and the heavy CP-even Higgs couplings to the gauge bosons are equal to zero. Therefore, the SM-like Higgs does not contribute to the τ → μγ , and the τ → μγ can not be corrected via the two-loop Barr–Zee diagrams with the W loop. The Br(τ → μγ ) in this model is given by   48π 3 α |A1L0 + A1Lc + A2L |2 + |A1R0 + A1Rc + A2R |2 BR(τ → μγ ) , (19) = BR(τ → μν¯ ν) G2F

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Fig. 1. The main Feynman diagrams of τ − → μ− μ− μ+ . The two μ− in final states can be exchanged. In the exact alignment limit Hk0 denotes A and H .

Fig. 2. The main Feynman diagrams of τ → μγ . In the exact alignment limit Hk0 denotes A and H .

where A1L0 , A1Lc , A1R0 and A1Rc are from the one-loop diagrams with the Higgs boson and τ lepton [6],      m2φ 3 yφ∗ τ μ yφ τ τ ∗ A1L0 = + , (20) yφ τ τ log 2 − 2 6 mτ 16π 2 m2φ φ=H, A (ρ e† ρ e )μτ , 192π 2 m2H −  A1R0 = A1L0 yφ∗ τ μ → yφ μτ , A1Lc = −

(21)  yφ τ τ ↔ yφ∗ τ τ ,

A1Rc = 0.

(22) (23)

The A2L and A2R are from the two-loop Barr–Zee diagrams with the third-generation fermion loop [6], ∗ NC Qf α yφ τ μ A2L = − 8π 3 mτ mf φ=H,A;f =t,b,τ      × Qf Re(yφ ff )FH xf φ − iIm(yφ ff )FA xf φ



+

2 )(2T 2 (1 − 4sW 3f − 4Qf sW )

2 c2 16sW W     ˜ , × Re(yφ ff )FH xf φ , xf Z − iIm(yφ ff )F˜A xf φ , xf Z   A2R = A2L yφ∗ τ μ → yφ μτ , i → −i ,



where T3f denotes the isospin of the fermion, and

(24)

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Fig. 3. The main Feynman diagram of τ → μπ 0 .

y FH (y) = 2

1 dx

1 − 2x(1 − x) x(1 − x) log (for φ = H ), x(1 − x) − y y

dx

1 x(1 − x) log (for φ = A), x(1 − x) − y y

0

FA (y) =

y 2

1 0

xFH (y) − yFH (x) , F˜H (x, y) = x −y xFA (y) − yFA (x) F˜A (x, y) = . x−y

(25)

The two terms of A2L come from the effective φγ γ vertex and φZγ vertex induced by the third-generation fermion loop. The current experimental data give an upper bound of Br(τ → μγ ) [32], Br(τ → μγ ) < 4.4 × 10−8 .

(26)

(iii) τ → μπ 0 . The τ can decay into a lepton and a pseudoscalar meson at the tree level via the CP-odd Higgs with the LFV couplings, such as τ → μπ 0 . The corresponding Feynman diagrams are shown in Fig. 3. The width of τ → μπ 0 is given as [33], (τ → μπ 0 ) =

fπ2 m4π mτ (|ρτ μ |2 + |ρμτ |2 )(κu + κd )2 . 512πm4A v 2

(27)

The current upper bound of Br(τ → μπ 0 ) is [34], Br(τ → μπ 0 ) < 1.1 × 10−7 .

(28)

(4) Muon g −2. The dominant contributions to the muon g −2 are from the one-loop diagrams with the Higgs LFV coupling [35], and the corresponding Feynman diagrams can be obtained by replacing the initial states τ with μ in Fig. 2(a) and Fig. 2(b). In the exact alignment limit, ⎡ ⎤ m2H m2A 3 3 log( − ) − ) (log 2 2 ⎥ mμ mτ ρμτ ρτ μ ⎢ m2τ m2τ LF V (29) δaμ1 = − ⎦. ⎣ 2 2 2 16π mH mA At the one-loop level, the diagrams with the Higgs LFC coupling can also give the contributions to the muon g − 2, especially for a large lepton Yukawa coupling [36]. The corresponding Feynman diagrams can be obtained by replacing τ in the initial state and loop with μ in Fig. 2(a)

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as well as replacing the initial state τ with μ and ντ in the loop with νμ in Fig. 2(b). The contributions from the one-loop diagrams with the Higgs LFC coupling are given as  1 LF C aμ1 = |yφμμ |2 rφμ fφ (rφμ ), (30) 2 8π ± φ=h, H, A, H

where rφμ = m2μ /m2φ and yH ± μμ = yAμμ . For rφμ 1, fh,H (r)  − ln r − 7/6,

fA (r)  ln r + 11/6,

fH ± (r)  −1/6.

(31)

The muon g − 2 can be corrected by the two-loop Barr–Zee diagrams with the fermions loops by replacing the initial state τ with μ in Fig. 2(c). Further replacing the fermion loop with W loop, we obtain the two-loop Barr–Zee diagrams with W loop which can contribute to muon g − 2 for the SM-like Higgs h as the mediator in the exact alignment limit. Using the well-known classical formulates in [37], the main contributions of two-loop Barr–Zee diagrams in the exact alignment limit are given as  αmμ δaμ2 = − 3 Nfc Q2f yφμμ yφff Fφ (xf φ ) 4π mf φ=h,H,A;f =t,b,τ    23   αmμ  y g FA xW φ + φμμ φW W 3FH xW φ + 3 4 8π v φ=h  2 mφ       3  + G xW φ + F H xW φ − F A x W φ , (32) 4 2m2W where xf φ = m2f /m2φ , xW φ = m2W /m2φ , ghW W = 1 and y G(y) = − 2

1 0

  1 y x(1 − x) dx 1− log . x(1 − x) − y x(1 − x) − y y

(33)

The experimental value of muon g − 2 excess is [38] δaμ = (26.2 ± 8.5) × 10−10 .

(34)

(5) Higgs searches experiments. (i) Non-observation of additional Higgs bosons. We employ HiggsBounds-4.3.1 [39] to implement the exclusion constraints from the neutral and charged Higgses searches at LEP, Tevatron and LHC at 95% confidence level. (ii) The global fit to the 125 GeV Higgs signal data. In the exact alignment limit, the SM-like Higgs has the same coupling to the gauge boson and fermions as the Higgs couplings in the SM, which is favored by the 125 GeV Higgs signal data. However, in order to explain the μτ excess around 125 GeV, we assume that the A (H ) almost degenerates with the SM-like Higgs at the 125 GeV. Since the mass splitting of A (H ) and h is smaller than the mass resolution of detector, A (H ) can affect the global fit to the 125 GeV Higgs signal data. Following the method in [40], we perform a global fit to the 125 GeV Higgs data of 29 channels, which are given in the Appendix A (Tables 1–5). The signal strength for a channel is defined as  i i μi =  i ˆ Rgg Hˆ +  i (35) ˆ RV BF Hˆ +  ˆ RV Hˆ +  ¯ ˆ Rt t¯Hˆ . Hˆ =h, φ

gg H

V BF H

VH

ttH

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Where Rj = (σ × BR)j /(σ × BR)SM with j denoting the partonic process gg Hˆ , V BF Hˆ , j i V Hˆ , or t t¯Hˆ . j denotes the assumed signal composition of the partonic process j . If A (H ) almost degenerates with the SM-like Higgs, φ denotes A (H ). For an uncorrelated observable i, χi2 =

exp 2 )

(μi − μi σi2

(36)

,

exp

where μi and σi denote the experimental central value and uncertainty for the i-channel. We retain the uncertainty asymmetry in the calculation. For the two correlated observables, we take   exp 2 exp 2 exp (μ − μexp ) (μ − μ ) − μ ) − μ ) (μ (μ j j 1 i i j j i i 2 χi,j = + − 2ρ , σi σj 1 − ρ2 σi2 σj2 (37) where ρ is the correlation coefficient. We sum over χ 2 in the 29 channels, and pay particular 2 ≤ 6.18, where χ 2 denotes the minimum attention to the surviving samples with χ 2 − χmin min 2 of χ . These samples correspond to the 95.4% confidence level region in any two-dimension plane of the model parameters when explaining the Higgs data (corresponding to the 2σ range). (iii) The Higgs decays into τ μ. In the exact alignment limit, the μτ excess around 125 GeV is from A (H ) → τ μ where A (H ) almost degenerates with the SM-like Higgs. The width of A (H ) → μτ is given by (A (H ) → μτ ) =

2 + ρ 2 )m (ρμτ A (H ) τμ

16π

.

(38)

We take the best fit value of Br(h → μτ ) = (0.84+0.39 −0.37 )% based on the CMS search for the h → μτ at the LHC run-I. Since the μτ excess is assumed to be from the A (H ), we require the production rates of pp → A (H ) → μτ to vary from σ (pp → h) × 0.1% to σ (pp → h) × 1.62%. In addition, the CMS Collaboration did not publish the bound on the heavy Higgs decaying into μτ . Ref. [7] gave the bound on the production rate of pp → φ → μτ by recasting results from the original h → μτ analysis of CMS. 3.2. Results and discussions In Fig. 4, we project the surviving samples on the planes of ρτ μ versus κ and κu versus ρτ μ . The lower panels show the κu is required to be smaller than 1 due to the constraints of B-meson decays and Rb . The upper panels show that there is a strong correlation between ρτ μ and κ , which is mainly due to the constraints of Br(τ → 3μ) on the product |ρτ μ × κ |, and obviously affected by the constraints of Br(τ → μγ ). For example, in the case of mA = 126 GeV, |ρτ μ | is required to be smaller than 0.06 for κ = −10. In the case of mA = 126 GeV, there are two different regions where the muon g − 2 anomaly can be explained. (i) ρτ μ = 0 and |κ | > 100: The Higgs LFV couplings are absent due to ρτ μ = 0, and the muon g − 2 can be only corrected via the diagrams with the Higgs LFC couplings. Without the contributions of top quark loops, the contributions of the CP-even (CP-odd) Higgs to

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Fig. 4. The surviving samples projected on the planes of ρτ μ versus κ and κu versus ρτ μ . The circles (green) are allowed by the “pre-muon g − 2” constraints: theoretical constraints, precision electroweak data, Rb , B meson decays, τ decays, the exclusion limits of Higgses, and the 125 GeV Higgs data; the pluses (red) allowed by the pre-muon g − 2 and muon g − 2 excess; the bullets (black) and triangles (blue) allowed by the pre-muon g − 2, the muon g − 2 anomaly and μτ excess around 125 GeV, and the triangles (blue) further allowed by the experimental constraints of the heavy Higgs decaying into μτ . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

muon g − 2 are negative (positive) at the two-loop level and positive (negative) at one-loop level. As m2f /m2μ could easily overcome the loop suppression factor α/π , the two-loop contributions may be larger than one-loop ones. Therefore, the muon g − 2 can obtain the positive contributions from A loop and negative contributions from H loop. For the enough mass splitting of H and A, the muon g − 2 can be sizably enhanced by the diagrams with the large Higgs LFC couplings. The corresponding κu is required to be smaller than 0.2 due to the constraints of the search for

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Fig. 5. Same as Fig. 4, but κu versus mH and κu versus mA .

gg → A → τ τ¯ at the LHC, see the pluses (red) with ρτ μ = 0 shown in the lower-left panel of Fig. 4. (ii) 0.04 < |ρτ μ | < 0.18 and −9 < κ < 3: The muon g − 2 can be corrected by the diagrams with the Higgs LFV interactions and the Higgs LFC interactions, and the contributions of the former dominate over those of the latter due to the small |κ |. For the diagrams with the Higgs LFV couplings, the muon g − 2 obtains the positive contributions from A loop and negative contributions from H loop due to ρμτ = −ρτ μ . For the enough mass splitting of H and A, the muon g − 2 can be sizably enhanced by the diagrams with the large Higgs LFV couplings, and slightly corrected by those with the Higgs LFC couplings. In the case of mH = 126 GeV, the contributions of the CP-even Higgs dominate over those of the CP-odd Higgs due to mA > mH . The muon g − 2 obtains the negative contributions from the diagrams with the Higgs LFC couplings and positive contributions from the diagrams with the LFV couplings due to ρμτ = ρτ μ . Therefore, a proper ρτ μ is required to explain the muon g − 2 excess, 0.04 < |ρτ μ | < 0.18 and −3 < κ < 8 as shown in the right panels of Fig. 4. In the case of mA = 126 GeV, 0.02 < κu < 0.1 is favored by the μτ excess around 125 GeV and allowed by the experimental constraints of the heavy Higgs decaying into μτ . In the case of mH = 126 GeV, 0.03 < κu < 0.15 is favored by the μτ excess around 125 GeV, but some samples with a relatively large κu are excluded by the experimental constraints of the heavy Higgs decaying into μτ . As well known, the effective ggA coupling is larger than the ggH coupling for the same Yukawa couplings and Higgs masses since the form factor of CP-odd Higgs loop is larger than the CP-even Higgs loop. Thus, κu in the case of mH = 126 GeV is required to be larger than that in the case of mA = 126 GeV in order to obtain the correct μτ excess around 125 GeV. σ (pp → A → μτ ) in the case of mH = 126 GeV (A as the heavy Higgs) is much larger than σ (pp → H → μτ ) in the case of mA = 126 GeV (H as the heavy Higgs) due to the enhancements of the large top Yukawa coupling and the form factor of the CP-odd Higgs. Therefore, the experimental data of the heavy Higgs decaying into μτ give more strong constraints on the case of mH = 126 GeV than the case of mA = 126 GeV. In Fig. 5, we project the surviving samples on the plane of κu versus mA (mH ) in the case of mH = 126 GeV (mA = 126 GeV). The upper bound of σ (pp → A/H → μτ ) is taken from

L. Wang et al. / Nuclear Physics B 919 (2017) 123–141

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Fig. 6. Same as Fig. 4, but ρτ μ versus mH and ρτ μ versus mA .

Ref. [7], which is obtained by recasting results from the original CMS h → μτ analysis for the heavy Higgs in the range of 125 GeV and 275 GeV. From the right panel, for the case of mH = 126 GeV we find that the experimental data of the heavy Higgs decaying into μτ can exclude most samples in the ranges of 0.07 < κu < 0.15 and mA < 230 GeV, which can explain the excesses of muon g − 2 and μτ around 125 GeV. For mA > 230 GeV, all the surviving samples which are consistent with the μτ excess around 125 GeV are allowed by the experimental constraints of the heavy Higgs decaying into μτ . As discussed before, the left panel shows that all the surviving samples are allowed by the experimental constraints of the heavy Higgs decaying into μτ in the case of mA = 126 GeV. Note that there is the κ asymmetry in the regions of 0.04 < |ρτ μ | < 0.18, −9 < κ < 3 and 0.02 < κu < 0.1 for mA = 126 GeV where muon g − 2 can be explained. The main reason is from the constraints of τ → μγ . In the above regions, the top quark can give sizable contributions to τ → μγ via the “A2L ” and “A2R ” terms as shown in Eq. (24), which have destructive (constructive) interferences with the “A1L0 ” of Eq. (20) and “A1R0 ” of Eq. (22) induced by the one-loop contributions of τ for κ < 0 (κ > 0). Therefore, |κ | for κ < 0 is allowed to be much larger than that for κ > 0. Similar reason is for the κ asymmetry in the case of mH = 126 GeV but the destructive (constructive) interferences for κ > 0 (κ < 0). In Fig. 6, we project the surviving samples on the planes of ρτ μ versus mH and ρτ μ versus mA in the cases of mA = 126 GeV and mH = 126 GeV, respectively. We find that ρτ μ is sensitive to the mass of heavy Higgs, and the absolute value decreases with increasing of the mass of heavy Higgs in order to explain the muon g − 2 anomaly and the μτ excess around 125 GeV. As we discussed above, there is an opposite sign between the contributions of the H loops and A loops to the muon g − 2. Therefore, with the decreasing of the mass splitting of H and A, the cancellation between the contributions of H and A loops becomes sizable so that a large absolute value of ρμτ is required to enhance the muon g − 2. In Fig. 7, we project the surviving samples on the planes of mH ± versus mH and mH ± versus mA in the cases of mA = 126 GeV and mH = 126 GeV, respectively. We find that the mass

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Fig. 7. Same as Fig. 4, but mH ± versus mH and mH ± versus mA .

splitting of H ± and H (A) decreases with increasing of mH ± in the case of mA = 126 GeV (mH = 126 GeV), which is due to the constraints of the oblique parameters and δρ. However, for mH ± < 130 GeV, mH (mA ) is allowed to be as large as 625 GeV in the case of mA = 126 GeV (mH = 126 GeV). In this paper we focus on the exact alignment limit. If the alignment limit is approximately realized, the μτ excess can be from the SM-like Higgs (h) in addition to H or A around the 125 GeV. Therefore, the upper limits of κu become more stringent. When the μτ excess is mainly from h, the lower limit of κu will disappear since the ht t¯ coupling hardly changes with κu , and the A(H )t t¯ coupling is (nearly) proportional to κu . In addition, the upper limit of ρμτ can become more strong for the proper deviation from the alignment limit. For example, for sin(β − α) = 0.996, Br(h → μτ ) < 1.62% will give an upper limit of |ρμτ | < 0.0408, which is much smaller than that in the exact alignment limit. In the exact alignment limit, the widths of H → hh, W W (∗) , ZZ (∗) and A → hZ are equal to zero, and increase with decreasing of | sin(β − α)|. Therefore, the searches for H → hh, W W (∗) , ZZ (∗) and A → hZ can be used to probe the deviation from the alignment limit. These signatures refer to the H or A whose mass is not near 125 GeV. Otherwise, its signal would be indistinguishable from that coming from the SM-like light Higgs, and even H → hh (A → hZ) is absent for H (A) near 125 GeV. Some similar studies have been done in the singlet extension of the SM [41]. In the previous studies, the μτ excess is assumed to be from the SM-like Higgs h. In this paper we discuss another interesting scenario where the μτ excess is from either H or A near the observed Higgs signal. There is no AV V coupling due to the CP-conserving. The H V V coupling is absent and the hV V coupling is the same as the SM value in the exact alignment limit. Therefore, the two scenarios can be distinguished by observing the μτ signal via the vector boson fusion production process at the LHC with high integrated luminosity. In other words, if the μτ signal excess is observed in the gluon fusion process and not observed in the vector boson fusion process, the scenario in this paper will be strongly favored. Even when sin(β − α) deviates from the alignment limit sizably, the production rates of μτ signal via the gluon fusion and vector boson fusion can still have different correlations in the two different scenarios.

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4. Conclusion In this paper we examine the muon g − 2 anomaly and the μτ excess around 125 GeV in the exact alignment limit of 2HDM. In the scenario, the SM-like Higgs couplings to the SM particles are the same as the Higgs couplings in the SM at the tree level, and the tree-level LFV coupling hμτ is absent. We assume the μτ signal excess observed by CMS to be respectively from the H and A, which almost degenerates with the SM-like Higgs at the 125 GeV. After imposing various relevant theoretical constraints and experimental constraints from precision electroweak data, B-meson decays, τ decays and Higgs searches, we obtain the following observations: For the case of mA = 126 GeV, the muon g − 2 anomaly can be explained in two different regions: (i) ρτ μ = 0 and |κ | > 100; (ii) 0.04 < |ρτ μ | < 0.18 (|ρτ μ | is sensitive to mH ) and −9 < κ < 3. Further, the μτ excess around 125 GeV can be explained in the region (ii) with 0.02 < κu < 0.1 where all the surviving samples are allowed by the experimental constraints of the heavy Higgs decaying into μτ . For the case of mH = 126 GeV, the muon g − 2 anomaly excludes the region of ρτ μ = 0, and can be only explained in the region with a proper ρτ μ . The muon g − 2 anomaly and μτ excess favor 0.04 < |ρτ μ | < 0.18 (|ρτ μ | is sensitive to mA ), −3 < κ < 8 and 0.03 < κu < 0.15. However, most samples in the ranges of 0.07 < κu < 0.15 and mA < 230 GeV are further excluded by the experimental constraints of the heavy Higgs decaying into μτ . Acknowledgements This work has been supported by the National Natural Science Foundation of China under grant Nos. 11575152, 11375248. Appendix A. The measured values of the signal strengths of 125 GeV Higgs at the LHC and Tevatron Table 1 The measured values of the signal strengths of h → γ γ at the LHC and Tevatron. The composition ji of each production mode in each data are given. Channel ATLAS (20.3 fb−1 μggH μV BF μW H μZH μttH

Signal strength μ

MH (GeV)

Production mode ggF

VBF

VH

ttH

at 8 TeV) [42] 1.32 ± 0.38 0.8 ± 0.7 1.0 ± 1.6 0.1+3.7 −0.1

125.40 125.40 125.40 125.40

100% – – –

– 100% – –

– – 100% 100%

– – – –

125.40

–

–

–

100%

124.70

100%

–

–

–

124.70

–

100%

–

–

124.70

–

–

100%

–

124.70

–

–

–

100%

125

78%

5%

17%

–

1.6+2.7 −1.8

CMS (19.7 fb−1 at 8 TeV) [43] μggH 1.12+0.37 −0.32 1.58+0.77 μV BF −0.68 −0.16+1.16 μV H −0.79 μttH 2.69+2.51 −1.81 Tevatron (10.0 fb−1 at 1.96 TeV) [44] Combined 6.14+3.25 −3.19

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Table 2 The same as Table 1 but for H → ZZ (∗) . MH (GeV)

Production mode ggF

VBF

VH

ttH

ATLAS (20.3 fb−1 at 8 TeV) [45,46] Inclusive 1.66+0.45 −0.38

124.51

87.5%

7.1%

4.9%

0.5%

CMS (19.7 fb−1 at 8 TeV) [47] Inclusive 0.93+0.29 −0.25

125.6

87.5%

7.1%

4.9%

0.5%

Channel

Signal strength μ

Table 3 The same as Table 1 but for H → W W (∗) . MH (GeV)

Production mode ggF

VBF

VH

ttH

125

87.5%

7.1%

4.9%

0.5%

CMS (19.4 fb−1 at 8 TeV) [49] 0/1 jet 0.74+0.22 −0.20

125.6

97%

3%

–

–

VBF tag

125.6

17%

83%

–

–

125.6

–

–

100%

–

125.6

–

–

100%

–

125

78%

5%

17%

–

Channel

Signal strength μ

ATLAS (20.7 fb−1 at 8 TeV) [48] Inclusive

VH tag (2l2ν2j ) WH tag (3l3ν)

0.99 ± 0.30

0.60+0.57 −0.46

0.39+1.97 −1.87 0.56+1.27 −0.95

Tevatron (10.0 fb−1 at 1.96 TeV) [44] Combined 0.85+0.88 −0.81

Table 4 ¯ The same as Table 1 but for H → bb. MH (GeV)

Production mode ggF

VBF

VH

125.5

–

–

100%

–

125.4

–

–

–

100%

CMS (18.9 fb−1 at 8 TeV) [52], (19.5 fb−1 at 8 TeV) [53] VH tag 1.0 ± 0.5 125 ttH tag 0.67+1.35 125 −1.33

– –

– –

100% –

– 100%

Tevatron (10.0 fb−1 at 1.96 TeV) [54] VH tag 1.59+0.69 −0.72

–

–

100%

–

Channel

Signal strength μ

ATLAS (20.3 fb−1 at 8 TeV) [50,51] VH tag 0.2+0.7 −0.6 ttH tag 1.8+1.66 −1.57

125

ttH

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139

Table 5 The same as Table 1 but for H → τ τ . The correlation for the τ τ data of ATLAS is ρ = −0.51. Channel

Signal strength μ

MH (GeV)

Production mode

ATLAS (20.3 fb−1 at 8 TeV) [55] μ(ggF ) 1.1+1.3 −1.0

125

100%

–

–

–

μ(V BF + V H )

125

–

59.6%

40.4%

–

125 125 125 125

96.9% 75.7% 19.6% –

1.0% 14.0% 80.4% –

2.1% 10.3% – 100%

– – – –

1.6+0.8 −0.7

CMS (19.7 fb−1 at 8 TeV) [56] 0 jet 0.34 ± 1.09 1 jet 1.07 ± 0.46 VBF tag 0.94 ± 0.41 VH tag −0.33 ± 1.02

ggF

VBF

VH

ttH

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