TToyama International Workshop on Higgs as a Probe of New Physics 2015, 11–15, February, 2015 Limiting two-Higgs-doublet models
Eung Jin Chun ∗ Korea Institute for Advanced Study, Seoul 130-722, Korea
Updating various theoretical and experimental constraints on the four different types of two-Higgs-doublet models (2HDMs), we find that only the “lepton-specific” (or “type X”) 2HDM canexplain the present muon (g-2) anomaly in the parameter region of large tan β , a light CP-odd Higgsboson, and heavier CP-even and charged Higgs bosons which are almost degenerate. The severeconstraints on the models come mainly from the consideration of vacuum stability and perturbativity,the electroweak precision data, B physics observables like b → sγ as well as the 125 GeV Higgs bosonproperties measured at the LHC. I. OUTLINE
Since the first measurement of the muon anomalous magnetic moment a µ = ( g − µ / σ discrepancy∆ a µ ≡ a EXP µ − a SM µ = +262 (85) × − (1)which is in a good agreement with the different group’s determinations. Since the 2001 announcement, therehave been quite a few studies in the context of 2HDMs [2–4] restricted only to the type I and II models. However,the type X model [5] has some unique features in explaining the a µ anomaly while evading all the experimentalconstraints.Among many recent experimental results further confirming the Standard Model (SM) predictions, the dis-covery of the 125 GeV Brout-Egnlert-Higgs boson, which is very much SM-like, particularly motivates us torevisit the issue of the muon g − − • The Barr-Zee two loop [10] can give a dominant (positive) contribution to the muon g − A and large tan β in the type II and X models. • In the type II model, a light A has a large bottom Yukawa coupling for large tan β , and thus is stronglyconstrained by the collider searches which have not been able to cover a small gap of 25 (40) GeV < M A <
70 GeV at the 2 (1) σ range of the muon (g-2) explanation [3]. • In the type II (and Y) model, the measured ¯ B → X s γ branching ratio pushes the charged Higgs boson H ± high up to 480 (358) GeV at 95 (99) % C.L. [11], which requires a large separation between M A and M H ± putting a strong limitation on the model due to the ρ parameter bound [4]. • Consideration of the electroweak precision data (EWPD) combined with the theoretical constraints fromthe vacuum stability and perturbativity requires the charged Higgs boson almost degenerate with theheavy Higgs boson H [12] (favoring M H ± > M H ) and lighter than about 250 GeV in “the SM limit”;cos( β − α ) →
0. This singles out the type X model in favor of the muon g − • In the favored low m A region, the 125 GeV Higgs decay h → AA has to suppressed kinematically or bysuppressing the trilinear coupling λ hAA which is generically order-one. This excludes the 1 σ range of themuon g − ∗ Electronic address: [email protected] a r X i v : . [ h e p - ph ] M a y oyama International Workshop on Higgs as a Probe of New Physics 2015, 11–15, February, 2015 hbb or hτ τ coupling) of 2HDMs [13]. • A cancellation in λ hAA can be arranged to suppress arbitrarily the h → AA decay only in the wrong-signlimit with the heavy Higgs masses in the range of M H ± ∼ M H ≈ −
600 GeV [7]. • The lepton universality affected by a large H + τ ν τ coupling turns out to severely constrain the large tan β and light H ± region of the type X (and II) model and thus only a very low M A and tan β region is allowedat 2 σ to explain the a µ anomaly [8]. II. FOUR TYPES OF 2HDMS
Non-observation of flavour changing neutral currents restricts 2HDMs to four different classes which differ byhow the Higgs doublets couple to fermions [14]. They are organized by a discrete symmetry Z under whichdifferent Higgs doublets and fermions carry different parities. These models are labeled as type I, II, “lepton-specific” (or X) and “flipped” (or Y). Having two Higgs doublets Φ , , the most general Z symmetric scalarpotential takes the form: V = m | Φ | + m | Φ | − m (Φ † Φ + Φ Φ † )+ λ | Φ | + λ | Φ | + λ | Φ | | Φ | + λ | Φ † Φ | + λ (cid:104) (Φ † Φ ) + (Φ Φ † ) (cid:105) , (2)where a (soft) Z breaking term m is introduced. Minimization of the scalar potential determines the vacuumexpectation values (cid:104) Φ , (cid:105) ≡ v , / √ , = (cid:20) η +1 , , √ (cid:0) v , + ρ , + iη , (cid:1)(cid:21) . (3)The model contains the five physical fields in mass eigenstates denoted by H ± , A, H and h . Assuming negligibleCP violation, H ± and A are given by H ± , A = s β η ± , − c β η ± , (4)where the angle β is determined from t β ≡ tan β = v /v , and their orthogonal combinations are the corre-sponding Goldstone modes G ± , . The neutral CP-even Higgs bosons are diagonalized as h = c α ρ − s α ρ , H = s α ρ + c α ρ (5)where h ( H ) denotes the lighter (heavier) state.The gauge couplings of h and H are given schematically by L gauge = g V m V (cid:0) s β − α h + c β − α H (cid:1) V V where V = W ± or Z . When h is the 125 GeV Higgs boson, the SM limit corresponds to s β − α →
1. Indeed, LHCfinds, c β − α (cid:28) y Au y Ad y Al y Hu y Hd y Hl y hu y hd y hl Type I cot β − cot β − cot β sin α sin β sin α sin β sin α sin β cos α sin β cos α sin β cos α sin β Type II cot β tan β tan β sin α sin β cos α cos β cos α cos β cos α sin β − sin α cos β − sin α cos β Type X cot β − cot β tan β sin α sin β sin α sin β cos α cos β cos α sin β cos α sin β − sin α cos β Type Y cot β tan β − cot β sin α sin β cos α cos β sin α sin β cos α sin β − sin α cos β cos α sin β TABLE I: The normalized Yukawa couplings for up- and down-type quarks and charged leptons. oyama International Workshop on Higgs as a Probe of New Physics 2015, 11–15, February, 2015 f by m f /v where v = (cid:112) v + v = 246GeV, we have the following Yukawa terms: − L = (cid:88) f = u,d,l m f v (cid:0) y hf h ¯ f f + y Hf H ¯ f f − iy Af A ¯ f γ f (cid:1) (6)+ (cid:104) √ V ud H + ¯ u (cid:16) m u v y Au P L + m d v y dA P R (cid:17) d + √ m l v y Al H + ¯ νP R l + h.c. (cid:105) where the normalized Yukawa coupligs y h,H,Af are summarized in Table I for each of these four types of 2HDMs.Let us now recall that the tau Yukawa coupling y τ ≡ y hl in Type X ( y b ≡ y hd in Type II) can be expressed as y τ = − s α c β = s β − α − t β c β − α (7)which allows us to have the wrong-sign limit y τ ∼ − c β − α ∼ /t β forlarge tan β favoured by the muon g −
2. Later we will see that a cancellation in λ hAA can be arranged only for y hτ < − h → AA decay. III. ELECTROWEAK CONSTRAINTS
FIG. 1: The parameter space allowed in the M A vs. ∆ M H = M H − M H ± plane by EW precision constraints. The green,yellow, gray regions satisfy ∆ χ EW ( M A , ∆ M ) < . , . , .
8, corresponding to 68.3, 95.4, and 99.7% confidence intervals,respectively.
Let us fist consider the constraints arising from EWPD on 2HDMs. In particular, we compare the theoretical2HDMs predictions for M W and sin θ lepteff with their present experimental values via a combined χ analysis.These quantities can be computed perturbatively by means of the following relations M W = M Z (cid:34) (cid:115) − πα em √ G F M Z − ∆ r (cid:35) (8)sin θ lepteff = k l (cid:0) M Z (cid:1) sin θ W , (9)where sin θ W = 1 − M W /M Z , and k l ( q ) = 1 + ∆ k l ( q ) is the real part of the vertex form factor Z → l ¯ l evaluated at q = M Z . We than use the following experimental values: M EXP W = 80 . ± .
015 GeV , sin θ lept , EXP eff = 0 . ± . . (10)The results of our analysis are displayed in Fig. 1 confirming a custodial symmetry limit of our interest M A (cid:28) M H ∼ M H ± (or M H (cid:28) M A ∼ M H ± ) [12]. oyama International Workshop on Higgs as a Probe of New Physics 2015, 11–15, February, 2015 IV. THEORETICAL CONSTRAINTS ON THE SPLITTING M A - M H + Although any value of M A is allowed by the EW precision tests in the limit of M H ∼ M H ± , a large separationbetween M H ± and M A is strongly constrained by theoretical requirements of vacuum stability, global minimum,and perturbativity: λ , > , λ > − (cid:112) λ λ , | λ | < λ + λ + (cid:112) λ λ , (11) m ( m − m (cid:112) λ /λ )(tan β − ( λ /λ ) / ) > , (12) | λ i | (cid:46) | λ max | = √ π, π, or 4 π. (13)Taking λ as a free parameter, one can have the following expressions for the other couplings in the large t β limit [9]: λ v ≈ s β − α M h (14) λ v ≈ M H ± − ( s β − α + s β − α y τ ) M H + s β − α y τ M h (15) λ v ≈ − M H ± + s β − α M H + M A (16) λ v ≈ s β − α M H − M A (17)where we have used the relation (7) neglecting the terms of O (1 /t β ). FIG. 2: Theoretical constraints on the M A - M H ± plane. The darker to lighter gray regions in the left panel correspondto the allowed regions for ∆ M ≡ M H − M H ± = { , , − } GeV and λ max = √ π . The allowed regions in the rightpanel correspond to λ max = {√ π, π, π } and vanishing ∆ M . Consideration of all the theoretical constraints mentioned above in the SM limit corresponding to s β − α = y τ = 1 gives us Fig. 2. One can see that for a light pseudoscalar with M A (cid:46)
100 GeV the charged Higgs bosonmass gets an upper bound of M H ± (cid:46)
250 GeV.
V. CONSTRAINTS FROM THE MUON G − Considering all the updated SM calculations of the muon g −
2, we obtain a SM µ = 116591829 (57) × − (18)comparing it with the experimental value a EXP µ = 116592091 (63) × − , one finds a deviation at 3.1 σ : ∆ a µ ≡ a EXP µ − a SM µ = +262 (85) × − . In the 2HDM, the one-loop contributions to a µ of the neutral and chargedHiggs bosons are δa µ (1loop) = G F m µ π √ (cid:88) j (cid:0) y jµ (cid:1) r jµ f j ( r jµ ) , (19) oyama International Workshop on Higgs as a Probe of New Physics 2015, 11–15, February, 2015 j = { h, H, A, H ± } , r jµ = m µ /M j , and f h,H ( r ) = (cid:90) dx x (2 − x )1 − x + rx , (20) f A ( r ) = (cid:90) dx − x − x + rx , (21) f H ± ( r ) = (cid:90) dx − x (1 − x )1 − (1 − x ) r . (22)These formula show that the one-loop contributions to a µ are positive for the neutral scalars h and H , andnegative for the pseudo-scalar and charged Higgs bosons A and H ± (for M H ± > m µ ). In the limit r (cid:28) f h,H ( r ) = − ln r − / O ( r ) , (23) f A ( r ) = + ln r + 11 / O ( r ) , (24) f H ± ( r ) = − / O ( r ) , (25)showing that in this limit f H ± ( r ) is suppressed with respect to f h,H,A ( r ). Now the two-loop Barr-Zee typediagrams with effective hγγ , Hγγ or Aγγ vertices generated by the exchange of heavy fermions gives δa µ (2loop − BZ) = G F m µ π √ α em π (cid:88) i,f N cf Q f y iµ y if r if g i ( r if ) , (26)where i = { h, H, A } , r if = m f /M i , and m f , Q f and N cf are the mass, electric charge and number of colordegrees of freedom of the fermion f in the loop. The functions g i ( r ) are g i ( r ) = (cid:90) dx N i ( x ) x (1 − x ) − r ln x (1 − x ) r , (27)where N h,H ( x ) = 2 x (1 − x ) − N A ( x ) = 1.Note the enhancement factor m f /m µ of the two-loop formula in Eq. (26) relative to the one-loop contri-bution in Eq. (19), which can overcome the additional loop suppression factor α/π , and makes the two-loopcontributions may become larger than the one-loop ones. Moreover, the signs of the two-loop functions g h,H (negative) and g A (positive) for the CP-even and CP-odd contributions are opposite to those of the functions f h,H (positive) and f A (negative) at one-loop. As a result, for small M A and large tan β in Type II and X, thepositive two-loop pseudoscalar contribution can generate a dominant contribution which can account for theobserved ∆ a µ discrepancy. The additional 2HDM contribution δa µ = δa µ (1loop) + δa µ (2loop − BZ)obtained adding Eqs. (19) and (26) (without the h contributions) is compared with ∆ a µ in Fig. 3.Finally, let us remark that the hAA coupling is generically order one and thus can leads to a sizable non-standard decay of h → AA which should be suppressed kinematically or by making | λ hAA /v | (cid:28) hAA coupling, λ hAA /v ≈ s β − α [ λ + λ − λ ], and thus λ hAA v/s β − α ≈ − (1 + s β − α y τ ) M H + s β − α y τ M h + 2 M A (28)where we have put s β − α = 1 [9]. It shows that, in the SM limit of s β − α y τ →
1, the condition λ hAA ≈ M H ∼ M h which is disfavoured, and thus one needs to have M A > M h /
2. On the other hand, one can arrangea cancellation for λ hAA ≈ s β − α y τ < y τ s β − α ≈ − M H − M A M H − M h . (29) VI. SUMMARY
The type X 2HDM provides a unique opportunity to explain the current ∼ σ deviation in the muon g − oyama International Workshop on Higgs as a Probe of New Physics 2015, 11–15, February, 2015 FIG. 3: The 1 σ , 2 σ and 3 σ regions allowed by ∆ a µ in the M A -tan β plane taking the limit of β − α = π/ M h ( H ) = 126(200) GeV in type II (left panel) and type X (right panel) 2HDMs. The regions below the dashed (dotted) lines areallowed at 3 σ (1.4 σ ) by ∆ a e . The vertical dashed line corresponds to M A = M h / for the muon g − σ is quite limited in the SM limit: tan β (cid:38)
30 and M A (cid:28) M H ∼ M H ± (cid:46)
250 GeV.However, consideration of the h → AA decay and lepton universality [8] rules out this region. On the otherhand, in the wrong-sign limit of y τ ∼ −
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