B0 → l +l− Decays in Two-Higgs Doublet Models

Nuclear Physics B Proceedings Supplement 00 (2014) 1–7

Supplement

B

→ `

`

Decays in Two-Higgs Doublet Models

Xin-Qiang Li

Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan, Hubei 430079, P. R. China State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,

Chinese Academy of Sciences, Beijing 100190, P. R. China

Jie Lu¹, Antonio Pich

IFIC, Universitat de Val`encia – CSIC, Apt. Correus 22085, E-46071 Val`encia, Spain

Abstract

We study the rare leptonic decaysB⁰_s,d → `<sup>+</sup>`⁻ within the general framework of the aligned two-Higgs doublet model [1]. A complete one-loop calculation of the relevant short-distance Wilson coefficients is presented, with a detailed technical summary of the results. The phenomenological constraints imposed by present data on the model parameters are also investigated.

Keywords: Rare decays, two-Higgs doublet model, Wilson coefficients,Z₂symmetry

1. Introduction

The discovery [2, 3] of a Higgs-like boson at the LHC has placed the last missing piece of the Standard Mod- el (SM), which is one of the greatest achievements of modern particle physics. However, it is widely believed that the SM cannot be the fundamental theory up to the Plank scale, and many theories beyond the SM (BSM) claim that new physics (NP) should appear around the TeV scale.

One of the simplest extensions of the SM is the ad- dition of an extra Higgs doublet [4]. Two scalar dou- blets are present in several BSM theories, for instance in supersymmetry. Two-Higgs doublet models (2HDM- s) with generic Yukawa couplings give rise to danger- ous tree-level flavour-changing neutral currents (FCNC- s) [5]. This can be avoided imposing discreteZ2 sym- metries [6] or, more generally, assuming the alignment in flavour space of the two Yukawa matrices for each type of right-handed fermions [7].

1Speaker

The leptonic decays B⁰_s,d →`<sup>+</sup>`⁻play a very special role in testing the SM and probing BSM physics. They are very sensitive to the mechanism of quark flavour mixing, and their branching ratios are extremely small due to the loop suppression and the helicity suppression factorm`/mb. Since the final state involves only leptons, the SM theoretical predictions are very clean [8]:

B(B⁰_s→µ⁺µ⁻)=(3.65±0.23)×10⁻⁹, (1) B(B⁰_d→µ⁺µ⁻)=(1.06±0.09)×10⁻¹⁰, (2) which include next-to-leading order (NLO) electroweak corrections [9] and next-to-next-to-leading order (NN- LO) QCD corrections [10].

The weighted world averages of the CMS [11] and LHCb [12] measurements [13]

B(B⁰_s→µ⁺µ⁻)exp.=(2.9±0.7)×10⁻⁹, (3) B(B⁰_d→µ⁺µ⁻)exp.=

3.6⁺_−1.4^1.6

×10⁻¹⁰, (4) are very close to the SM predictions and put stringent constraints on BSM physics.

2. The aligned two-Higgs doublet model

It is convenient to define the 2HDM in the “Higgs ba- sis” where only one scalar doublet gets a nonzero vacu- um expectation valuev=(√

2G_F)^−1/2'246 GeV:

Φ1=

" G⁺

√1

2(v+S₁+i G⁰)

#

, (5)

Φ2=

" H⁺

√1

2(S2+i S3)

#

. (6)

The first doublet contains the Goldstone fieldsG^±and G⁰. The five physical degrees of freedom are given by the two charged fieldsH^±(x) and three neutral scalars ϕ⁰_i(x) = {h(x),H(x),A(x)}. The latter are related with the Si fields through an orthogonal transformation R, which defines the neutral mass eigenstates:

R M R^T =diag

M_h²,M_H²,M²_A

, ϕ⁰_i =R_{i j}Sj. (7) The mass matrixMof the neutral scalars is fixed by the scalar potential:

V =µ1

Φ^†₁Φ1

+µ2

Φ^†₂Φ2

+h µ3

Φ^†₁Φ2

+µ^∗₃ Φ^†₂Φ1

i

+λ1

Φ^†₁Φ1

2

+λ2

Φ^†₂Φ2

2

+λ₃

Φ^†₁Φ1 Φ^†₂Φ2

+λ₄

Φ^†₁Φ2 Φ^†₂Φ1

(8)

+h

λ5Φ^†₁Φ2+ λ6Φ^†₁Φ1+λ7Φ^†₂Φ2 Φ^†₁Φ2

+h.c.i , whereµ1,µ2andλ_1,2,3,4are real, whileµ3andλ_5,6,7can be complex.

In the CP-conserving limit, the neutral Higgs spec- trum contains a CP-odd fieldA=S3and two CP-even scalarshandHwhich mix through the two-dimensional rotation matrix:

h H

!

=

"

cos ˜α sin ˜α

−sin ˜α cos ˜α

# S1

S2

!

. (9)

We use the conventions M_h ≤ M_H, and 0 ≤α˜ ≤ πso that sin ˜αis always positive.

2.1. Yukawa sector

The 2HDM Yukawa sector is given by L_Y =−

√ 2 v

hQ¯⁰_L(M_d⁰Φ1+Y_d⁰Φ2)d_R⁰ +Q¯⁰_L(M⁰_uΦ˜1+Y_u⁰Φ˜2)u⁰_R +L¯⁰_L(M_{`</sub><sup>0</sup>Φ1+Y<sub>`}⁰Φ2)`_R⁰i

+ h.c. , (10) with ˜Φi(x) = iτ2Φ^∗_i(x) the charge-conjugated scalar doublets with hyperchargeY =−¹₂. Q⁰_L andL⁰_Ldenote

the SM left-handed quark and lepton doublets, respec- tively, and u⁰_R, d⁰_R and`⁰_R are the corresponding right- handed singlets, in the weak interaction basis.

The Yukawa couplings M⁰_f andY⁰_f (f = u,d, `) are complex 3×3 matrices which, in general, cannot be diagonalized simultaneously, generating FCNCs at tree level. This can be avoided by assuming that M⁰_f and Y⁰_f are proportional to each other [7]. In the mass- eigenstate fermion basis with diagonal matricesMf, one has then

Yd,`=ςd,`Md,`, Yu=ς<sup>∗</sup><sub>u</sub>Mu, (11) with arbitrary complex parameters ςf (f = d,u, `), which introduce new sources of CP violation. The aligned 2HDM (A2HDM) Yukawa Lagrangian reads

L_Y=−

√ 2 v H⁺n

¯ uh

ςdV MdPR−ςuM_u^†V PL

id (12) +ς_{`</sub>νM¯ <sub>`}PR`o

−1 v

X

ϕ⁰_i,f

y^ϕ

0 i

f ϕ⁰_i hf M¯ fPRfi +h.c. ,

whereP_R,L≡ ^1±γ₂⁵,Vis the CKM quark-mixing matrix and the neutral Yukawa couplings are given by

y^ϕ

0

d,`i =R<sub>i1</sub>+(R<sub>i2</sub>+iR<sub>i3</sub>)ς<sub>d,`</sub>, (13) y^ϕ

0

ui =R_i1+(Ri2−iR_i3)ς_u^∗. (14) The usualZ2symmetric models can be recovered with specific assignments of the alignment parameters.

2.2. Flavour misalignment

The alignment conditions (11) presumably hold at some high-energy scale ΛA and are spoiled by radia- tive corrections which induce a misalignment of the Yukawa matrices. However, the flavour symmetries of the A2HDM tightly constrain the possible FCNC struc- tures, keeping their effects well below the present ex- perimental bounds. The only FCNC local structures in- duced at one loop take the form [7, 14],

L_FCNC= C

4π²v³ 1+ς^∗_uς_d (15)

×X

i

ϕ⁰_in

(Ri2+iR_i3) (ς_d−ς_u)hd¯_LV^†M_uM^†_uV M_dd_Ri

−(Ri2−iR_i3) (ς_d^∗−ς^∗_u)h

¯

uLV MdM^†_dV^†MuuR

i o+h.c. . The renormalization of the misalignment parameterCis determined to be [14]

C=C_R(µ)+1 2

(2µ^D−4

D−4 +γ_E−ln (4π) )

, (16) and absorbs the UV divergences from one-loop Higgs- penguin diagrams inB⁰_s,d→`<sup>+</sup>`⁻decays [1].

3. Effective Hamiltonian

The low-energy effective Hamiltonian describing B⁰_s,d →`<sup>+</sup>`⁻decays is given by [15, 16, 17]

H_eff = − G_Fα

√2πs²_W









 VtbV_tq^∗

10,S,P

X

i

CiO_i+h.c.









 ,

O₁₀ = ( ¯qγµPLb) ( ¯`γ<sup>µ</sup>γ5`), O_S = m`mb

M²_W ( ¯qPRb) ( ¯``), O_P = m_`mb

M²_W ( ¯qP_Rb) ( ¯`γ<sub>5</sub>`), (17) where` =e, µ, τ;q =d,s, andmb =mb(µ) denotes the b-quark MS running mass. Other possible operators are neglected because their contributions are either zero or proportional to the light-quark massmq.

The anomalous dimension ofO₁₀ is zero due to the conservation of the (V−A) quark current in the mass- less quark limit. The operatorsO_S andO_Palso have ze- ro anomalous dimensions because theµdependences of mb(µ) and the scalar current ( ¯qPRb)(µ) cancel each oth- er. Therefore the Wilson coefficientsCido not receive additional renormalization from QCD corrections.

4. Calculation of the Wilson coefficientsC_10,S,P The Wilson coefficientsC10,S,P are obtained by re- quiring the equality of one-particle irreducible ampu- tated Green functions in the full and in the effective the- ories. The relevant Feynman diagrams for a given pro- cess can be created by the packageFeynArts[18], with the model files provided byFeynRules[19]. The gen- erated decay amplitudes are evaluated either with the help of FeynCalc[20], or using standard techniques such as the Feynman parametrization to combine prop- agators. We found full agreement between the results obtained with these two methods. Throughout the w- hole calculation, we set the light-quark massesm_d,s to zero; while form_b, we keep it up to linear order.

In general, the Wilson coefficientsCi are functions of the internal up-type quark masses, together with the corresponding CKM factors [21]:

C_i = X

j=u,c,t

V^∗_jqV_jbF_i(x_j), (18) wherexj = m²_j/M²_W, andFi(xj) denote the loop func- tions. In deriving the effective Hamiltonian (17), the limitmu,c→0 and the unitarity of the CKM matrix,

V_uq^∗V_ub+V_cq^∗V_cb+V_tq^∗V_tb = 0, (19)

have to be exploited. This implies that we need only to calculate explicitly the contributions from internal top quarks, while those from up and charm quarks are tak- en into account by means of simply omitting the mass- independent terms in the basic functionsF_i(x_t).

The relevant Feynman diagrams are split into vari- ous box, penguin and self-energy diagrams, which are mediated by the top quark, gauge bosons, and Higgs scalars. In order to check the gauge independence of the final results, we perform the calculation both in the Feynman (ξ=1) and in the unitary (ξ=∞) gauges.

4.1. Wilson coefficients in the SM

In the SM, the dominant contribution to the decays B⁰_s,d → `<sup>+</sup>`⁻ comes from the Wilson coefficient C₁₀, which arises fromW-box andZ-penguin diagrams:

C^SM₁₀ = −η^EW_Y η^QCD_Y Y0(xt), (20) where

Y0(xt) = xt

8

"

xt−4

xt−1+ 3xt

(xt−1)²lnxt

#

(21) is the one-loop Inami-Lim function [22]. The factors η^EW_Y andη^QCD_Y account for the NLO electroweak [9] and the NNLO QCD corrections [10], respectively.

The coefficientsCS andCPreceive SM contributions from box, Z penguin, Goldstone-boson (GB) penguin and Higgs (h) penguin diagrams:

C^SM_S = C_S^box,SM+Ch penguin,SM

S , (22)

C^SM_P = C^box,_P ^SM+CZ penguin,SM

P +CGB penguin,SM

P . (23)

The Goldstone contribution is of course absent in the unitary gauge. Explicit expressions can be found in [1].

4.2. Wilson coefficients in the A2HDM

In the A2HDM there are additional contributions from box and Z penguin diagrams, involving H^± ex- changes, and from Higgs penguin diagrams. The only new contribution toC10comes fromZpenguin diagram- s and is gauge independent by itself:

C^A2HDM₁₀ =CZ penguin,A2HDM

10 . (24)

TheZ penguin diagrams also generate contributions toC_P. The sum ofZpenguin diagrams and Goldstone- boson penguin diagrams is gauge independent:

CZ penguin,A2HDM

P,Unitary =CZ penguin,A2HDM

P,Feynman +CGB penguin,A2HDM

P,Feynman .

(25) The contributions from box diagrams withH^±bosons, C_S^box,_,P^A2HDM, are gauge dependent.

Neutral scalar exchanges induce both tree and loop diagrams. The loop contributions consist of the Higgs- penguin and self-energy diagrams governed by the Yukawa couplings (12), whereas the tree ones are given by the misalignment couplings (15). The sum of these contributions can be written as:

C^ϕ_S^0i,A2HDM = X

ϕ⁰_i

Re(y^ϕ_`⁰ⁱ) ˆC^ϕ0i , (26) C^ϕ

0 i,A2HDM

P = i X

ϕ⁰_i

Im(y^ϕ

0

`i) ˆC^ϕ0i , (27)

with Cˆ^ϕ0i =xt

((ς_u−ς_d) (1+ς_u^∗ς_d) 2x_ϕ0

i

(Ri2+iRi3)C_R(MW) + v²

M_ϕ²₀

i

λ^ϕ_H⁰ⁱ_+H−g₀ +

3

X

j=1

R_{i j}ξj

g^(a)_j 2x_ϕ0 i

+g^(b)_j )

, (28)

whereλ^ϕ_H⁰ⁱ_+H− = λ₃R_i1+λ^R₇R_i2−λ^I₇R_i3,ξ₁ = ξ₂ = 1 andξ3 = i. When ς_u,d → 0, x_H,A → ∞, x_h → x_hSM, R_i2,i3→0 andR₁₁→1, this reproduces the SM result.

The coefficientsg₀,g^(a)_j andg^(b)_j are functions ofx_t, xH⁺,ςuandςd. g₀ andg^(a)_j are gauge independent be- cause they do not involve any gauge bosons, whileg^(b)_j are all related to Goldstone-boson vertices and, there- fore, are identically zero in the unitary gauge. These g^(b)_j contributions cancel the gauge dependence from the box diagrams:

C^box,SM_S,Unitary − C_S^box,_,Feynman^SM = xtg^(b)₁ , (29) C^box,A2HDM_S,Unitary − C_S^box,_,Feynman^A2HDM =

xt

hRe(ς`)g<sup>(b)</sup><sub>2</sub> −iIm(ς`)g^(b)₃ i

, (30) C^box,A2HDM_P,Unitary − C^box,_P,Feynman^A2HDM =

xt

hiIm(ς_{`</sub>)g<sup>(b)</sup><sub>2</sub> −Re(ς<sub>`})g^(b)₃ i

. (31) The loop contributions with neutral scalar exchanges generate UV divergences which are cancelled by the renormalization of the misalignment coupling in (16).

Theµdependence of the results is reabsorbed into the combinationC_R(MW)=C_R(µ)−ln (MW/µ).

5. Phenomenological analysis

Currently, onlyB_s →µ⁺µ⁻is observed with a signal significance of∼4.0σ[13]. Thus we shall investigate the allowed parameter space of the A2HDM under the

constraint from B(B⁰_s → µ⁺µ⁻). With updated input parameters, the SM prediction reads

B(B⁰_s→µ⁺µ⁻)_SM=(3.67±0.25)×10⁻⁹. (32) In order to explore constraints on the model parameters, it is convenient to introduce the ratio [15, 16]

Rs` ≡ B(B<sup>0</sup><sub>s</sub>→`⁺`<sup>−</sup>) B(B<sup>0</sup><sub>s</sub> →`⁺`⁻)_SM =

|P|²+ 1−∆Γs

Γ^s_L |S|²

,

(33) whereΓ^s_L(H) denote the lighter (heavier) eigenstate de- cay width of theBs meson, and∆Γs = Γ^s_L−Γ^s_H. The quantitiesS andPare defined as

P ≡ C10

C^SM₁₀ + M²_B

s

2M_W² mb

m_b+m_s

! CP−C^SM_P

C₁₀^SM , (34) S ≡

vt 1−4m²_`

M_B²

s

M²_B

s

2M²_W mb

mb+ms

! CS −C^SM_S C^SM₁₀ , (35) where the Wilson coefficients are given by:

C10 = C₁₀^SM+CZ penguin,A2HDM

10 ,

CS = C_S^box,SM+C^box,A2HDM_S +C^ϕ

0 i,A2HDM

S ,

CP = C^SM_P +C^box,A2HDM_P +C^ϕ

0 i,A2HDM P

+ C^{Z penguin,}_P ^A2HDM+CGB penguin,A2HDM

P . (36)

Combining the SM prediction (32) with the latest exper- imental result (3), we get

Rsµ=0.79±0.20. (37) 5.1. Model parameters

We consider the CP-conserving limit and assume that the lightest CP-even scalar h corresponds to the ob- served neutral boson with Mh ' 126 GeV. We have then 10 free parameters: 3 alignment couplings ςf, 3 scalar masses (MH, MA, MH^±), 2 scalar-potential cou- plings (λ3,λ7), the mixing angle ˜αand the misalignmen- t parameterCR(MW). Four of them ( ˜α,λ3,7,CR(MW)) have minor impacts onRsµ, compared to the others. In order to simplify the analysis, we assign them the fol- lowing values, using the bounds from earlier studies:

λ₃=λ₇=1, cos ˜α=0.95, CR(M_W)=0. (38) The C_S,P contributions to (34) and (35) are sup- pressed by a factorM²_B

s/M²_W. Therefore, unless there are large enhancements from theςfparameters, the branch- ing ratio shall be dominated byC10, where the A2HDM

Figure 1:Dependence of ¯Rsµonςu(left) and resulting upper bounds onςu(right), as function ofMH^±, for|ςd,`|.|ςu| ≤2.

contribution depends only on|ς_u|² andM_H^±. We shall then discuss two possible scenarios: 1)|ς<sub>d,`</sub>|.|ςu| ≤2, whereC10dominates, and 2)|ςd,`| |ςu|, whereCS and CPcould play a significant role.

5.2. Smallς_d,`

The only relevant NP contribution isC^A2HDM₁₀ which involves two parameters,ςu andMH^±. The constraints imposed by ¯Rsµare shown in Fig. 1. In the left panel, we chooseMH^± = 80, 200 and 500 GeV (upper, mid- dle and lower curves, respectively). The shaded hor- izontal bands denote the allowed experimental region at 1σ (dark green), 2σ (green), and 3σ (light green), respectively. The right panel shows the resulting up- per bounds onς_u, as function ofM_H^±. A 95% CL up- per bound|ςu| ≤ 0.49 (0.97) is obtained for MH^± = 80 (500) GeV. SinceC₁₀^A2HDM ∼ |ςu|², this constraint is independent of any assumption about CP. For larg- er masses the constraint becomes weaker since theH^± contribution starts to decouple.

5.3. Largeς_d,`

In this case,CS andCP can induce a significant im- pact on the branching ratio. We varyς_dandς_`within the range [−50,50], and choose three representative values forς_u =0,±1. We also take three different representa- tive sets of scalar masses:

Mass1 : M_H^±=M_A=80 GeV, M_H=130 GeV, Mass2 : MH^±=MA=MH=200 GeV,

Mass3 : M_H^±=M_A=M_H=500 GeV. (39) In Fig. 2, we show the allowed regions in theς_d–ς_{`</sub> plane under the constraint from ¯R<sub>sµ</sub>. The regions with largeς<sub>d</sub> andς<sub>`} are already excluded, especially when they have the same sign. The impact ofς_uis significant:

a nonzeroςu will exclude most of the regions allowed in the case withςu = 0, and changing the sign ofςu

Figure 2: Allowed regions (at 1σ, 2σand 3σ) in theςd– ς` plane under the constraint from ¯Rsµ, with three different assignments of the scalar masses andςu=0,±1.

Model ςd ςu ςl

Type I cotβ cotβ cotβ

Type II −tanβ cotβ −tanβ Type X (lepton-specific) cotβ cotβ −tanβ Type Y (flipped) −tanβ cotβ cotβ

Inert 0 0 0

Table 1: 2HDMs based on discreteZ2symmetries.

will also flip that of ς_`. The allowed regions expand with increasing scalar masses, as expected, since the NP contributions gradually decouple from the SM.

5.4. 2HDMs with discrete Z₂symmetries

The usualZ2symmetric models are recovered for the values ofςf indicated in Table 1. In these models, the ratio ¯Rsµonly involves seven free parameters:MH^±,MH, MA,λ3,λ7, cos ˜αand tanβ. In the particular case of the type-II 2HDM at large tanβ, our results agree with the ones calculated in Ref. [23]. It is also interesting to note that the B⁰_s,d → `<sup>+</sup>`⁻branching ratios depend only on the charged-Higgs mass and tanβin this case.

Fig. 3 shows the dependence of ¯Rsµon tanβ, for three representative charged-Higgs masses: M_H^± = 80, 200 and 500 GeV. The other two neutral scalar masses have been fixed atM_H =M_A =500 GeV. The four differen- t panels correspond to the models of types I, II, X and Y, respectively. A lower bound tanβ >1.6 is obtained

Figure 3:Dependence of ¯Rsµon tanβfor the 2HDMs of types I, II, X and Y. The upper, middle and lower curves correspond toMH^± =80, 200 and 500 GeV, respectively. The horizon- tal bands denote the allowed experimental region at 1σ(dark green), 2σ(green), and 3σ(light green).

at 95% CL under the constraint from the current exper- imental data on ¯Rsµ. This impliesςu = cotβ < 0.63, which is stronger than the bounds obtained previously from other sources [14, 24].

6. Conclusions

We have studied the rare decaysB⁰_s,d →`<sup>+</sup>`⁻within the general framework of the A2HDM. A complete one- loop calculation of the Wilson coefficientsC10,CS and CP has been performed. They arise from various box, penguin and self-energy diagrams, as well as tree-level FCNC diagrams induced by the flavour misalignment interaction (15). The gauge independence of the results has been checked through separate calculations in the Feynman and unitary gauges, and the gauge relations among different diagrams have been examined in detail.

We have also investigated the impact of the current B(B⁰_s→µ⁺µ⁻) data on the model parameters, especial- ly the resulting constraints on the three alignment cou- plingsςf. This information is complementary to the one obtained from collider physics and will be useful for fu- ture global data fits within the A2HDM.

Acknowledgements

Work supported by the NSFC [contracts 11005032 and 11435003], the Spanish Government [FPA2011- 23778] and Generalitat Valenciana [PROMETEOI- I/2013/007]. X. Li was also supported by the Scientif- ic Research Foundation for the Returned Overseas Chi- nese Scholars, State Education Ministry. J. Lu is grate-

ful for the hospitality of Center for Future High Energy Physics in Beijing while this proceeding was written.

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